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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topos_theory">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="higher_geometry">Higher geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_terms_of_differential_operators'>In terms of differential operators</a></li> <li><a href='#in_terms_of_sheaves_on_the_derham_space'>In terms of sheaves on the deRham space</a></li> </ul> <li><a href='#meaning_and_usage'>Meaning and usage</a></li> <li><a href='#PositiveCharacteristics'>Positive characteristic</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#SixOperationsYoga'>Six operations yoga</a></li> <li><a href='#RelationToGeometricRepresentationTheory'>Relation to geometric representation theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <h3 id="in_terms_of_differential_operators">In terms of differential operators</h3> <p>A <em>D-module</em> (introduced by <a class="existingWikiWord" href="/nlab/show/Mikio+Sato">Mikio Sato</a>) is a <a class="existingWikiWord" href="/nlab/show/abelian+sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/regular+differential+operators">regular differential operators</a> on a ‘variety’ <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (the latter notion depends on whether we work over a <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a>, <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, analytic complex manifold etc.), which is quasicoherent as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-module. As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math> is a subsheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math> consisting of the zeroth-order differential operators (multiplications by the sections of structure sheaf), every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-module is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-module. Moreover, the (quasi)coherence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-modules implies the (quasi)coherence of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-module regarded as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-module (but not vice versa).</p> <h3 id="in_terms_of_sheaves_on_the_derham_space">In terms of sheaves on the deRham space</h3> <p>The category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>-modules on a smooth scheme <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> may equivalently be identified with the category of <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaves</a> on its <a class="existingWikiWord" href="/nlab/show/deRham+space">deRham space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dR</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">dR(X)</annotation></semantics></math> (in non-smooth case one needs to work in derived setting, with de Rham stack instead).</p> <p>(<a href="#LurieCrystal">Lurie, above theorem 0.4</a>, <a href="#GaitsgoryRozenblyum11">Gaitsgory-Rozenblyum 11, 2.1.1</a>)</p> <p>Remembering, from this discussion there, that</p> <ul> <li> <p>the deRham space is the decategorification of the <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal path groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>inf</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_{inf}(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a quasicoherent sheaf on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dR</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">dR(X)</annotation></semantics></math> is a generalization of a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a vector bundle with a flat <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> is an <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a> vector bundle on the infinitesimal path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Π</mi> <mi>inf</mi></msup></mrow><annotation encoding="application/x-tex">\Pi^{inf}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> </li> </ul> <p>this shows pretty manifestly how D-modules are “sheaves of modules with flat connection”, as described more below.</p> <h2 id="meaning_and_usage">Meaning and usage</h2> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-modules are useful as a means of applying the methods of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> and <a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">sheaf theory</a> to the study of analytic systems of partial differential equations.</p> <p>Insofar as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math>-module on a <a class="existingWikiWord" href="/nlab/show/ringed+site">ringed site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>O</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, O)</annotation></semantics></math> can be interpreted as a generalization of the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of sections of a vector bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, a D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">-</annotation></semantics></math>module can be interpreted as a generalization of the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of sections of a vector bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> <em>with flat <a class="existingWikiWord" href="/nlab/show/connection">connection</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>. The idea is that the action of the differential operation given by a vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> on a section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of the sheaf (over some patch <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>) is to be thought of as the covariant derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>↦</mo><msub><mo>∇</mo> <mi>v</mi></msub><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma \mapsto \nabla_v \sigma</annotation></semantics></math> with respect to the flat connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>.</p> <p>In fact when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a complex analytic manifold, any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-module which is coherent as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-module is isomorphic to the sheaf of sections of some holomorphic vector bundle with flat connection. Furthermore, the subcategory of nonsingular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-modules coherent as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-modules is equivalent to the category of <a class="existingWikiWord" href="/nlab/show/local+systems">local systems</a>.</p> <h2 id="PositiveCharacteristics">Positive characteristic</h2> <div class="query"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>: it would be nice to have a little more explanation about how not every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-module that is coherent as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math>-module is coherent as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-module. If I understand correctly, this may be the same question as how not every holomorphic vector bundle with flat connection is a local system. Perhaps the answer can be found under <a class="existingWikiWord" href="/nlab/show/local+system">local system</a>? Apparently not. Perhaps the point is that not every flat connection on a holomorphic vector bundle is locally holomorphically trivializable? If so, this is different than how it works in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math> category, which might explain my puzzlement.</p> </div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/variety">variety</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/positive+characteristic">positive characteristic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, the terms “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-coherent coherent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-module” and “vector bundle with flat connection” are not interchangeable, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math> no longer is the <a class="existingWikiWord" href="/nlab/show/enveloping+algebra">enveloping algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mtext>Der</mtext> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>O</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>O</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{Der}_X(O_X,O_X)</annotation></semantics></math>. Indeed, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is smooth over a base field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, the ring of Grothendieck differential operators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math> will not be Noetherian, instead being generated by operators like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><msup><mi>p</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><mo stretchy="false">(</mo><mfrac><mi>d</mi><mi>dt</mi></mfrac><msup><mo stretchy="false">)</mo> <mrow><msup><mi>p</mi> <mi>k</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\frac{1}{(p^k)!} (\frac{d}{dt})^{p^k}</annotation></semantics></math>. Thus an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-coherent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-module will never be coherent over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>.</p> <p>A theorem by Katz states that for smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-coherent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">D_X</annotation></semantics></math>-modules is equivalent to the category with objects sequences <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_0, E_1,\ldots)</annotation></semantics></math> of locally free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-modules together with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>E</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>F</mi> <mo>*</mo></msup><msub><mi>E</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma_i: E_i\rightarrow F^* E_{i+1}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the Frobenius endomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> <a href="#Gieseker75">Gieseker ‘75, Theorem 1.3</a>.</p> <p>In contrast, modules over the ring of <a class="existingWikiWord" href="/nlab/show/crystalline+differential+operator">crystalline differential operators</a> are tautologically <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">O_X</annotation></semantics></math>-modules equipped with an integrable connection. These have a different flavor than in characteristic zero because of the existence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-curvature, or equivalently, because the ring of crystalline differential operators is an Azumaya algebra.</p> <h2 id="properties">Properties</h2> <h3 id="SixOperationsYoga">Six operations yoga</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/six+operations+yoga">six operations yoga</a> for pull-push of (<a class="existingWikiWord" href="/nlab/show/coherent+D-module">coherent</a>, <a class="existingWikiWord" href="/nlab/show/holonomic+D-module">holonomic</a>) D-modules is in (<a href="#Bernstein">Bernstein, around p. 18</a>). This is reviewed for instance in (<a href="#Etingof">Etingof</a>, <a href="#BenZviNadler09">Ben-Zvi & Nadler 09</a>).</p> <p>The most efficient and intuitive way to define the <a class="existingWikiWord" href="/nlab/show/six+operations">six operations</a> on D-modules is to transfer them from <span class="newWikiWord">Ω-modules<a href="/nlab/new/%CE%A9-modules">?</a></span> (i.e., modules over the differential graded algebra of differential forms) using <a class="existingWikiWord" href="/nlab/show/Koszul+duality">Koszul duality</a>. The <a class="existingWikiWord" href="/nlab/show/six+operations">six operations</a> on <span class="newWikiWord">Ω-modules<a href="/nlab/new/%CE%A9-modules">?</a></span> can be defined in the standard way using the fact that differential forms can be pulled back, unlike differential operators. See the article <a class="existingWikiWord" href="/nlab/show/Koszul+duality">Koszul duality</a> for more information.</p> <h3 id="RelationToGeometricRepresentationTheory">Relation to geometric representation theory</h3> <p>For the moment see at <em><a class="existingWikiWord" href="/nlab/show/Harish+Chandra+transform">Harish Chandra transform</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherent+D-module">coherent D-module</a>, <a class="existingWikiWord" href="/nlab/show/holonomic+D-module">holonomic D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+D-module">arithmetic D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+differential+equation">linear differential equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-scheme">D-scheme</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-algebra">D-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemann-Hilbert+correspondence">Riemann-Hilbert correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weyl+algebra">Weyl algebra</a>, <a class="existingWikiWord" href="/nlab/show/regular+differential+operator">regular differential operator</a>, <a class="existingWikiWord" href="/nlab/show/local+system">local system</a>, <a class="existingWikiWord" href="/nlab/show/differential+bimodule">differential bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+connection">Grothendieck connection</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+analysis">algebraic analysis</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hecke+category">Hecke category</a>, <a class="existingWikiWord" href="/nlab/show/Harish+Chandra+transform">Harish Chandra transform</a></p> </li> </ul> <h2 id="references">References</h2> <p>A comprehensive account is in chapter 2 of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Alexander Beilinson</a> and <a class="existingWikiWord" href="/nlab/show/Vladimir+Drinfeld">Vladimir Drinfeld</a>, chapter 2 of <em><a class="existingWikiWord" href="/nlab/show/Chiral+Algebras">Chiral Algebras</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Armand+Borel">Armand Borel</a> et al., <em>Algebraic D-modules</em>, Perspectives in Mathematics, Academic Press, 1987 (<a href="http://www.math.columbia.edu/~scautis/dmodules/boreletal.djvu">djvu</a>)</p> </li> <li> <p>R. Hotta, K. Takeuchi, T. Tanisaki, <em>D-modules, perverse sheaves, and representation theory</em>, Progress in Mathematics <strong>236</strong>, Birkhäuser (<a href="http://www.math.columbia.edu/~scautis/dmodules/hottaetal.pdf">pdf</a>)</p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a> is in</p> <ul> <li id="GaitsgoryRozenblyum11"><a class="existingWikiWord" href="/nlab/show/Dennis+Gaitsgory">Dennis Gaitsgory</a>, <a class="existingWikiWord" href="/nlab/show/Nick+Rozenblyum">Nick Rozenblyum</a>, <em>Crystals and D-modules</em>, Pure and Applied Mathematics Quarterly Volume 10 (2014) Number 1 (<a href="http://arxiv.org/abs/1111.2087">arXiv:1111.2087</a>, <a href="http://www.intlpress.com/site/pub/pages/journals/items/pamq/content/vols/0010/0001/a002/index.html">publisher</a>)</li> </ul> <p>Lecture notes:</p> <ul> <li id="LurieCrystal"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em>Notes on crystals and algebraic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>-modules</em> (2010) [<a class="existingWikiWord" href="/nlab/files/Lurie-NotesOnCrystals.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Severino+C.+Coutinho">Severino C. Coutinho</a>, <em>A primer of algebraic D-modules</em>, London Math. Soc. Stud. Texts <strong>33</strong>, Cambridge University Press (1995) [<a href="https://doi.org/10.1017/CBO9780511623653">doi:10.1017/CBO9780511623653</a>]</p> </li> <li id="Bernstein"> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Bernstein">Joseph Bernstein</a>, <em>Algebraic theory of D-modules</em> (<a class="existingWikiWord" href="/nlab/files/BernsteinDModule.pdf" title="pdf">pdf</a>, <a href="http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.ps">ps</a>, <a href="http://www.math.uchicago.edu/~mitya/langlands/Bernstein/Bernstein-dmod.dvi">dvi</a>)</p> </li> <li> <p>Peter Schneiders’ <a href="http://www.analg.ulg.ac.be/jps/rec/idm.pdf">notes</a>,</p> </li> <li> <p>Dragan Miličić‘s <a href="http://www.math.utah.edu/~milicic/Eprints/dmodules.pdf">notes</a>, , <a href="http://www.math.utah.edu/~milicic/Eprints/book.pdf">Localization and representation theory of reductive Lie groups</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+Ginzburg">Victor Ginzburg</a>‘s 1998 Chicago notes <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/Ginzburg.pdf">pdf</a>; A.</p> </li> <li> <p>Braverman-T. Chmutova, <em>Lectures on algebraic D-modules</em>, <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/Dmod_brav.pdf">pdf</a></p> </li> <li> <p>R. Bezrukavnikov, MIT course notes, <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/bezr_notes.pdf">pdf</a></p> </li> <li> <p>Notes in Gaitsgory’s seminar <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf">pdf</a></p> </li> <li> <p>A. <a class="existingWikiWord" href="/nlab/show/Beilinson">Beĭlinson</a>, J. Bernstein, <em>A proof of Jantzen’s conjectures</em>, I. M. Gelʹfand Seminar, 1–50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, <a href="http://www.math.harvard.edu/~gaitsgde/grad_2009/BB%20-%20Jantzen.pdf">pdf</a></p> </li> </ul> <p>See also</p> <ul> <li id="Saito89"> <p><a class="existingWikiWord" href="/nlab/show/Morihiko+Saito">Morihiko Saito</a>, <em>Induced D-modules and differential complexes</em>, Bull. Soc. Math. France 117 (1989), 361–387, <a href="http://smf4.emath.fr/Publications/Bulletin/117/pdf/smf_bull_117_361-387.pdf">pdf</a></p> </li> <li id="Gieseker75"> <p>D. Gieseker, <em>Flat vector bundles and the fundamental group in non-zero characteristics</em>, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 1, 1–31.</p> </li> <li> <p>J.-E. Björk, <em>Rings of differential operators</em>, North-Holland Math. Library 21. North-Holland Publ. 1979. xvii+374 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">M. Kashiwara</a>, W.Schmid, <em>Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups</em>, Lie Theory and Geometry, in Honor of Bertram Kostant, Progress in Mathematics, Birkhäuser, 1994, pp. 457–488</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M.+Kashiwara">M. Kashiwara</a>, <em>D-modules and representation theory of Lie groups</em>,</p> <p>Annales de l’institut Fourier, 43 no. 5 (1993), p. 1597-1618, <a href="http://aif.cedram.org/item?id=AIF_1993__43_5_1597_0">article</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=95b:22033">MR95b:22033</a></p> </li> <li> <p>P. Maisonobe, C. Sabbah, <em>D-modules cohérents et holonomes</em>, Travaux en cours, Hermann, Paris 1993. (collection of lecture notes)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Donu+Arapura">Donu Arapura</a>, <em>Notes on D-modules and connection with Hodge theory</em>, <a href="http://www.math.purdue.edu/~dvb/preprints/dmod.pdf">pdf</a></p> </li> <li> <p>Nero Budur, <em>On the V-filtration of D-modules</em>, <a href="http://arxiv.org/abs/math/0409123">math.AG/0409123</a>, in “Geometric methods in algebra and number theory” Proc. 2003 conf. Univ. of Miami, edited by F. Bogomolov, Yu. Tschinkel</p> </li> </ul> <p>Review of <a class="existingWikiWord" href="/nlab/show/six+operations+yoga">six operations yoga</a> for D-modules is in</p> <ul> <li id="Etingof"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <em>Formalism of six functors on all (coherent) D-modules</em> (<a href="http://www-math.mit.edu/~etingof/dmodfactsheet.pdf">pdf</a>)</p> </li> <li id="BenZviNadler09"> <p><a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>, <a class="existingWikiWord" href="/nlab/show/David+Nadler">David Nadler</a>, section 3 of <em>The Character Theory of a Complex Group</em> (<a href="http://arxiv.org/abs/0904.1247">arXiv:0904.1247</a>)</p> </li> </ul> <p>See also</p> <ul> <li> <p>A. Beilinson, I. N. Bernstein, <em>A proof of Jantzen conjecture</em>, Adv. in Soviet Math. 16, Part 1 (1993), 1-50. MR95a:22022</p> </li> <li> <p>Secret Blogging Seminar <a href="http://sbseminar.wordpress.com/2007/07/07/musings-on-d-modules/">Musings on D-modules</a>, <a href="http://sbseminar.wordpress.com/2007/07/14/musings-on-d-modules-part-2/">Musings on D-modules, part 2</a></p> </li> <li> <p>The Everything Seminar <a href="http://cornellmath.wordpress.com/2007/09/06/d-module-basics-i/">D-module Basics I</a>, <a href="http://cornellmath.wordpress.com/2007/09/09/d-module-basics-ii/">D-Module Basics II</a>.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 27, 2024 at 13:54:45. 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