Sheaf of modules - Wikipedia

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<span>Operations</span> </div> </a> <ul id="toc-Operations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sheaf_associated_to_a_module" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sheaf_associated_to_a_module"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sheaf associated to a module</span> </div> </a> <ul id="toc-Sheaf_associated_to_a_module-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sheaf_associated_to_a_graded_module" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sheaf_associated_to_a_graded_module"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Sheaf associated to a graded module</span> </div> </a> <ul id="toc-Sheaf_associated_to_a_graded_module-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computing_sheaf_cohomology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computing_sheaf_cohomology"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Computing sheaf cohomology</span> </div> </a> <ul id="toc-Computing_sheaf_cohomology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sheaf_extension" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sheaf_extension"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Sheaf extension</span> </div> </a> <button aria-controls="toc-Sheaf_extension-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Sheaf extension subsection</span> </button> <ul id="toc-Sheaf_extension-sublist" class="vector-toc-list"> <li id="toc-Locally_free_resolutions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Locally_free_resolutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Locally free resolutions</span> </div> </a> <ul id="toc-Locally_free_resolutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Examples</span> </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> <li id="toc-Hypersurface" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hypersurface"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2.1</span> <span>Hypersurface</span> </div> </a> <ul id="toc-Hypersurface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Union_of_smooth_complete_intersections" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Union_of_smooth_complete_intersections"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2.2</span> <span>Union of smooth complete intersections</span> </div> </a> <ul id="toc-Union_of_smooth_complete_intersections-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Sheaf consisting of modules on a ringed space; generalizing vector bundles</div><style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Sheaf_of_modules&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">November 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In mathematics, a <b>sheaf of <i>O</i>-modules</b> or simply an <b><i>O</i>-module</b> over a <a href="/wiki/Ringed_space" title="Ringed space">ringed space</a> (<i>X</i>, <i>O</i>) is a <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> <i>F</i> such that, for any open subset <i>U</i> of <i>X</i>, <i>F</i>(<i>U</i>) is an <i>O</i>(<i>U</i>)-module and the restriction maps <i>F</i>(<i>U</i>) → <i>F</i>(<i>V</i>) are compatible with the restriction maps <i>O</i>(<i>U</i>) → <i>O</i>(<i>V</i>): the restriction of <i>fs</i> is the restriction of <i>f</i> times the restriction of <i>s</i> for any <i>f</i> in <i>O</i>(<i>U</i>) and <i>s</i> in <i>F</i>(<i>U</i>). </p><p>The standard case is when <i>X</i> is a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> and <i>O</i> its structure sheaf. If <i>O</i> is the <a href="/wiki/Constant_sheaf" title="Constant sheaf">constant sheaf</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\mathbf {Z} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>_</mo> </munder> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\mathbf {Z} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16ec565233e46a4c1f7107e2e86e4b222f4d80f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-bottom: -0.827ex; width:1.636ex; height:3.176ex;" alt="{\displaystyle {\underline {\mathbf {Z} }}}"></span>, then a sheaf of <i>O</i>-modules is the same as a <a href="/wiki/Sheaf_of_abelian_groups" class="mw-redirect" title="Sheaf of abelian groups">sheaf of abelian groups</a> (i.e., an <b>abelian sheaf</b>). </p><p>If <i>X</i> is the <a href="/wiki/Prime_spectrum" class="mw-redirect" title="Prime spectrum">prime spectrum</a> of a ring <i>R</i>, then any <i>R</i>-module defines an <i>O</i><sub><i>X</i></sub>-module (called an <b>associated sheaf</b>) in a natural way. Similarly, if <i>R</i> is a <a href="/wiki/Graded_ring" title="Graded ring">graded ring</a> and <i>X</i> is the <a href="/wiki/Proj_construction" title="Proj construction">Proj</a> of <i>R</i>, then any graded module defines an <i>O</i><sub><i>X</i></sub>-module in a natural way. <i>O</i>-modules arising in such a fashion are examples of <a href="/wiki/Quasi-coherent_sheaves" class="mw-redirect" title="Quasi-coherent sheaves">quasi-coherent sheaves</a>, and in fact, on affine or projective schemes, all quasi-coherent sheaves are obtained this way. </p><p>Sheaves of modules over a ringed space form an <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Moreover, this category has <a href="/wiki/Enough_injectives" class="mw-redirect" title="Enough injectives">enough injectives</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and consequently one can and does define the <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} ^{i}(X,-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} ^{i}(X,-)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b83561219741404a98a6d963359e0c566faa91bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.174ex; height:3.176ex;" alt="{\displaystyle \operatorname {H} ^{i}(X,-)}"></span> as the <i>i</i>-th <a href="/wiki/Right_derived_functor" class="mw-redirect" title="Right derived functor">right derived functor</a> of the <a href="/wiki/Global_section_functor" class="mw-redirect" title="Global section functor">global section functor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (X,-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (X,-)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e34602234200472f45f1562a7efe02104ac4d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.084ex; height:2.843ex;" alt="{\displaystyle \Gamma (X,-)}"></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Given a ringed space (<i>X</i>, <i>O</i>), if <i>F</i> is an <i>O</i>-submodule of <i>O</i>, then it is called the sheaf of ideals or <a href="/wiki/Ideal_sheaf" title="Ideal sheaf">ideal sheaf</a> of <i>O</i>, since for each open subset <i>U</i> of <i>X</i>, <i>F</i>(<i>U</i>) is an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of the ring <i>O</i>(<i>U</i>).</li> <li>Let <i>X</i> be a <a href="/wiki/Smooth_variety" class="mw-redirect" title="Smooth variety">smooth variety</a> of dimension <i>n</i>. Then the <a href="/wiki/Tangent_sheaf" class="mw-redirect" title="Tangent sheaf">tangent sheaf</a> of <i>X</i> is the dual of the <a href="/wiki/Cotangent_sheaf" title="Cotangent sheaf">cotangent sheaf</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63cd0e0ecbeae43109b323b4ee9d030f25c3c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.31ex; height:2.509ex;" alt="{\displaystyle \Omega _{X}}"></span> and the <a href="/wiki/Canonical_sheaf" class="mw-redirect" title="Canonical sheaf">canonical sheaf</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42947d0b86d4d98a3172ca71d3f1595d296b5a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.078ex; height:2.009ex;" alt="{\displaystyle \omega _{X}}"></span> is the <i>n</i>-th exterior power (<a href="/wiki/Determinant_line_bundle" class="mw-redirect" title="Determinant line bundle">determinant</a>) of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63cd0e0ecbeae43109b323b4ee9d030f25c3c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.31ex; height:2.509ex;" alt="{\displaystyle \Omega _{X}}"></span>.</li> <li>A <a href="/wiki/Sheaf_of_algebras" title="Sheaf of algebras">sheaf of algebras</a> is a sheaf of module that is also a sheaf of rings.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Operations">Operations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=2" title="Edit section: Operations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let (<i>X</i>, <i>O</i>) be a ringed space. If <i>F</i> and <i>G</i> are <i>O</i>-modules, then their tensor product, denoted by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\otimes _{O}G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\otimes _{O}G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bb5dbdf475930cd461c8962d9f76adcabd8d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.894ex; height:2.509ex;" alt="{\displaystyle F\otimes _{O}G}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\otimes G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊗</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\otimes G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d05a23b9f86e0360521b77f83c249d3098e4b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.408ex; height:2.343ex;" alt="{\displaystyle F\otimes G}"></span>,</dd></dl> <p>is the <i>O</i>-module that is the sheaf associated to the presheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto F(U)\otimes _{O(U)}G(U).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>G</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mapsto F(U)\otimes _{O(U)}G(U).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90658ea6fec9cea315f28a3ab31d472965046993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.661ex; height:3.176ex;" alt="{\displaystyle U\mapsto F(U)\otimes _{O(U)}G(U).}"></span> (To see that sheafification cannot be avoided, compute the global sections of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)\otimes O(-1)=O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)\otimes O(-1)=O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb0746ddf7119a3b8e34792647944b216c54d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.01ex; height:2.843ex;" alt="{\displaystyle O(1)\otimes O(-1)=O}"></span> where <i>O</i>(1) is <a href="/wiki/Serre%27s_twisting_sheaf" class="mw-redirect" title="Serre's twisting sheaf">Serre's twisting sheaf</a> on a projective space.) </p><p>Similarly, if <i>F</i> and <i>G</i> are <i>O</i>-modules, then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}om_{O}(F,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mi>o</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}om_{O}(F,G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5aa15d2018b01766218f7b131c5b79f213af3c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.029ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}om_{O}(F,G)}"></span></dd></dl> <p>denotes the <i>O</i>-module that is the sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>,</mo> <mi>G</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8575b787a0e24830169c6a9b932f4cbd05ac7ed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.059ex; height:3.176ex;" alt="{\displaystyle U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})}"></span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In particular, the <i>O</i>-module </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}om_{O}(F,O)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mi>o</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>O</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}om_{O}(F,O)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d356d35597284d65bff0dd969d12b6c3a97e2820" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.975ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}om_{O}(F,O)}"></span></dd></dl> <p>is called the <b>dual module</b> of <i>F</i> and is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\check {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d18b3e659fc1b78444b00a6ef3323589d5b6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.805ex; height:2.676ex;" alt="{\displaystyle {\check {F}}}"></span>. Note: for any <i>O</i>-modules <i>E</i>, <i>F</i>, there is a canonical homomorphism </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {E}}\otimes F\to {\mathcal {H}}om_{O}(E,F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> <mo>⊗</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mi>o</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\check {E}}\otimes F\to {\mathcal {H}}om_{O}(E,F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d934c45a3c7d3f5dc77fc146852683b1e6c2a56c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.948ex; height:3.176ex;" alt="{\displaystyle {\check {E}}\otimes F\to {\mathcal {H}}om_{O}(E,F)}"></span>,</dd></dl> <p>which is an isomorphism if <i>E</i> is a <a href="/wiki/Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free sheaf</a> of finite rank. In particular, if <i>L</i> is locally free of rank one (such <i>L</i> is called an <a href="/wiki/Invertible_sheaf" title="Invertible sheaf">invertible sheaf</a> or a <a href="/wiki/Line_bundle" title="Line bundle">line bundle</a>),<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> then this reads: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {L}}\otimes L\simeq O,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>L</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> <mo>⊗</mo> <mi>L</mi> <mo>≃</mo> <mi>O</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\check {L}}\otimes L\simeq O,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c58284db48e24558fae650804aa313ca22bb7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.525ex; height:3.009ex;" alt="{\displaystyle {\check {L}}\otimes L\simeq O,}"></span></dd></dl> <p>implying the isomorphism classes of invertible sheaves form a group. This group is called the <a href="/wiki/Picard_group" title="Picard group">Picard group</a> of <i>X</i> and is canonically identified with the first cohomology group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} ^{1}(X,{\mathcal {O}}^{*})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} ^{1}(X,{\mathcal {O}}^{*})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b708554e32217dd59df7e40e2313b82c019f4014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.525ex; height:3.176ex;" alt="{\displaystyle \operatorname {H} ^{1}(X,{\mathcal {O}}^{*})}"></span> (by the standard argument with <a href="/wiki/%C4%8Cech_cohomology" title="Čech cohomology">Čech cohomology</a>). </p><p>If <i>E</i> is a locally free sheaf of finite rank, then there is an <i>O</i>-linear map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {E}}\otimes E\simeq \operatorname {End} _{O}(E)\to O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">ˇ</mo> </mover> </mrow> </mrow> <mo>⊗</mo> <mi>E</mi> <mo>≃</mo> <msub> <mi>End</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\check {E}}\otimes E\simeq \operatorname {End} _{O}(E)\to O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fadb082858230fe26ad7720febdb98feb8cfd3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.116ex; height:3.176ex;" alt="{\displaystyle {\check {E}}\otimes E\simeq \operatorname {End} _{O}(E)\to O}"></span> given by the pairing; it is called the <a href="/wiki/Trace_map" class="mw-redirect" title="Trace map">trace map</a> of <i>E</i>. </p><p>For any <i>O</i>-module <i>F</i>, the <a href="/wiki/Tensor_algebra" title="Tensor algebra">tensor algebra</a>, <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> and <a href="/wiki/Symmetric_algebra" title="Symmetric algebra">symmetric algebra</a> of <i>F</i> are defined in the same way. For example, the <i>k</i>-th exterior power </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge ^{k}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mover> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge ^{k}F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/789567557e307d144d615fc5ea88f306fa168b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.709ex; height:5.676ex;" alt="{\displaystyle \bigwedge ^{k}F}"></span></dd></dl> <p>is the sheaf associated to the presheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle U\mapsto \bigwedge _{O(U)}^{k}F(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <munderover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mi>F</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle U\mapsto \bigwedge _{O(U)}^{k}F(U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d03f4b3a36dfa77a0f0568874dfd507783e00b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.078ex; height:3.843ex;" alt="{\textstyle U\mapsto \bigwedge _{O(U)}^{k}F(U)}"></span>. If <i>F</i> is locally free of rank <i>n</i>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \bigwedge ^{n}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \bigwedge ^{n}F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881c6e76ef939df8ba8fc118f37830e014eb0eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.282ex; height:3.009ex;" alt="{\textstyle \bigwedge ^{n}F}"></span> is called the <a href="/wiki/Determinant_line_bundle" class="mw-redirect" title="Determinant line bundle">determinant line bundle</a> (though technically <a href="/wiki/Invertible_sheaf" title="Invertible sheaf">invertible sheaf</a>) of <i>F</i>, denoted by det(<i>F</i>). There is a natural perfect pairing: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge ^{r}F\otimes \bigwedge ^{n-r}F\to \det(F).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </mover> <mi>F</mi> <mo>⊗</mo> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>r</mi> </mrow> </mover> <mi>F</mi> <mo stretchy="false">→</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge ^{r}F\otimes \bigwedge ^{n-r}F\to \det(F).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d2b9a7ff04afa84352e2c988fa9772f84b8e8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.725ex; height:5.509ex;" alt="{\displaystyle \bigwedge ^{r}F\otimes \bigwedge ^{n-r}F\to \det(F).}"></span></dd></dl> <p>Let <i>f</i>: (<i>X</i>, <i>O</i>) →(<i>X<span class="nowrap" style="padding-left:0.1em;">'</span></i>, <i>O<span class="nowrap" style="padding-left:0.1em;">'</span></i>) be a morphism of ringed spaces. If <i>F</i> is an <i>O</i>-module, then the <a href="/wiki/Direct_image_sheaf" class="mw-redirect" title="Direct image sheaf">direct image sheaf</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd073a4715fd3af6312e3e0adeb187c8aa29470" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.934ex; height:2.509ex;" alt="{\displaystyle f_{*}F}"></span> is an <i>O<span class="nowrap" style="padding-left:0.1em;">'</span></i>-module through the natural map <i>O<span class="nowrap" style="padding-left:0.1em;">'</span></i> →<i>f</i><sub>*</sub><i>O</i> (such a natural map is part of the data of a morphism of ringed spaces.) </p><p>If <i>G</i> is an <i>O<span class="nowrap" style="padding-left:0.1em;">'</span></i>-module, then the module inverse image <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90efe94494f6eafa739300fb468a884c40f7324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.201ex; height:2.676ex;" alt="{\displaystyle f^{*}G}"></span> of <i>G</i> is the <i>O</i>-module given as the tensor product of modules: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}G\otimes _{f^{-1}O'}O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>G</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>O</mi> <mo>′</mo> </msup> </mrow> </msub> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}G\otimes _{f^{-1}O'}O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c752649aee7d1519545758d2711fd8009e7d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:14.915ex; height:3.676ex;" alt="{\displaystyle f^{-1}G\otimes _{f^{-1}O'}O}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8225198a5752abae7c8062c85ef16330bdbddfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.48ex; height:3.009ex;" alt="{\displaystyle f^{-1}G}"></span> is the <a href="/wiki/Inverse_image_sheaf" class="mw-redirect" title="Inverse image sheaf">inverse image sheaf</a> of <i>G</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}O'\to O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>O</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}O'\to O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3f87fc3c14ef96ec06bdded757a6bc5aa6f662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.499ex; height:3.009ex;" alt="{\displaystyle f^{-1}O'\to O}"></span> is obtained from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O'\to f_{*}O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>O</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O'\to f_{*}O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf47c6a210d01f7b397498d26589c2e3014c20f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.039ex; height:2.843ex;" alt="{\displaystyle O'\to f_{*}O}"></span> by <a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjuction</a>. </p><p>There is an adjoint relation between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903e97c260c5d157bff320b5e38e0e4b42cd20ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{*}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190a73fde235865b8d2a783334f90194331c7f19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:2.676ex;" alt="{\displaystyle f^{*}}"></span>: for any <i>O</i>-module <i>F</i> and <i>O'</i>-module <i>G</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{O}(f^{*}G,F)\simeq \operatorname {Hom} _{O'}(G,f_{*}F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>G</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>O</mi> <mo>′</mo> </msup> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{O}(f^{*}G,F)\simeq \operatorname {Hom} _{O'}(G,f_{*}F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03779103ec37a6d2544e34c53f3d2ab438640e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.675ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{O}(f^{*}G,F)\simeq \operatorname {Hom} _{O'}(G,f_{*}F)}"></span></dd></dl> <p>as abelian group. There is also the <a href="/wiki/Projection_formula" title="Projection formula">projection formula</a>: for an <i>O</i>-module <i>F</i> and a locally free <i>O'</i>-module <i>E</i> of finite rank, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}(F\otimes f^{*}E)\simeq f_{*}F\otimes E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo>⊗</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>E</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>F</mi> <mo>⊗</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{*}(F\otimes f^{*}E)\simeq f_{*}F\otimes E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae27719cf482d89de11867739a788bb1810fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.03ex; height:2.843ex;" alt="{\displaystyle f_{*}(F\otimes f^{*}E)\simeq f_{*}F\otimes E.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=3" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="Generated_by_global_sections"></span> Let (<i>X</i>, <i>O</i>) be a ringed space. An <i>O</i>-module <i>F</i> is said to be <b>generated by global sections</b> if there is a surjection of <i>O</i>-modules: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{i\in I}O\to F\to 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <mi>O</mi> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigoplus _{i\in I}O\to F\to 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aef841a55f71eb5f8d357802535bf5f40bacdec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.449ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{i\in I}O\to F\to 0.}"></span></dd></dl> <p>Explicitly, this means that there are global sections <i>s</i><sub><i>i</i></sub> of <i>F</i> such that the images of <i>s</i><sub><i>i</i></sub> in each stalk <i>F</i><sub><i>x</i></sub> generates <i>F</i><sub><i>x</i></sub> as <i>O</i><sub><i>x</i></sub>-module. </p><p>An example of such a sheaf is that associated in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> to an <i>R</i>-module <i>M</i>, <i>R</i> being any <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, on the <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a> <i>Spec</i>(<i>R</i>). Another example: according to <a href="/wiki/Cartan%27s_theorem_A" class="mw-redirect" title="Cartan's theorem A">Cartan's theorem A</a>, any <a href="/wiki/Coherent_sheaf" title="Coherent sheaf">coherent sheaf</a> on a <a href="/wiki/Stein_manifold" title="Stein manifold">Stein manifold</a> is spanned by global sections. (cf. Serre's theorem A below.) In the theory of <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a>, a related notion is <a href="/wiki/Ample_line_bundle" title="Ample line bundle">ample line bundle</a>. (For example, if <i>L</i> is an ample line bundle, some power of it is generated by global sections.) </p><p>An injective <i>O</i>-module is <a href="/wiki/Flasque_sheaf" class="mw-redirect" title="Flasque sheaf">flasque</a> (i.e., all restrictions maps <i>F</i>(<i>U</i>) → <i>F</i>(<i>V</i>) are surjective.)<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Since a flasque sheaf is acyclic in the category of abelian sheaves, this implies that the <i>i</i>-th right derived functor of the global section functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (X,-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (X,-)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e34602234200472f45f1562a7efe02104ac4d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.084ex; height:2.843ex;" alt="{\displaystyle \Gamma (X,-)}"></span> in the category of <i>O</i>-modules coincides with the usual <i>i</i>-th sheaf cohomology in the category of abelian sheaves.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Sheaf_associated_to_a_module">Sheaf associated to a module</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=4" title="Edit section: Sheaf associated to a module"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> be a module over a ring <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. Put <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\operatorname {Spec} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\operatorname {Spec} (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65a7698ca0bfc532fab9f9109a0d7ddb593bb34a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.281ex; height:2.843ex;" alt="{\displaystyle X=\operatorname {Spec} (A)}"></span> and write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed209cfddafaab71225e00d5372ca6cc65168bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.223ex; height:3.176ex;" alt="{\displaystyle D(f)=\{f\neq 0\}=\operatorname {Spec} (A[f^{-1}])}"></span>. For each pair <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(f)\subseteq D(g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(f)\subseteq D(g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb9bed1dd4c1681603f4f921799193b08626771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.96ex; height:2.843ex;" alt="{\displaystyle D(f)\subseteq D(g)}"></span>, by the universal property of localization, there is a natural map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{g,f}:M[g^{-1}]\to M[f^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>:</mo> <mi>M</mi> <mo stretchy="false">[</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{g,f}:M[g^{-1}]\to M[f^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70cb90947ee387662b3ba064e0c0b726f2f98538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.712ex; height:3.343ex;" alt="{\displaystyle \rho _{g,f}:M[g^{-1}]\to M[f^{-1}]}"></span></dd></dl> <p>having the property that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{g,f}=\rho _{g,h}\circ \rho _{h,f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo>,</mo> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{g,f}=\rho _{g,h}\circ \rho _{h,f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/509ec7ff940fe3f0aa3842d271380fc8bbe91999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.248ex; height:2.343ex;" alt="{\displaystyle \rho _{g,f}=\rho _{g,h}\circ \rho _{h,f}}"></span>. Then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(f)\mapsto M[f^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>M</mi> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(f)\mapsto M[f^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c5fbf31be70749ad8d44a4effb32c573b25f5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.015ex; height:3.176ex;" alt="{\displaystyle D(f)\mapsto M[f^{-1}]}"></span></dd></dl> <p>is a contravariant functor from the category whose objects are the sets <i>D</i>(<i>f</i>) and morphisms the inclusions of sets to the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a>. One can show<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> it is in fact a <a href="/wiki/B-sheaf" class="mw-redirect" title="B-sheaf">B-sheaf</a> (i.e., it satisfies the gluing axiom) and thus defines the sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301b4dae6caafe7d093e43c068d75940062f7fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.699ex; height:2.843ex;" alt="{\displaystyle {\widetilde {M}}}"></span> on <i>X</i> called the sheaf associated to <i>M</i>. </p><p>The most basic example is the structure sheaf on <i>X</i>; i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9051bcad65f3b0fd714c68791f8cf6f8caaa0229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.421ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}"></span>. Moreover, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301b4dae6caafe7d093e43c068d75940062f7fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.699ex; height:2.843ex;" alt="{\displaystyle {\widetilde {M}}}"></span> has the structure of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9051bcad65f3b0fd714c68791f8cf6f8caaa0229" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.421ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}_{X}={\widetilde {A}}}"></span>-module and thus one gets the <a href="/wiki/Exact_functor" title="Exact functor">exact functor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\mapsto {\widetilde {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\mapsto {\widetilde {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85a1a64a7876abbeb1699e3ce4217ba3a984185" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.755ex; height:2.843ex;" alt="{\displaystyle M\mapsto {\widetilde {M}}}"></span> from Mod<sub><i>A</i></sub>, the <a href="/wiki/Category_of_modules" title="Category of modules">category of modules</a> over <i>A</i> to the category of modules over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fed6a46b79218af44f23e5d6f487fb7e0d6cd01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}"></span>. It defines an equivalence from Mod<sub><i>A</i></sub> to the category of <a href="/wiki/Quasi-coherent_sheaves" class="mw-redirect" title="Quasi-coherent sheaves">quasi-coherent sheaves</a> on <i>X</i>, with the inverse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (X,-)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (X,-)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e34602234200472f45f1562a7efe02104ac4d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.084ex; height:2.843ex;" alt="{\displaystyle \Gamma (X,-)}"></span>, the <a href="/wiki/Global_section_functor" class="mw-redirect" title="Global section functor">global section functor</a>. When <i>X</i> is <a href="/wiki/Noetherian_scheme" title="Noetherian scheme">Noetherian</a>, the functor is an equivalence from the category of finitely generated <i>A</i>-modules to the category of coherent sheaves on <i>X</i>. </p><p>The construction has the following properties: for any <i>A</i>-modules <i>M</i>, <i>N</i>, and any morphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :M\to N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :M\to N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7dd52f475d4d090b5f59f745e773f63d3176a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.577ex; height:2.676ex;" alt="{\displaystyle \varphi :M\to N}"></span>, </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b54f210365bb77a31f233cee1fc02581d3a2f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.12ex; height:3.843ex;" alt="{\displaystyle M[f^{-1}]^{\sim }={\widetilde {M}}|_{D(f)}}"></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>For any prime ideal <i>p</i> of <i>A</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {M}}_{p}\simeq M_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≃</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}_{p}\simeq M_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50eb5752e43c3960245541033f2c34cf669bc09b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.17ex; height:3.509ex;" alt="{\displaystyle {\widetilde {M}}_{p}\simeq M_{p}}"></span> as <i>O</i><sub><i>p</i></sub> = <i>A</i><sub><i>p</i></sub>-module.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M\otimes _{A}N)^{\sim }\simeq {\widetilde {M}}\otimes _{\widetilde {A}}{\widetilde {N}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>N</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>~</mo> </mover> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M\otimes _{A}N)^{\sim }\simeq {\widetilde {M}}\otimes _{\widetilde {A}}{\widetilde {N}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dadd238eac780a9379a6258a257456e48b564a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.043ex; height:3.676ex;" alt="{\displaystyle (M\otimes _{A}N)^{\sim }\simeq {\widetilde {M}}\otimes _{\widetilde {A}}{\widetilde {N}}}"></span>.<sup id="cite_ref-Corollary_1.3.12._10-0" class="reference"><a href="#cite_note-Corollary_1.3.12.-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>If <i>M</i> is <a href="/wiki/Finitely_presented_module" class="mw-redirect" title="Finitely presented module">finitely presented</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{A}(M,N)^{\sim }\simeq {\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mi>o</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>~</mo> </mover> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{A}(M,N)^{\sim }\simeq {\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa698dc32b48fc24e99d5c82d6b258509efb9838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.212ex; height:3.676ex;" alt="{\displaystyle \operatorname {Hom} _{A}(M,N)^{\sim }\simeq {\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}})}"></span>.<sup id="cite_ref-Corollary_1.3.12._10-1" class="reference"><a href="#cite_note-Corollary_1.3.12.-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{A}(M,N)\simeq \Gamma (X,{\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mi>o</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>~</mo> </mover> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>N</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{A}(M,N)\simeq \Gamma (X,{\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f58dcb952139ea21f9a89406730287e22dcb5a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.978ex; height:3.676ex;" alt="{\displaystyle \operatorname {Hom} _{A}(M,N)\simeq \Gamma (X,{\mathcal {H}}om_{\widetilde {A}}({\widetilde {M}},{\widetilde {N}}))}"></span>, since the equivalence between Mod<sub><i>A</i></sub> and the category of quasi-coherent sheaves on <i>X</i>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\varinjlim M_{i})^{\sim }\simeq \varinjlim {\widetilde {M_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo>≃</mo> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\varinjlim M_{i})^{\sim }\simeq \varinjlim {\widetilde {M_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e36fe190145bef24be662688ec0d9ee79e60cb9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:20.02ex; height:4.509ex;" alt="{\displaystyle (\varinjlim M_{i})^{\sim }\simeq \varinjlim {\widetilde {M_{i}}}}"></span>;<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> in particular, taking a direct sum and ~ commute.</li> <li>A sequence of <i>A</i>-modules is exact if and only if the induced sequence by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"></span> is exact. In particular, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\ker(\varphi ))^{\sim }=\ker({\widetilde {\varphi }}),(\operatorname {coker} (\varphi ))^{\sim }=\operatorname {coker} ({\widetilde {\varphi }}),(\operatorname {im} (\varphi ))^{\sim }=\operatorname {im} ({\widetilde {\varphi }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo>=</mo> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>coker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo>=</mo> <mi>coker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mi>im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo>=</mo> <mi>im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>φ</mi> <mo>~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\ker(\varphi ))^{\sim }=\ker({\widetilde {\varphi }}),(\operatorname {coker} (\varphi ))^{\sim }=\operatorname {coker} ({\widetilde {\varphi }}),(\operatorname {im} (\varphi ))^{\sim }=\operatorname {im} ({\widetilde {\varphi }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64e3b4ae85d2f714695a5e4bfd392ff4ec9b524c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.78ex; height:2.843ex;" alt="{\displaystyle (\ker(\varphi ))^{\sim }=\ker({\widetilde {\varphi }}),(\operatorname {coker} (\varphi ))^{\sim }=\operatorname {coker} ({\widetilde {\varphi }}),(\operatorname {im} (\varphi ))^{\sim }=\operatorname {im} ({\widetilde {\varphi }})}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Sheaf_associated_to_a_graded_module">Sheaf associated to a graded module</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=5" title="Edit section: Sheaf associated to a graded module"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a graded analog of the construction and equivalence in the preceding section. Let <i>R</i> be a graded ring generated by degree-one elements as <i>R</i><sub>0</sub>-algebra (<i>R</i><sub>0</sub> means the degree-zero piece) and <i>M</i> a graded <i>R</i>-module. Let <i>X</i> be the <a href="/wiki/Proj_construction" title="Proj construction">Proj</a> of <i>R</i> (so <i>X</i> is a <a href="/wiki/Projective_scheme" class="mw-redirect" title="Projective scheme">projective scheme</a> if <i>R</i> is Noetherian). Then there is an <i>O</i>-module <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301b4dae6caafe7d093e43c068d75940062f7fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.699ex; height:2.843ex;" alt="{\displaystyle {\widetilde {M}}}"></span> such that for any homogeneous element <i>f</i> of positive degree of <i>R</i>, there is a natural isomorphism </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {M}}|_{\{f\neq 0\}}\simeq (M[f^{-1}]_{0})^{\sim }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> <mo>≃</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}|_{\{f\neq 0\}}\simeq (M[f^{-1}]_{0})^{\sim }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e3c8df95226bf9fa13c36b7656d599552f28b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:23.088ex; height:3.843ex;" alt="{\displaystyle {\widetilde {M}}|_{\{f\neq 0\}}\simeq (M[f^{-1}]_{0})^{\sim }}"></span></dd></dl> <p>as sheaves of modules on the affine scheme <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{f\neq 0\}=\operatorname {Spec} (R[f^{-1}]_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f\neq 0\}=\operatorname {Spec} (R[f^{-1}]_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/967fed6f4464945e714e78d26ad7b4f64641ba3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.187ex; height:3.176ex;" alt="{\displaystyle \{f\neq 0\}=\operatorname {Spec} (R[f^{-1}]_{0})}"></span>;<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> in fact, this defines <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>M</mi> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widetilde {M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301b4dae6caafe7d093e43c068d75940062f7fe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.699ex; height:2.843ex;" alt="{\displaystyle {\widetilde {M}}}"></span> by gluing. </p><p><b>Example</b>: Let <i>R</i>(1) be the graded <i>R</i>-module given by <i>R</i>(1)<sub><i>n</i></sub> = <i>R</i><sub><i>n</i>+1</sub>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(1)={\widetilde {R(1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>~</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(1)={\widetilde {R(1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12914cec316a4111f0b65e3694f2633114d843a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.579ex; height:3.676ex;" alt="{\displaystyle O(1)={\widetilde {R(1)}}}"></span> is called <a href="/wiki/Serre%27s_twisting_sheaf" class="mw-redirect" title="Serre's twisting sheaf">Serre's twisting sheaf</a>, which is the dual of the <a href="/wiki/Tautological_line_bundle" class="mw-redirect" title="Tautological line bundle">tautological line bundle</a> if <i>R</i> is finitely generated in degree-one. </p><p>If <i>F</i> is an <i>O</i>-module on <i>X</i>, then, writing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(n)=F\otimes O(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo>⊗</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(n)=F\otimes O(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa09e0313a657d6157213a03e497c2b1a55e1735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.602ex; height:2.843ex;" alt="{\displaystyle F(n)=F\otimes O(n)}"></span>, there is a canonical homomorphism: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\bigoplus _{n\geq 0}\Gamma (X,F(n))\right)^{\sim }\to F,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </msup> <mo stretchy="false">→</mo> <mi>F</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\bigoplus _{n\geq 0}\Gamma (X,F(n))\right)^{\sim }\to F,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b538dc061a4dc4024c4d3a2e750d714d8ffb69c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.312ex; height:7.509ex;" alt="{\displaystyle \left(\bigoplus _{n\geq 0}\Gamma (X,F(n))\right)^{\sim }\to F,}"></span></dd></dl> <p>which is an isomorphism if and only if <i>F</i> is quasi-coherent. </p> <div class="mw-heading mw-heading2"><h2 id="Computing_sheaf_cohomology">Computing sheaf cohomology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=6" title="Edit section: Computing sheaf cohomology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Sheaf_of_modules&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">January 2016</span>)</i></span></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a></div> <p>Sheaf cohomology has a reputation for being difficult to calculate. Because of this, the next general fact is fundamental for any practical computation: </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret"> — </span>Let <i>X</i> be a topological space, <i>F</i> an abelian sheaf on it and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {U}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.057ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {U}}}"></span> an open cover of <i>X</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} ^{i}(U_{i_{0}}\cap \cdots \cap U_{i_{p}},F)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mo>∩</mo> <mo>⋯</mo> <mo>∩</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} ^{i}(U_{i_{0}}\cap \cdots \cap U_{i_{p}},F)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0995cfb4d54a4358fc11dad4e94225e80da31e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.718ex; height:3.509ex;" alt="{\displaystyle \operatorname {H} ^{i}(U_{i_{0}}\cap \cdots \cap U_{i_{p}},F)=0}"></span> for any <i>i</i>, <i>p</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i_{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i_{j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e2f7576b1a414cf75467ab4475e8357bf35d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.101ex; height:3.009ex;" alt="{\displaystyle U_{i_{j}}}"></span>'s in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {U}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">U</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {U}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042ea9f1b7b1e435446133c8cc8bc5a204766da5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.057ex; width:1.603ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {U}}}"></span>. Then for any <i>i</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {H} ^{i}(C^{\bullet }({\mathfrak {U}},F))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">U</mi> </mrow> </mrow> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {H} ^{i}(C^{\bullet }({\mathfrak {U}},F))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c50d04d679e3862aa56bba04b7a56c3aa22fec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.539ex; height:3.176ex;" alt="{\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {H} ^{i}(C^{\bullet }({\mathfrak {U}},F))}"></span></dd></dl> <p>where the right-hand side is the <i>i</i>-th <a href="/wiki/%C4%8Cech_cohomology" title="Čech cohomology">Čech cohomology</a>. </p> </div> <p><a href="/wiki/Serre%27s_vanishing_theorem" class="mw-redirect" title="Serre's vanishing theorem">Serre's vanishing theorem</a><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> states that if <i>X</i> is a projective variety and <i>F</i> a coherent sheaf on it, then, for sufficiently large <i>n</i>, the <a href="/wiki/Serre_twist" class="mw-redirect" title="Serre twist">Serre twist</a> <i>F</i>(<i>n</i>) is generated by finitely many global sections. Moreover, </p> <ol type="a"> <li> For each <i>i</i>, H<sup><i>i</i></sup>(<i>X</i>, <i>F</i>) is finitely generated over <i>R</i><sub>0</sub>, and</li> <li> There is an integer <i>n</i><sub>0</sub>, depending on <i>F</i>, such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} ^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} ^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2b765b74575c138212778c3bc86dee9a4a9d95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.679ex; height:3.176ex;" alt="{\displaystyle \operatorname {H} ^{i}(X,F(n))=0,\,i\geq 1,n\geq n_{0}.}"></span></li> </ol><p><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><div class="mw-heading mw-heading2"><h2 id="Sheaf_extension">Sheaf extension</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=7" title="Edit section: Sheaf extension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let (<i>X</i>, <i>O</i>) be a ringed space, and let <i>F</i>, <i>H</i> be sheaves of <i>O</i>-modules on <i>X</i>. An <b>extension</b> of <i>H</i> by <i>F</i> is a <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of <i>O</i>-modules </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi>F</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2581c21a1b7040bf4139e7f5cc3af9881c440ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.059ex; height:2.176ex;" alt="{\displaystyle 0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.}"></span></dd></dl> <p>As with group extensions, if we fix <i>F</i> and <i>H</i>, then all equivalence classes of extensions of <i>H</i> by <i>F</i> form an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> (cf. <a href="/wiki/Baer_sum" class="mw-redirect" title="Baer sum">Baer sum</a>), which is isomorphic to the <a href="/wiki/Ext_functor" title="Ext functor">Ext group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Ext</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a384d1b5247f446cfdcc42cab91f98cb5d8990e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.849ex; height:3.343ex;" alt="{\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}"></span>, where the identity element in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Ext</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a384d1b5247f446cfdcc42cab91f98cb5d8990e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.849ex; height:3.343ex;" alt="{\displaystyle \operatorname {Ext} _{O}^{1}(H,F)}"></span> corresponds to the trivial extension. </p><p>In the case where <i>H</i> is <i>O</i>, we have: for any <i>i</i> ≥ 0, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {Ext} _{O}^{i}(O,F),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>Ext</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>O</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {Ext} _{O}^{i}(O,F),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8475792d43e14ac0a8a793a42f6393d4aad319d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.411ex; height:3.343ex;" alt="{\displaystyle \operatorname {H} ^{i}(X,F)=\operatorname {Ext} _{O}^{i}(O,F),}"></span></dd></dl> <p>since both the sides are the right derived functors of the same functor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (X,-)=\operatorname {Hom} _{O}(O,-).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>O</mi> <mo>,</mo> <mo>−</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma (X,-)=\operatorname {Hom} _{O}(O,-).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0e0a9931e992767d3da82fdbd24ebb3478b67c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.582ex; height:2.843ex;" alt="{\displaystyle \Gamma (X,-)=\operatorname {Hom} _{O}(O,-).}"></span> </p><p><b>Note</b>: Some authors, notably Hartshorne, drop the subscript <i>O</i>. </p><p>Assume <i>X</i> is a projective scheme over a Noetherian ring. Let <i>F</i>, <i>G</i> be coherent sheaves on <i>X</i> and <i>i</i> an integer. Then there exists <i>n</i><sub>0</sub> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Ext</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mi>x</mi> <msubsup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>n</mi> <mo>≥</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82ea239689c9ca8c1302af4cd3b397e8a46abf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.135ex; height:3.343ex;" alt="{\displaystyle \operatorname {Ext} _{O}^{i}(F,G(n))=\Gamma (X,{\mathcal {E}}xt_{O}^{i}(F,G(n))),\,n\geq n_{0}}"></span>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></dd></dl> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Local-to-global_Ext_spectral_sequence" class="mw-redirect" title="Local-to-global Ext spectral sequence">local-to-global Ext spectral sequence</a></div> <div class="mw-heading mw-heading3"><h3 id="Locally_free_resolutions">Locally free resolutions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=8" title="Edit section: Locally free resolutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Ext}}({\mathcal {F}},{\mathcal {G}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">x</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">t</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Ext}}({\mathcal {F}},{\mathcal {G}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0913f4970eb57b5e83482d11ce3a91896038858e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.643ex; height:2.843ex;" alt="{\displaystyle {\mathcal {Ext}}({\mathcal {F}},{\mathcal {G}})}"></span> can be readily computed for any coherent sheaf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span> using a locally free resolution:<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> given a complex </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdots \to {\mathcal {L}}_{2}\to {\mathcal {L}}_{1}\to {\mathcal {L}}_{0}\to {\mathcal {F}}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋯</mo> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdots \to {\mathcal {L}}_{2}\to {\mathcal {L}}_{1}\to {\mathcal {L}}_{0}\to {\mathcal {F}}\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5c3bd0e8df0872cd9b849d78fae3c7145f5b94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.857ex; height:2.509ex;" alt="{\displaystyle \cdots \to {\mathcal {L}}_{2}\to {\mathcal {L}}_{1}\to {\mathcal {L}}_{0}\to {\mathcal {F}}\to 0}"></span></dd></dl> <p>then </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {RHom}}({\mathcal {F}},{\mathcal {G}})={\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">o</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">m</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">o</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">m</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {RHom}}({\mathcal {F}},{\mathcal {G}})={\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6938f3293dd697cda4486b261d6408d182d2b0f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.389ex; height:2.843ex;" alt="{\displaystyle {\mathcal {RHom}}({\mathcal {F}},{\mathcal {G}})={\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}})}"></span></dd></dl> <p>hence </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Ext}}^{k}({\mathcal {F}},{\mathcal {G}})=h^{k}({\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">x</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">t</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">o</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">m</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∙</mo> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">G</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Ext}}^{k}({\mathcal {F}},{\mathcal {G}})=h^{k}({\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bd4809759a60cf6133e4c7c58c53590893136ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.092ex; height:3.176ex;" alt="{\displaystyle {\mathcal {Ext}}^{k}({\mathcal {F}},{\mathcal {G}})=h^{k}({\mathcal {Hom}}({\mathcal {L}}_{\bullet },{\mathcal {G}}))}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=9" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Hypersurface">Hypersurface</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=10" title="Edit section: Hypersurface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider a smooth hypersurface <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> of degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>. Then, we can compute a resolution </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(-d)\to {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(-d)\to {\mathcal {O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9041a18352a94dcbf258dcfba43f2192392d5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.147ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(-d)\to {\mathcal {O}}}"></span></dd></dl> <p>and find that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})=h^{i}({\mathcal {Hom}}({\mathcal {O}}(-d)\to {\mathcal {O}},{\mathcal {F}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">x</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">t</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">o</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">m</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})=h^{i}({\mathcal {Hom}}({\mathcal {O}}(-d)\to {\mathcal {O}},{\mathcal {F}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa52ef7825a0f62db8facc09ac8cb96943ab7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.627ex; height:3.176ex;" alt="{\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})=h^{i}({\mathcal {Hom}}({\mathcal {O}}(-d)\to {\mathcal {O}},{\mathcal {F}}))}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Union_of_smooth_complete_intersections">Union of smooth complete intersections</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=11" title="Edit section: Union of smooth complete intersections"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the scheme </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} [x_{0},\ldots ,x_{n}]}{(f)(g_{1},g_{2},g_{3})}}\right)\subseteq \mathbb {P} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Proj</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>⊆</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} [x_{0},\ldots ,x_{n}]}{(f)(g_{1},g_{2},g_{3})}}\right)\subseteq \mathbb {P} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc2605ff5732feb4d573b72e7f9d80e84d80053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.284ex; height:6.509ex;" alt="{\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} [x_{0},\ldots ,x_{n}]}{(f)(g_{1},g_{2},g_{3})}}\right)\subseteq \mathbb {P} ^{n}}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f,g_{1},g_{2},g_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f,g_{1},g_{2},g_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4d5630cc9317d11e9d9ba7783b63b142c51511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.68ex; height:2.843ex;" alt="{\displaystyle (f,g_{1},g_{2},g_{3})}"></span> is a smooth complete intersection and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(f)=d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f)=d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155f37ef1e223f3fa834ffbfed323cc2b60b2d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.89ex; height:2.843ex;" alt="{\displaystyle \deg(f)=d}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \deg(g_{i})=e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(g_{i})=e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d505ea6787006f00a60169573822a0580a426b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.187ex; height:2.843ex;" alt="{\displaystyle \deg(g_{i})=e_{i}}"></span>. We have a complex </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(-d-e_{1}-e_{2}-e_{3}){\xrightarrow {\begin{bmatrix}g_{3}\\-g_{2}\\-g_{1}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1}-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{1}-e_{3})\\\oplus \\{\mathcal {O}}(-d-e_{2}-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}g_{2}&g_{3}&0\\-g_{1}&0&-g_{3}\\0&-g_{1}&g_{2}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1})\\\oplus \\{\mathcal {O}}(-d-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}fg_{1}&fg_{2}&fg_{3}\end{bmatrix}}}{\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mpadded> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⊕</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⊕</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mpadded> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⊕</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⊕</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi>d</mi> <mo>−</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>f</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>f</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi>f</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mpadded> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(-d-e_{1}-e_{2}-e_{3}){\xrightarrow {\begin{bmatrix}g_{3}\\-g_{2}\\-g_{1}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1}-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{1}-e_{3})\\\oplus \\{\mathcal {O}}(-d-e_{2}-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}g_{2}&g_{3}&0\\-g_{1}&0&-g_{3}\\0&-g_{1}&g_{2}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1})\\\oplus \\{\mathcal {O}}(-d-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}fg_{1}&fg_{2}&fg_{3}\end{bmatrix}}}{\mathcal {O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9dc0fd1a4586b37c9969a94803b4c89dbdbf45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; margin-top: -0.415ex; width:93.224ex; height:16.676ex;" alt="{\displaystyle {\mathcal {O}}(-d-e_{1}-e_{2}-e_{3}){\xrightarrow {\begin{bmatrix}g_{3}\\-g_{2}\\-g_{1}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1}-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{1}-e_{3})\\\oplus \\{\mathcal {O}}(-d-e_{2}-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}g_{2}&g_{3}&0\\-g_{1}&0&-g_{3}\\0&-g_{1}&g_{2}\end{bmatrix}}}{\begin{matrix}{\mathcal {O}}(-d-e_{1})\\\oplus \\{\mathcal {O}}(-d-e_{2})\\\oplus \\{\mathcal {O}}(-d-e_{3})\end{matrix}}{\xrightarrow {\begin{bmatrix}fg_{1}&fg_{2}&fg_{3}\end{bmatrix}}}{\mathcal {O}}}"></span></dd></dl> <p>resolving <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa7132f92984d15c60982967988e64dcedd52b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.129ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X},}"></span> which we can use to compute <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">x</mi> <mi class="MJX-tex-caligraphic" mathvariant="script">t</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4e04b65fd29ab00f61091d237009bb6c593b1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.532ex; height:3.176ex;" alt="{\displaystyle {\mathcal {Ext}}^{i}({\mathcal {O}}_{X},{\mathcal {F}})}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/D-module" title="D-module">D-module</a> (in place of <i>O</i>, one can also consider <i>D</i>, the sheaf of differential operators.)</li> <li><a href="/wiki/Fractional_ideal" title="Fractional ideal">fractional ideal</a></li> <li><a href="/wiki/Holomorphic_vector_bundle" title="Holomorphic vector bundle">holomorphic vector bundle</a></li> <li><a href="/wiki/Generic_freeness" class="mw-redirect" title="Generic freeness">generic freeness</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=13" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Vakil, <a rel="nofollow" class="external text" href="http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf">Math 216: Foundations of algebraic geometry</a>, 2.5.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne">Hartshorne</a>, Ch. III, Proposition 2.2.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">This cohomology functor coincides with the right derived functor of the global section functor in the category of abelian sheaves; cf. <a href="#CITEREFHartshorne">Hartshorne</a>, Ch. III, Proposition 2.6.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">There is a canonical homomorphism: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}om_{O}(F,O)_{x}\to \operatorname {Hom} _{O_{x}}(F_{x},O_{x}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mi>o</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo>,</mo> <mi>O</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}om_{O}(F,O)_{x}\to \operatorname {Hom} _{O_{x}}(F_{x},O_{x}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d88913de03b3e069fde40831345df04689657251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.12ex; height:2.843ex;" alt="{\displaystyle {\mathcal {H}}om_{O}(F,O)_{x}\to \operatorname {Hom} _{O_{x}}(F_{x},O_{x}),}"></span></dd></dl> which is an isomorphism if <i>F</i> is of finite presentation (EGA, Ch. 0, 5.2.6.)</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">For coherent sheaves, having a tensor inverse is the same as being locally free of rank one; in fact, there is the following fact: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\otimes G\simeq O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊗</mo> <mi>G</mi> <mo>≃</mo> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\otimes G\simeq O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36aba8e82341a348f0e600c91372b9aa44845245" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.28ex; height:2.343ex;" alt="{\displaystyle F\otimes G\simeq O}"></span> and if <i>F</i> is coherent, then <i>F</i>, <i>G</i> are locally free of rank one. (cf. EGA, Ch 0, 5.4.3.)</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne">Hartshorne</a>, Ch III, Lemma 2.4.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">see also: <a rel="nofollow" class="external free" href="https://math.stackexchange.com/q/447234">https://math.stackexchange.com/q/447234</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne">Hartshorne</a>, Ch. II, Proposition 5.1.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFEGA_I">EGA I</a>, Ch. I, Proposition 1.3.6.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFEGA_I (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-Corollary_1.3.12.-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Corollary_1.3.12._10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Corollary_1.3.12._10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFEGA_I">EGA I</a>, Ch. I, Corollaire 1.3.12.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFEGA_I (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFEGA_I">EGA I</a>, Ch. I, Corollaire 1.3.9.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFEGA_I (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>)</span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne">Hartshorne</a>, Ch. II, Proposition 5.11.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/01X8">"Section 30.2 (01X8): Čech cohomology of quasi-coherent sheaves—The Stacks project"</a>. <i>stacks.math.columbia.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-12-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=stacks.math.columbia.edu&rft.atitle=Section+30.2+%2801X8%29%3A+%C4%8Cech+cohomology+of+quasi-coherent+sheaves%E2%80%94The+Stacks+project&rft_id=https%3A%2F%2Fstacks.math.columbia.edu%2Ftag%2F01X8&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFCostaMiró-RoigPons-Llopis2021">Costa, Miró-Roig & Pons-Llopis 2021</a>, Theorem 1.3.1</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">"Links with sheaf cohomology". <i>Local Cohomology</i>. Cambridge Studies in Advanced Mathematics. Cambridge University Press. 2012. pp. 438–479. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9781139044059.023">10.1017/CBO9781139044059.023</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521513630" title="Special:BookSources/9780521513630"><bdi>9780521513630</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Links+with+sheaf+cohomology&rft.btitle=Local+Cohomology&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pages=438-479&rft.pub=Cambridge+University+Press&rft.date=2012&rft_id=info%3Adoi%2F10.1017%2FCBO9781139044059.023&rft.isbn=9780521513630&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFSerre1955">Serre 1955</a>, §.66 Faisceaux algébriques cohérents sur les variétés projectives.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne">Hartshorne</a>, Ch. III, Proposition 6.9.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne" class="citation book cs1">Hartshorne, Robin. <i>Algebraic Geometry</i>. pp. 233–235.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.pages=233-235&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sheaf_of_modules&action=edit&section=14" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrothendieckDieudonné1960" class="citation journal cs1"><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexandre</a>; <a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1960). <a rel="nofollow" class="external text" href="http://www.numdam.org/item/PMIHES_1960__4__5_0">"Éléments de géométrie algébrique: I. Le langage des schémas"</a>. <i><a href="/wiki/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a></i>. <b>4</b>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02684778">10.1007/bf02684778</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0217083">0217083</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publications+Math%C3%A9matiques+de+l%27IH%C3%89S&rft.atitle=%C3%89l%C3%A9ments+de+g%C3%A9om%C3%A9trie+alg%C3%A9brique%3A+I.+Le+langage+des+sch%C3%A9mas&rft.volume=4&rft.date=1960&rft_id=info%3Adoi%2F10.1007%2Fbf02684778&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0217083%23id-name%3DMR&rft.aulast=Grothendieck&rft.aufirst=Alexandre&rft.au=Dieudonn%C3%A9%2C+Jean&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%2FPMIHES_1960__4__5_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <i><a href="/wiki/Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a></i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90244-9" title="Special:BookSources/978-0-387-90244-9"><bdi>978-0-387-90244-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=978-0-387-90244-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0463157%23id-name%3DMR&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCostaMiró-RoigPons-Llopis2021" class="citation book cs1">Costa, Laura; Miró-Roig, Rosa María; Pons-Llopis, Joan (2021). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=v9IuEAAAQBAJ&pg=PT22"><i>Ulrich Bundles</i></a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110647686">10.1515/9783110647686</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783110647686" title="Special:BookSources/9783110647686"><bdi>9783110647686</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ulrich+Bundles&rft.date=2021&rft_id=info%3Adoi%2F10.1515%2F9783110647686&rft.isbn=9783110647686&rft.aulast=Costa&rft.aufirst=Laura&rft.au=Mir%C3%B3-Roig%2C+Rosa+Mar%C3%ADa&rft.au=Pons-Llopis%2C+Joan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dv9IuEAAAQBAJ%26pg%3DPT22&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1">"Links with sheaf cohomology". <i>Local Cohomology</i>. Cambridge Studies in Advanced Mathematics. Cambridge University Press. 2012. pp. 438–479. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9781139044059.023">10.1017/CBO9781139044059.023</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521513630" title="Special:BookSources/9780521513630"><bdi>9780521513630</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Links+with+sheaf+cohomology&rft.btitle=Local+Cohomology&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pages=438-479&rft.pub=Cambridge+University+Press&rft.date=2012&rft_id=info%3Adoi%2F10.1017%2FCBO9781139044059.023&rft.isbn=9780521513630&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre1955" class="citation cs2"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1955), <a rel="nofollow" class="external text" href="https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf">"Faisceaux algébriques cohérents (§.66 Faisceaux algébriques cohérents sur les variétés projectives.)"</a> <span class="cs1-format">(PDF)</span>, <i>Annals of Mathematics</i>, <b>61</b> (2): 197–278, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969915">10.2307/1969915</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969915">1969915</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0068874">0068874</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Faisceaux+alg%C3%A9briques+coh%C3%A9rents+%28%C2%A7.66+Faisceaux+alg%C3%A9briques+coh%C3%A9rents+sur+les+vari%C3%A9t%C3%A9s+projectives.%29&rft.volume=61&rft.issue=2&rft.pages=197-278&rft.date=1955&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0068874%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969915%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1969915&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Fwww.college-de-france.fr%2Fmedia%2Fjean-pierre-serre%2FUPL5435398796951750634_Serre_FAC.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASheaf+of+modules" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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