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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+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> <ul> <li><a href='#the_chain_complex_for_cyclic_homology'>The chain complex for cyclic homology</a></li> <li><a href='#ordinary_cohomology_of__and_cyclic_homology_of_'>Ordinary cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}X/S^1</annotation></semantics></math> and cyclic homology 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></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#lodayquillentsygan_theorem'>Loday-Quillen-Tsygan theorem</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a> may be understood as the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>s (as described there). These free loop space objects are canonically equipped with a <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>-<a class="existingWikiWord" href="/nlab/show/action">action</a> that rotates the loops. <em>Cyclic homology</em> is the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> of free loop space objects.</p> <p>Like <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a> is an <em>additive invariant</em> of <a class="existingWikiWord" href="/nlab/show/dg-categories">dg-categories</a> or <a class="existingWikiWord" href="/nlab/show/stable+infinity-categories">stable infinity-categories</a>, in the sense of <a class="existingWikiWord" href="/nlab/show/noncommutative+motives">noncommutative motives</a>. It also admits a <a class="existingWikiWord" href="/nlab/show/Dennis+trace+map">Dennis trace map</a> from <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a>, and has been successful in allowing computations of the latter.</p> <p>There are several definitions for the cyclic homology of an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</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> (over a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</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/Alain+Connes">Alain Connes</a> originally defined cyclic homology over <a class="existingWikiWord" href="/nlab/show/fields">fields</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>, as the <a class="existingWikiWord" href="/nlab/show/homology">homology</a> groups of a cyclic variant of the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> computing <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>. <a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a> and <a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a> gave a definition via a certain <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> (for arbitrary commutative rings). <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Connes</a> gave another definition by associating to <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 <a class="existingWikiWord" href="/nlab/show/cyclic+object">cyclic</a> <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>♯</mo></msup></mrow><annotation encoding="application/x-tex">A^\sharp</annotation></semantics></math>, and showing that the cyclic homology 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> may be computed as via <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a>-groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>♯</mo></msup><mo>,</mo><msup><mi>k</mi> <mo>♯</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^*(A^\sharp, k^\sharp)</annotation></semantics></math>. A fourth definition was given by <a class="existingWikiWord" href="/nlab/show/Christian+Kassel">Christian Kassel</a>, who showed that the cyclic homology groups may be computed as the homology groups of a certain <a class="existingWikiWord" href="/nlab/show/mixed+complex">mixed complex</a> associated to <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>Following <a class="existingWikiWord" href="/nlab/show/Alexandre+Grothendieck">Alexandre Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a> gave a definition of <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a> (and <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>) for <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a>, using <a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a>. On the other hand, the definition of <a class="existingWikiWord" href="/nlab/show/Christian+Kassel">Christian Kassel</a> via <a class="existingWikiWord" href="/nlab/show/mixed+complexes">mixed complexes</a> was extended by <a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a> to <a class="existingWikiWord" href="/nlab/show/linear+categories">linear categories</a> and <a class="existingWikiWord" href="/nlab/show/dg-categories">dg-categories</a>, and he showed that the cyclic homology of the <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a> of <a class="existingWikiWord" href="/nlab/show/perfect+complexes">perfect complexes</a> on a (nice) <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> <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> coincides with the cyclic homology 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> in the sense of <a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Weibel</a>.</p> <p>There are closely related variants called <span class="newWikiWord">periodic cyclic homology<a href="/nlab/new/periodic+cyclic+homology">?</a></span> and <span class="newWikiWord">negative cyclic homology<a href="/nlab/new/negative+cyclic+homology">?</a></span>. There is a version for <a class="existingWikiWord" href="/nlab/show/ring+spectra">ring spectra</a> called <a class="existingWikiWord" href="/nlab/show/topological+cyclic+homology">topological cyclic homology</a>.</p> <h2 id="definition">Definition</h2> <h3 id="the_chain_complex_for_cyclic_homology">The chain complex for cyclic homology</h3> <p>Let <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> be an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</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> of characteristic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet(A,A)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a> <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> 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> with coefficients in <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>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda : C_n(A,A) \to C_n(A,A)</annotation></semantics></math> be the <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>-linear map that cyclically permutes the elements and introduces a sign:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>:</mo><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo stretchy="false">(</mo><msub><mi>a</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lambda : (a_0, a_1, \cdots, a_{n-1}, a_n) \mapsto (-1)^n (a_n, a_0 , \cdots, a_{n-1}) \,. </annotation></semantics></math></div> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>The <strong>cyclic homology complex</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mo>•</mo> <mi>λ</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\lambda_\bullet(A)</annotation></semantics></math> 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> is the quotient of the Hochschild homology complex 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> by cyclic permutations:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>C</mi> <mo>•</mo> <mi>λ</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>Id</mi><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_\bullet^\lambda(A) := C_\bullet(A,A)/im(Id-\lambda) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/homology">homology</a> of the cyclic complex, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>HC</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msubsup><mi>C</mi> <mo>•</mo> <mi>λ</mi></msubsup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> HC_n(A) := H_n( C_\bullet^\lambda(A) ) </annotation></semantics></math></div> <p>is called the <strong>cyclic homology</strong> 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>.</p> </div> <div class="un_defn"> <h6 id="definition_3">Definition</h6> <p>The <strong>cyclic cohomology</strong> groups 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> Are the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> groups of the dual <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>HC</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">HC^n(A)</annotation></semantics></math>.</p> </div> <div class="un_defn"> <h6 id="definition_4">Definition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">I\subset A</annotation></semantics></math> is an ideal, then the <strong>relative cyclic homology</strong> groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>HC</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">HC_n(A,I)</annotation></semantics></math> are the homology groups of the complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet(A,I) = ker(C_\bullet(A)\to C_\bullet(A/I))</annotation></semantics></math>.</p> </div> <h3 id="ordinary_cohomology_of__and_cyclic_homology_of_">Ordinary cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}X/S^1</annotation></semantics></math> and cyclic homology 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></h3> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">H^\bullet</annotation></semantics></math> of its free loop space is the <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>HH</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">HH_\bullet</annotation></semantics></math> of its <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular chains</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\bullet(X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>HH</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^\bullet(\mathcal{L}X) \simeq HH_\bullet( C^\bullet(X) ) \,. </annotation></semantics></math></div> <p>Moreover the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-equivariant cohomology of the loop space, hence the ordinary cohomology of the <a class="existingWikiWord" href="/nlab/show/cyclic+loop+space">cyclic loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mi>X</mi><msup><mo stretchy="false">/</mo> <mi>h</mi></msup><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathcal{L}X/^h S^1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>HC</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">HC_\bullet</annotation></semantics></math> of the singular chains:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><msup><mo stretchy="false">/</mo> <mi>h</mi></msup><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>HC</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( C^\bullet(X) ) </annotation></semantics></math></div> <p>(<a href="#Jones87">Jones 87, Thm. A</a>, review in <a href="#Loday92">Loday 92, Cor. 7.3.14</a>, <a href="#Loday11">Loday 11, Sec 4</a>)</p> <p>If the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> are <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational</a>, and <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 of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> then this may be computed by the <em><a class="existingWikiWord" href="/nlab/show/Sullivan+model+for+free+loop+spaces">Sullivan model for free loop spaces</a></em>, see there the section on <em><a href="Sullivan+model+of+free+loop+space#RelationToHochschildHomology">Relation to Hochschild homology</a></em>.</p> <p>In the special case that the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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> carries the structure of a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, then the singular cochains 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> are equivalent to the <a class="existingWikiWord" href="/nlab/show/dgc-algebra">dgc-algebra</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> (the <a class="existingWikiWord" href="/nlab/show/de+Rham+algebra">de Rham algebra</a>) and hence in this case the statement becomes that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>HH</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,. </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℒ</mi><mi>X</mi><msup><mo stretchy="false">/</mo> <mi>h</mi></msup><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>HC</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( \Omega^\bullet(X) ) \,. </annotation></semantics></math></div> <p>This is known as <em><a class="existingWikiWord" href="/nlab/show/Jones%27+theorem">Jones' theorem</a></em> (<a href="#Jones87">Jones 87</a>)</p> <p>An <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory">infinity-category theoretic</a> proof of this fact is indicated at <em><a href="Hochschild+cohomology#JonesTheorem">Hochschild cohomology – Jones’ theorem</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="lodayquillentsygan_theorem">Loday-Quillen-Tsygan theorem</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/Loday-Quillen-Tsygan+theorem">Loday-Quillen-Tsygan theorem</a></em> (<a href="#LodayQuillen84">Loday-Quillen 84</a>, <a href="#Tsygan83">Tsygan 83</a>) states that for any <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</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> in <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>, the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+homology">Lie algebra homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_\bullet(\mathfrak{gl}(A))</annotation></semantics></math> of the infinite <a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{gl}(A)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <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, up to a degree shift, the <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>HC</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\wedge(HC_{\bullet - 1}(A))</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>HC</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">HC_{\bullet - 1}(A)</annotation></semantics></math> 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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>HC</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_\bullet(\mathfrak{gl}(A)) \;\simeq\; \wedge( HC_{\bullet - 1}(A) ) </annotation></semantics></math></div> <p>(see e.g <a href="#Loday07">Loday 07, theorem 1.1</a>).</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+category">cyclic category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+object">cyclic object</a></p> <p><a class="existingWikiWord" href="/nlab/show/cyclic+set">cyclic set</a><a class="existingWikiWord" href="/nlab/show/cyclic+space">cyclic space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+loop+space">cyclic loop space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cyclic+homology">topological cyclic homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>, <a class="existingWikiWord" href="/nlab/show/topological+Hochschild+homology">topological Hochschild homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+free+loop+space">Sullivan model of free loop space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dihedral+homology">dihedral homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Loday-Quillen-Tsygan+theorem">Loday-Quillen-Tsygan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+K-theory">additive K-theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>Foundational articles:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A.+Connes">A. Connes</a>, <em>Noncommutative differential geometry, Part I, the Chern character in <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>-homology</em>, Preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1982; <em>Part II, de Rham homology and noncommutative algebra</em>, Preprint, IHÉS 1983; <em>Cohomologie cyclique et foncteurs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Ext^n</annotation></semantics></math></em>, C. R. Acad. Sci. Paris <strong>296</strong>, (1983), pp. 953–958, <a href="http://www.ams.org/mathscinet-getitem?mr=777584">MR86d:18007</a></p> </li> <li id="Connes83"> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <em>Cohomologie cyclique et foncteurs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Ext^n</annotation></semantics></math></em>, C.R.A.S. <strong>296</strong> (1983), Série I, 953-958 (<a href="https://alainconnes.org/wp-content/uploads/n83.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Connes_CohomologieCyclique.pdf" title="pdf">pdf</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B.+L.+Tsygan">B. L. Tsygan</a>, <em>The homology of matrix Lie algebras over rings and the Hochschild homology</em>, Uspekhi Mat. Nauk, 38:2(230) (1983), 217–218.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>Cyclic homology and the Lie algebra homology of matrices</em>, Comment. Math. Helvetici 59 (1984) 565-591 (<a href="https://link.springer.com/book/10.1007/978-3-662-21739-9">doi:10.1007/978-3-662-21739-9</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Christian+Kassel">Christian Kassel</a>, <em>Cyclic homology, comodules and mixed complexes</em>, J. Alg. 107 (1987), 195–216 (<a href="https://doi.org/10.1016/0021-8693(87)90086-X">doi:10.1016/0021-8693(87)90086-X</a>)</p> </li> </ul> <p>Monographs:</p> <ul> <li id="Loday98"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <em>Cyclic homology</em>, Grundlehren Math.Wiss. <strong>301</strong>, Springer (1998)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <em>Noncommutative geometry</em>, Acad. Press 1994, 661 p. <a href="http://www.alainconnes.org/docs/book94bigpdf.pdf">PDF</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>Homologie cyclique et K-théorie</em>, Astérique <strong>149</strong>, Société Mathématique de France (1987).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ib+Madsen">Ib Madsen</a>, <em>Algebraic K-theory and traces</em>, Current Developments in Mathematics, 1995.</p> </li> </ul> <p>Lecture notes:</p> <ul> <li id="Loday07"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <em>Cyclic Homology Theory, Part II</em>, notes taken by Pawel Witkowsk (2007) (<a href="https://www.impan.pl/swiat-matematyki/notatki-z-wyklado~/loday_cht_2.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D.+Kaledin">D. Kaledin</a>, <em>Tokyo lectures “Homological methods in non-commutative geometry”</em>, (<a href="http://imperium.lenin.ru/~kaledin/tokyo/final.pdf">pdf</a>, <a href="http://imperium.lenin.ru/~kaledin/tokyo/final.tex">TeX</a>); and related but different <a href="http://imperium.lenin.ru/~kaledin/seoul">Seoul lectures</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Masoud+Khalkhali">Masoud Khalkhali</a>, <em>A short survey of cyclic cohomology</em>, <a href="http://arxiv.org/abs/1008.1212">arxiv/1008.1212</a></p> </li> </ul> <p>Some modern treatments:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a>, <em>On the cyclic homology of ringed spaces and schemes</em>, Doc. Math. J. DMV 3 (1998), 231-259, <a href="http://www.math.jussieu.fr/~keller/publ/KellerHCSchemes.pdf">pdf</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a>, <em>On the cyclic homology of exact categories</em>, Journal of Pure and Applied Algebra 136 (1999), 1-56, <a href="http://www.math.jussieu.fr/~keller/publ/cyex.pdf">pdf</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a>, <em>Invariance and Localization for Cyclic Homology of DG algebras</em>, Journal of Pure and Applied Algebra, 123 (1998), 223-273, <a href="http://www.math.jussieu.fr/~keller/publ/ilc.pdf">pdf</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Weibel">Charles Weibel</a>, <em>Cyclic homology for schemes</em>, Proceedings of the AMS, 124 (1996), 1655-1662, <a href="http://www.math.uiuc.edu/K-theory/0043/">web</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D.+Kaledin">D. Kaledin</a>, <em>Cyclic homology with coefficients</em>, <a href="http://arxiv.org/abs/math.KT/0702068">math.KT/0702068</a>, to appear in Yu. Manin’s 70th anniversary volume.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E.+Getzler">E. Getzler</a>, <a class="existingWikiWord" href="/nlab/show/M.+Kapranov">M. Kapranov</a>, <em>Cyclic operads and cyclic homology</em>, in: “Geometry, Topology and Physics for R. Bott”,</p> <p>ed. S.-T. Yau, p. 167-201, International Press, Cambridge MA, 1995, <a href="http://www.math.northwestern.edu/~getzler/Papers/cyclic.pdf">pdf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Teimuraz+Pirashvili">Teimuraz Pirashvili</a>, <a class="existingWikiWord" href="/nlab/show/Birgit+Richter">Birgit Richter</a>, <em>Hochschild and cyclic homology via functor homology</em>, K-Theory <strong>25</strong> (2002), no. 1, 39–49, <a href="http://www.ams.org/mathscinet-getitem?mr=1899698">MR2003c:16011</a>, <a href="http://dx.doi.org/10.1023/A:1015064621329">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jolanta+S%C5%82omi%C5%84ska">Jolanta Słomińska</a>, <em>Decompositions of the category of noncommutative sets and Hochschild and cyclic homology</em>, Cent. Eur. J. Math. <strong>1</strong> (2003), no. 3, 327–331 (<a href="http://dx.doi.org/10.2478/BF02475213">doi:10.2478/BF02475213</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=1992895">MR2004f:16011</a>)</p> </li> </ul> <p>Some influential original references from 1980s:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Boris+Tsygan">Boris Tsygan</a>, <a class="existingWikiWord" href="/nlab/show/Boris+Feigin">Boris Feigin</a>, <em><a class="existingWikiWord" href="/nlab/show/Additive+K-theory">Additive K-theory</a></em>, in LNM 1289 (1987), edited by Yu. I. Manin, pp. 67–209, seminar 1984-1986 in Moscow), <a href="http://www.ams.org/mathscinet-getitem?mr=923136">MR89a:18017</a>; <p><em>Аддитивная K-теория и кристальные когомологии</em>, Функц. анализ и его прил., 19:2 (1985), 52–-62, <a href="http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1358&what=fullt&option_lang=rus">pdf</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=800920">MR88e:18008</a>; Engl. transl. in B. L. Feĭgin, B. L. Tsygan, <em>Additive <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>-theory and crystalline cohomology</em>, Functional Analysis and Its Applications, 1985, 19:2, 124–132.</p> </li> <li><a class="existingWikiWord" href="/nlab/show/T.+Goodwillie">T. Goodwillie</a>, <em>Cyclic homology, derivations, and the free loopspace</em>, Topology <strong>24</strong> (1985), <p>no. 2, 187–215, <a href="http://www.ams.org/mathscinet-getitem?mr=793184">MR87c:18009</a>, <a href="http://dx.doi.org/10.1016/0040-9383(85)90055-2">doi</a></p> </li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/cyclic+loop+spaces">cyclic loop spaces</a>:</p> <ul> <li id="Jones87"><a class="existingWikiWord" href="/nlab/show/John+D.S.+Jones">John D.S. Jones</a>, <em>Cyclic homology and equivariant homology</em>, Invent. Math. <strong>87</strong>, 403-423 (1987) (<a href="https://math.berkeley.edu/~nadler/jones.pdf">pdf</a>, <a href="https://doi.org/10.1007/BF01389424">doi:10.1007/BF01389424</a>)</li> </ul> <p>reviewed in:</p> <ul> <li id="Loday92"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <em>Cyclic Spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-Equivariant Homology</em> (<a href="https://link.springer.com/chapter/10.1007/978-3-662-21739-9_7">doi:10.1007/978-3-662-21739-9_7</a>)</p> <p>Chapter 7 in: <em>Cyclic Homology</em>, Grundlehren <strong>301</strong>, Springer 1992 (<a href="https://link.springer.com/book/10.1007/978-3-662-21739-9">doi:10.1007/978-3-662-21739-9</a>)</p> </li> <li id="Loday11"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, Section 4 of: <em>Free loop space and homology</em>, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): <em>Free Loop Spaces in Geometry and Topology</em>, IRMA Lectures in Mathematics and Theoretical Physics <strong>24</strong>, EMS 2015 (<a href="https://arxiv.org/abs/1110.0405">arXiv:1110.0405</a>, <a href="https://bookstore.ams.org/emsilmtp-24/">ISBN:978-3-03719-153-8</a>)</p> </li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of <a class="existingWikiWord" href="/nlab/show/cyclic+loop+spaces">cyclic loop spaces</a> (“string cohomology”) computed directly (instead of via the isomorphism to cyclic homology):</p> <ul> <li id="BokstedtOttosen05"><a class="existingWikiWord" href="/nlab/show/Marcel+B%C3%B6kstedt">Marcel Bökstedt</a>, <a class="existingWikiWord" href="/nlab/show/Iver+Ottosen">Iver Ottosen</a>, <em>A spectral sequence for string cohomology</em>, Topology Volume 44, Issue 6, November 2005, Pages 1181-1212 (<a href="https://arxiv.org/abs/math/0411571">arXiv:math/0411571</a>, <a href="https://doi.org/10.1016/j.top.2005.04.006">doi:10.1016/j.top.2005.04.006</a>)</li> </ul> <p>and specifically for <a class="existingWikiWord" href="/nlab/show/complex+projective+spaces">complex projective spaces</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marcel+B%C3%B6kstedt">Marcel Bökstedt</a>, <a class="existingWikiWord" href="/nlab/show/Iver+Ottosen">Iver Ottosen</a>, <em>String cohomology groups of complex projective spaces</em>, Algebr. Geom. Topol. 7(4): 2165-2238 (2007). (<a href="https://arxiv.org/abs/math/0605754">arXiv:math/0605754</a>, <a href="https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-7/issue-4/String-cohomology-groups-of-complex-projective-spaces/10.2140/agt.2007.7.2165.full">doi:10.2140/agt.2007.7.2165</a>)</li> </ul> <p>On the cyclic homology of the <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebra">groupoid convolution algebra</a> of <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+groupoid">étale</a> <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a>, hence of <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Luc+Brylinski">Jean-Luc Brylinski</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Nistor">Victor Nistor</a>, <em>Cyclic cohomology of étale groupoids</em>, K-theory 8.4 (1994): 341-365 (<a href="https://www.researchgate.net/profile/Victor-Nistor/publication/226520448_Cyclic_cohomology_of_etale_groupoids/links/5509d9d30cf20ed529e227b0/Cyclic-cohomology-of-etale-groupoids.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BrylinskiNistor_CyclicCohomology.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marius+Crainic">Marius Crainic</a>, <em>Cyclic Cohomology of Étale Groupoids; The General Case</em>, K-Theory 17: 319–362, 1999 (<a href="https://arxiv.org/abs/funct-an/9712001">arXiv:funct-an/9712001</a>, <a href="https://access.portico.org/Portico/rest/au/getAUContent?auId=ark:/27927/pgg1zfpvw2x&fileName=pgg1zfprgx2.pdf&viewType=PDF">published pdf</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Loday-Quillen-Tsygan+theorem">Loday-Quillen-Tsygan theorem</a> is originally due, independently, to</p> <ul> <li id="LodayQuillen84"><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>Cyclic homology and the Lie algebra homology of matrices</em> Comment. Math. Helv., 59(4):569–591, 1984 (<a href="https://doi.org/10.1007/BF02566367">doi:10.1007/BF02566367</a>)</li> </ul> <p>and</p> <ul> <li id="Tsygan83"><a class="existingWikiWord" href="/nlab/show/Boris+Tsygan">Boris Tsygan</a>, <em>Homology of matrix algebras over rings and the Hochschild homology</em>, Uspeki Math. Nauk., 38:217–218, 1983.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 8, 2024 at 12:23:14. 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