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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#base_ring_extension'>Base ring extension</a></li> <li><a href='#morphisms_of_corings_over_varying_bases'>Morphisms of corings over varying bases</a></li> <ul> <li><a href='#comodules_over_a_coring'>Comodules over a coring</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#canonical_sweedler_coring'>Canonical (Sweedler) coring</a></li> <li><a href='#coring_of_a_projective_module'>Coring of a projective module</a></li> <li><a href='#matrix_coring'>Matrix coring</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The notion of coring is a generalization of a <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>-<a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>. While for a coalgebra <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> must be a <strong>commutative</strong> ring (often a <a class="existingWikiWord" href="/nlab/show/field">field</a>), a coring is defined over a general noncommutative ring <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> or even an associative algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Whereas a coalgebra structure is defined on a <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>-module (if <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> is a field, it is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>) – which may be regarded as a central <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>-<a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a> – a coring structure is defined on a general bimodule over a general <a class="existingWikiWord" href="/nlab/show/ring">ring</a>.</p> <h2 id="definition">Definition</h2> <p>An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-coring</strong> is a <a class="existingWikiWord" href="/nlab/show/comonoid">comonoid</a> in the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/bimodule"> bimodules</a> over a fixed (typically noncommutative) unital <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>This generalizes the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/coalgebra"> coalgebras</a> which are defined only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is commutative and where the bimodules in question are <a class="existingWikiWord" href="/nlab/show/central+bimodule">central</a>.</p> <h2 id="base_ring_extension">Base ring extension</h2> <p>More generally, fix a ground commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. Corings will be now over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-algebras. So a coring will mean a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,C)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-algebra and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-coring.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha:A\to B</annotation></semantics></math> be a morphism of rings and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-coring. Then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">B\otimes_A C\otimes_A B</annotation></semantics></math> has an induced structure of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-coring with comultiplication</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mover><mo>⟶</mo><mrow><mi>B</mi><mo>⊗</mo><msub><mi>Δ</mi> <mi>C</mi></msub><mo>⊗</mo><mi>B</mi></mrow></mover><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mover><mo>⟶</mo><mrow><mi>B</mi><mo>⊗</mo><mi>C</mi><mo>⊗</mo><msub><mn>1</mn> <mi>B</mi></msub><mo>⊗</mo><mi>C</mi><mo>⊗</mo><mi>B</mi></mrow></mover><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mo>≅</mo><mo stretchy="false">(</mo><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B\otimes_A C\otimes_A B \stackrel{B\otimes \Delta_C\otimes B}\longrightarrow B\otimes_A C\otimes_A C\otimes_A B \stackrel{B\otimes C\otimes 1_B\otimes C\otimes B}\longrightarrow B\otimes_A C\otimes_A B\otimes_A C\otimes_A B \cong (B\otimes_A C\otimes_A B)\otimes_B (B \otimes_A C\otimes_A B) </annotation></semantics></math></div> <p>and the counit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mover><mo>⟶</mo><mrow><mi>B</mi><mo>⊗</mo><msub><mi>ϵ</mi> <mi>C</mi></msub><mo>⊗</mo><mi>B</mi></mrow></mover><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>A</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mover><mo>⟶</mo><mrow><mi>B</mi><mo>⊗</mo><mi>ϕ</mi><mo>⊗</mo><mi>B</mi></mrow></mover><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mover><mo>⟶</mo><mi>mult</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex"> B\otimes_A C\otimes_A B \stackrel{B\otimes\epsilon_C\otimes B}\longrightarrow B\otimes_A A\otimes_A B \stackrel{B\otimes\phi\otimes B}\longrightarrow B\otimes_A B\otimes_A B \stackrel{mult}\longrightarrow B </annotation></semantics></math></div> <h2 id="morphisms_of_corings_over_varying_bases">Morphisms of corings over varying bases</h2> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,C)\to (B,D)</annotation></semantics></math> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\alpha,\gamma)</annotation></semantics></math> where</p> <p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\alpha : A\to B</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-algebra morphism; by restriction this makes <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> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-bimodule by restriction. 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xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∘</mo><msub><mi>ϵ</mi> <mi>C</mi></msub><mo>=</mo><msub><mi>ϵ</mi> <mi>D</mi></msub><mo>∘</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">\alpha \circ \epsilon_C = \epsilon_D\circ \gamma</annotation></semantics></math></p> <p>(iv) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mo stretchy="false">(</mo><mi>γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>γ</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>Δ</mi> <mi>C</mi></msub><mo>=</mo><msub><mi>Δ</mi> <mi>D</mi></msub><mo>∘</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">p\circ (\gamma\otimes_A\gamma)\circ \Delta_C = \Delta_D\circ \gamma</annotation></semantics></math>, or diagramatically, <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="226.936" height="95.004" viewBox="0 0 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fill-opacity="1"> <use xlink:href="#oU7HNqHD1ieLDPMtcMp4zp8gnMM=-glyph-6-1" x="194.669" y="90.222"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#oU7HNqHD1ieLDPMtcMp4zp8gnMM=-glyph-1-4" x="203.968" y="92.015"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#oU7HNqHD1ieLDPMtcMp4zp8gnMM=-glyph-0-2" x="213.903" y="90.222"></use> </g> </svg> Map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>⊗</mo><mi>c</mi><mo>⊗</mo><mi>b</mi><mo>′</mo><mo>↦</mo><mi>b</mi><mi>γ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mi>b</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">b\otimes c\otimes b'\mapsto b\gamma(c)b'</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">B\otimes_A C\otimes_A B\to D</annotation></semantics></math> from the base extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to <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> is by construction a map of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-bimodules (externally we just use the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-actions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and on <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> which are compatible by action axioms) and the conditions (iii),(iv) express that this map of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-bimodules is a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-corings.</p> <p>Morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\alpha,\gamma)</annotation></semantics></math> factorizes into a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-corings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>A</mi></msub><msub><mi>C</mi> <mi>A</mi></msub><mo>→</mo><msub><mrow></mrow> <mi>A</mi></msub><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>B</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">{}_A C_A\to {}_A B\otimes_A C\otimes_A B_A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>↦</mo><mn>1</mn><mo>⊗</mo><mi>c</mi><mo>⊗</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">c\mapsto 1\otimes c\otimes 1</annotation></semantics></math> into the base extension coring (determined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>), followed by a morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-corings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>B</mi></msub><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>B</mi> <mi>B</mi></msub><mo>→</mo><msub><mrow></mrow> <mi>B</mi></msub><msub><mi>D</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">{}_B B\otimes_A C\otimes_A B_B\to {}_B D_B</annotation></semantics></math>.</p> <p>Every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\alpha,\gamma)</annotation></semantics></math> as above induces an <strong>induction functor</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mi>C</mi></msup><mi>ℳ</mi><mo>→</mo><msup><mrow></mrow> <mi>D</mi></msup><mi>ℳ</mi></mrow><annotation encoding="application/x-tex">{}^C\mathcal{M}\to{}^D\mathcal{M}</annotation></semantics></math>, among the categories of (say, left) comodules, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>↦</mo><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>M</mi></mrow><annotation encoding="application/x-tex">M\mapsto B\otimes_A M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↦</mo><msub><mi>id</mi> <mi>B</mi></msub><mo>⊗</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">f\mapsto id_B\otimes f</annotation></semantics></math> with, which for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-coaction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ρ</mi> <mi>M</mi></msup><mo>:</mo><mi>M</mi><mo>→</mo><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>M</mi><mo>,</mo><mi>m</mi><mo>↦</mo><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><msub><mi>m</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><msub><mi>m</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\rho^M: M\to C\otimes_A M, m\mapsto \sum m_{(-1)}\otimes m_{(0)}</annotation></semantics></math> gives <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>-comodule structure</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>M</mi><mo>→</mo><mi>D</mi><msub><mo>⊗</mo> <mi>B</mi></msub><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>M</mi><mo>≅</mo><mi>D</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>M</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>b</mi><mo>⊗</mo><mi>m</mi><mo>↦</mo><mi>b</mi><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>m</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>m</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> B\otimes_A M \to D\otimes_B B\otimes_A M\cong D\otimes_A M,\,\,\, b\otimes m \mapsto b\gamma(m_{(-1)}) \otimes m_{(0)}. </annotation></semantics></math></div> <p>Similarly, one defines the induction functor for right comodules, and in particular coaction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>′</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mo>→</mo><mi>M</mi><mo>′</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><msub><mo>⊗</mo> <mi>B</mi></msub><mi>D</mi><mo>≅</mo><mi>M</mi><mo>′</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mi>D</mi></mrow><annotation encoding="application/x-tex">M'\otimes_A B\to M'\otimes_A B\otimes_B D\cong M'\otimes_A D</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>′</mo><mo>⊗</mo><mi>b</mi><mo>↦</mo><msub><mi>m</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>m</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">m'\otimes b\mapsto m_{(0)}\otimes\gamma(m_{(1)})b</annotation></semantics></math>. In particular, one can start with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\Delta_C</annotation></semantics></math> and induce <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>B</mi><mo>→</mo><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>D</mi></mrow><annotation encoding="application/x-tex">C\otimes_A B\to C\otimes_A D</annotation></semantics></math> via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>⊗</mo><mi>b</mi><mo>↦</mo><msub><mi>c</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mi>γ</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">c\otimes b\mapsto c_{(1)}\otimes \gamma(c_{(2)})b</annotation></semantics></math>.</p> <p>Under the assumption that the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\alpha,\gamma)</annotation></semantics></math> is a pure morphism of corings (left hand version), which is for example satisfied automatically when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is flat as a right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module, the induction functor has a right adjoint, the coinduction functor. It is given by a cotensor product which is a comodule under the purity condition. 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x="377.331" y="19.987"></use> </g> </svg> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>D</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>N</mi><mo>→</mo><mi>D</mi><msub><mo>⊗</mo> <mi>B</mi></msub><mi>N</mi></mrow><annotation encoding="application/x-tex">p:D\otimes_A N\to D\otimes_B N</annotation></semantics></math> is the canonical projection.</p> <h3 id="comodules_over_a_coring">Comodules over a coring</h3> <p>Given an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-coring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow></mrow> <mi>A</mi></msub><msub><mi>C</mi> <mi>A</mi></msub><mo>,</mo><mi>Δ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C = ({}_A C_A, \Delta,\epsilon)</annotation></semantics></math>, a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-comodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module with a map of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mrow></mrow> <mi>A</mi></msub><msub><mi>C</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\nu: M\to M\otimes_A {}_A C_A</annotation></semantics></math> (left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-coaction), such that the composition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mover><mo>→</mo><mi>ρ</mi></mover><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><mover><mo>→</mo><mrow><mi>ρ</mi><mo>⊗</mo><mi>C</mi></mrow></mover><mo stretchy="false">(</mo><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">M\stackrel{\rho}\to M\otimes_A C \stackrel{\rho\otimes C}\to (M\otimes_A C)\otimes_A C</annotation></semantics></math> after rebracketing becomes identical to the composition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mover><mo>→</mo><mi>ρ</mi></mover><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><mover><mo>→</mo><mrow><mi>M</mi><mo>⊗</mo><msub><mi>Δ</mi> <mi>C</mi></msub></mrow></mover><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>C</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M\stackrel{\rho}\to M\otimes_A C \stackrel{M\otimes \Delta_C}\to M\otimes_A (C \otimes_A C)</annotation></semantics></math> and the counitality axiom holds.</p> <p>A morphism of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-comodules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><msup><mi>ρ</mi> <mi>M</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><msup><mi>ρ</mi> <mi>N</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f:(M,\rho^M)\to(N,\rho^N)</annotation></semantics></math> is a morphism of underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">f:M\to N</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msup><mi>ρ</mi> <mi>M</mi></msup><mo>=</mo><msup><mi>ρ</mi> <mi>N</mi></msup><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(M\otimes_A f)\circ\rho^M = \rho^N\circ f</annotation></semantics></math>. Thus there is a category of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-comodules equipped with a forgetful functor to the category of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules.</p> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>M</mi><mo>↦</mo><mi>M</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">F: M\mapsto M\otimes_A C</annotation></semantics></math> is canonically a comonad on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>A</mi></msub><mi>Mod</mi></mrow><annotation encoding="application/x-tex">{}_A Mod</annotation></semantics></math> with comultiplication <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mi>F</mi></msup><mo>=</mo><mi>Id</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>Δ</mi> <mi>C</mi></msub><mo>:</mo><mi>F</mi><mo>→</mo><mi>F</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">\Delta^F = Id\otimes_A\Delta_C : F\to F F</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a> of the comonad <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 isomorphic to the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-comodules.</p> <p>Analogously, one considers right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-comodules as right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-comodules with right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-coactions.</p> <h2 id="examples">Examples</h2> <h3 id="canonical_sweedler_coring">Canonical (Sweedler) coring</h3> <p>The classical example of a coring is the canonical or <a class="existingWikiWord" href="/nlab/show/Sweedler+coring">Sweedler coring</a> corresponding to an extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>↪</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">R\hookrightarrow S</annotation></semantics></math> of unital rings. The category of <a class="existingWikiWord" href="/nlab/show/descent">descent</a> data for this ring extension is equivalent to the category of <a class="existingWikiWord" href="/nlab/show/comodule"> comodules</a> over the canonical coring.</p> <p>Corings are in general useful for the treatment of <a class="existingWikiWord" href="/nlab/show/descent+in+noncommutative+algebraic+geometry">descent in noncommutative algebraic geometry</a>.</p> <h3 id="coring_of_a_projective_module">Coring of a projective module</h3> <p>Suppose we are given ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>,</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">R,S</annotation></semantics></math> and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule, finitely generated as right projective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>=</mo><msub><mrow></mrow> <mi>S</mi></msub><msub><mi>M</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">M = {}_S M_R</annotation></semantics></math>. Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mo>*</mo></msup><mo>=</mo><msub><mi>Hom</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M^* = Hom_R(M,R)</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-bimodule. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> be given as a direct summand of a free module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><msub><mo>⊕</mo> <mi>i</mi></msub><msub><mi>R</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">F = \oplus_i R_i</annotation></semantics></math>; the decomposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><mi>M</mi><mo>⊕</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">F = M\oplus L</annotation></semantics></math> induces projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>F</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">p:F\to M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>r</mi> <mi>i</mi></msub><mo>↦</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msub><mi>r</mi> <mi>i</mi></msub><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum_i r_i\mapsto \sum_i r_i x_i</annotation></semantics></math>, a section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>M</mi><mo>↪</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">s: M\hookrightarrow F</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msubsup><mi>x</mi> <mi>i</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mo>⊕</mo> <mi>i</mi></msub><msub><mi>R</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">s(m) = \sum_i x_i^*(m)\in\oplus_i R_i</annotation></semantics></math>, and <a class="existingWikiWord" href="/nlab/show/dual+basis">dual basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">x_1,\ldots,x_n\in M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>x</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>,</mo><mi>…</mi><mo>,</mo><msubsup><mi>x</mi> <mi>n</mi> <mo>*</mo></msubsup><mo>∈</mo><msub><mi>Hom</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_1^*,\ldots,x_n^*\in Hom_R(M,R)</annotation></semantics></math> characterized by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></msub><msubsup><mi>x</mi> <mi>i</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m = \sum_i x_i^*(m) x_i</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">m\in M</annotation></semantics></math>. Then define an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-coring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-bimodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>S</mi></msub><mi>M</mi></mrow><annotation encoding="application/x-tex">M^*\otimes_S M</annotation></semantics></math> with comultiplication</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo>:</mo><msup><mi>M</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>S</mi></msub><mi>M</mi><mo>→</mo><mo stretchy="false">(</mo><msup><mi>M</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>S</mi></msub><mi>M</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>R</mi></msub><mo stretchy="false">(</mo><msup><mi>M</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>S</mi></msub><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta : M^*\otimes_S M\to (M^*\otimes_S M)\otimes_R(M^*\otimes_S M) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>f</mi><mo>⊗</mo><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mi>f</mi><mo>⊗</mo><msubsup><mi>x</mi> <mi>i</mi> <mo>*</mo></msubsup><mo>⊗</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>⊗</mo><mi>m</mi></mrow><annotation encoding="application/x-tex"> \Delta(f\otimes m) = \sum_{i=1}^n f\otimes x_i^*\otimes x_i\otimes m </annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>:</mo><msup><mi>M</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>S</mi></msub><mi>M</mi><mo>,</mo><mi>f</mi><mo>⊗</mo><mi>m</mi><mo>↦</mo><mi>f</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\epsilon:M^*\otimes_S M, f\otimes m\mapsto f(m)\in R</annotation></semantics></math>.</p> <h3 id="matrix_coring">Matrix coring</h3> <p>Another major class of examples are the so-called <span class="newWikiWord"> matrix corings<a href="/nlab/new/matrix+coring">?</a></span>.</p> <h2 id="references">References</h2> <p>The notion of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-coring is introduced by M. Sweedler and recently lived through a renaissance in works of <a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, R. Wisbauer, <a class="existingWikiWord" href="/nlab/show/G.+B%C3%B6hm">G. Böhm</a>, L. Kaoutit, Gómez-Torrecillas, S. Caenepeel, J. Y. Abuhlail, J. Vercruysse and others, including the creation of Galois theory for corings. Some prefer to speak about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cocategory">cocategories</a>.</p> <p>There is already a monograph:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, R. Wisbauer, <em>Corings and comodules</em>, London Math. Soc. Lec. Note Series <strong>309</strong>, Cambridge 2003.</li> </ul> <p>Special topics:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, <em>Descent cohomology and corings</em>, Comm. Algebra 36:1894-1900, 2008, <a href="http://arxiv.org/abs/math.RA/0601491">arxiv:math.RA/0601491</a></p> </li> <li> <p>L. El Kaoutit, J. Gomez-Torrecillas, <em>On the set of grouplikes of a coring</em>, <a href="http://arxiv.org/abs/0901.4291">arxiv/0901.4291</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, <em>Flat connections and (co)modules</em>, [in:] New Techniques in Hopf Algebras and Graded Ring Theory, S Caenepeel and F Van Oystaeyen (eds), Universa Press, Wetteren, 2007 pp. 35-52 <a href="http://arxiv.org/abs/math.QA/0608170">arxiv:math.QA/0608170</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, <em>The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties</em>, Algebras and Representation Theory <strong>5</strong> (2002) 389-410, <a href="http://arxiv.org/abs/math.QA/0002105">math.QA/0002105</a></p> </li> <li> <p>Lars Kadison, <em>Depth two and Galois coring</em>, <a href="http://arxiv.org/abs/math.RA/0408155">math.RA/0408155</a></p> </li> <li id="BergmanHausknecht96"> <p><a class="existingWikiWord" href="/nlab/show/George+M.+Bergman">George M. Bergman</a>, Adam O. Hausknecht, <em>Cogroups and co-rings in categories of associative rings</em>, A.M.S. Math. Surveys and Monographs <strong>45</strong> (1996) [ISBN 0-8218-0495-2, <a href="https://bookstore.ams.org/surv-45">ams:surv-45</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=97k:16001">MR 97k:16001</a>, <a href="http://math.berkeley.edu/~gbergman/papers/updates/coalg.html">errata and updates</a>]</p> <blockquote> <p>(discussion <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/T.+Brzezi%C5%84ski">T. Brzeziński</a>, L. Kadison, <a class="existingWikiWord" href="/nlab/show/R.+Wisbauer">R. Wisbauer</a>, <em>On coseparable and biseparable corings</em>, Hopf algebras in noncommutative geometry and physics, 71–87, Lecture Notes in Pure and Appl. Math., 239, Dekker, New York, 2005.</p> </li> <li> <p>T. Brzeziński, L. El Kaoutit, J. Gómez-Torrecillas, <em>The bicategories of corings</em>, J. Pure & Appl. Alg. <strong>205</strong>:3 (2006) 510-541 <a href="https://doi.org/10.1016/j.jpaa.2005.07.013">doi:10.1016/j.jpaa.2005.07.013</a> <a href="https://arxiv.org/abs/math/0408042">math.RA/0408042</a></p> </li> </ul> <p>There is a generalization of corings:</p> <ul> <li>Jawad Y. Abuhlail, <em>Semicorings and semicomodules</em>, <a href="http://arxiv.org/abs/1303.3924">arxiv/1303.3924</a></li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/algebra">algebra</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on January 13, 2024 at 05:25:51. 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