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class="existingWikiWord" 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> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <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> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#statement'>Statement</a></li> <ul> <li><a href='#InCohomology'>For ordinary cohomology</a></li> <ul> <li><a href='#in_terms_of_homology'>In terms of homology</a></li> <li><a href='#UCTForOrdinaryCohomologyInTermsOfCohomology'>In terms of cohomology</a></li> </ul> <li><a href='#InHomology'>For ordinary homology</a></li> <li><a href='#ForGeneralizedCohomology'>For generalised cohomology theories</a></li> <ul> <li><a href='#a_special_case'>A Special Case</a></li> <li><a href='#freeness_and_flatness'>Freeness and Flatness</a></li> </ul> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#InTopology'>For singular cohomology</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#for_ordinary_cohomology_2'>For ordinary (co)homology</a></li> <li><a href='#ForGeneralizedCoHomology'>For generalized (co)homology</a></li> <li><a href='#ReferencesForKK'>For KK-theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>universal coefficient theorem</em> states how <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a>/<a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> determines <a class="existingWikiWord" href="/nlab/show/homology">homology</a>/<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> with arbitrary <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> (of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>) and <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> a <a class="existingWikiWord" href="/nlab/show/field">field</a> (the <em>coefficient field</em>), the <a class="existingWikiWord" href="/nlab/show/homology+group">homology group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_p(C,F)</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(C,F)</annotation></semantics></math> are indeed related by <a class="existingWikiWord" href="/nlab/show/dualization">dualization</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^p(C,F) \simeq Hom_F(H_p(C,F), F)</annotation></semantics></math>. If the coefficients are not a <a class="existingWikiWord" href="/nlab/show/field">field</a> but an arbitrary <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, then this relationship receives a correction by an <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a>-group. This is discussed below in <em><a href="#InCohomology">For ordinary cohomology</a></em>.</p> <p>Dually, again if <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 a field then there is an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>F</mi><mo>≃</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>⊗</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_n(C) \otimes F \simeq H_n(C \otimes F)</annotation></semantics></math> and for more general <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> this is corrected by a <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>-group. This is discussed below in <em><a href="#InHomology">For ordinary homology</a></em>.</p> <p>More generally, under suitable conditions there are universal coefficient theorems that relate <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> to the dual of <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a>. This is discussed below in <em><a href="#ForGeneralizedCohomology">For generalized cohomology</a></em>.</p> <p>There is also a version of the theorem for Kasparov’s <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>, see the <a href="#ReferencesForKK">references</a>.</p> <h2 id="statement">Statement</h2> <h3 id="InCohomology">For ordinary cohomology</h3> <h4 id="in_terms_of_homology">In terms of homology</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/free+abelian+groups">free abelian groups</a>. 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 arbitrary <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>.</p> <p>Write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo>≔</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\bullet \coloneqq Hom_{Ab}(C_\bullet, A)</annotation></semantics></math> for the dual <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> with respect 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> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_n(C)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(C,A)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">C^\bullet</annotation></semantics></math> hence for the cochain cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</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>.</p> </li> </ul> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>There is a canonical morphism of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int_{(-)}(-) : H^n(C,A) \to Hom_{Ab}(H_n(C,\mathbb{Z}), A) </annotation></semantics></math></div> <p>given by sending a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> to <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> of that cocycle on a <a class="existingWikiWord" href="/nlab/show/chain">chain</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ω</mi><mo stretchy="false">]</mo><mo>↦</mo><mrow><mo>(</mo><mo stretchy="false">[</mo><mi>σ</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mo>∫</mo> <mi>σ</mi></msub><mi>ω</mi><mo>≔</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> [\omega] \mapsto \left( [\sigma] \mapsto \int_{\sigma} \omega \coloneqq \omega(\sigma) \right) </annotation></semantics></math></div></div> <div class="num_theorem" id="OrdinaryStatementInCohomology"> <h6 id="theorem">Theorem</h6> <p><strong>(universal coefficient theorem in ordinary cohomology)</strong></p> <p>The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is surjective and its <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> is the <a class="existingWikiWord" href="/nlab/show/Ext+group">Ext group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^1(H_{n-1}(C, \mathbb{Z}), A)</annotation></semantics></math>. In other words, there is a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to Ext^1(H_{n-1}(C), A) \to H^n(C, A) \to Hom_{Ab}(H_n(C), A) \to 0 </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to Ext^1(H_{n-1}(C), A) \to H^n(Hom_{Ab}(C_\bullet,A)) \to Hom_{Ab}(H_n(C), A) \to 0 </annotation></semantics></math></div> <p>Moreover, this sequence <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">splits</a> (non-canonically).</p> </div> <p>We reproduce the direct proof given for instance in (<a href="#Boardman">Boardman</a>).</p> <div class="num_lemma" id="LemmaForUCTInOrdinaryCohomology"> <h6 id="lemma">Lemma</h6> <p>Given a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>f</mi></mover><msub><mi>A</mi> <mn>2</mn></msub><mover><mo>→</mo><mi>g</mi></mover><msub><mi>A</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">A_1 \stackrel{f}{\to} A_2 \stackrel{g}{\to} A_3</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> together with a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><msub><mi>A</mi> <mn>3</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">s : A_3 \to A_2</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>coker</mi><mi>f</mi><mover><mo>→</mo><mrow><mi>g</mi><mo>′</mo></mrow></mover><mi>coker</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>coker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to coker f \stackrel{g'}{\to} coker(g \circ f) \to coker(g) \to 0 \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Since we work in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>, all the <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a> appearing here (as discussed there) may be expressed as <a class="existingWikiWord" href="/nlab/show/quotients">quotients</a>, e.g <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>coker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">coker(f) \simeq A_2/im(f)</annotation></semantics></math>.</p> <p>The sequence of inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>A</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">im(g \circ f) \hookrightarrow im(g) \hookrightarrow A_3</annotation></semantics></math> induces the canonical <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mfrac><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>→</mo><mfrac><mrow><msub><mi>A</mi> <mn>3</mn></msub></mrow><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>→</mo><mfrac><mrow><msub><mi>A</mi> <mn>3</mn></msub></mrow><mrow><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \frac{im(g)}{im(g \circ f)} \to \frac{A_3}{im(g \circ f)} \to \frac{A_3}{im(g)} \to 0 </annotation></semantics></math></div> <p>and we claim that this is already isomorphic to the one stated in the lemma. This is manifestly true for the two terms on the right. For the term on the left observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> induces a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo><mo>:</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>A</mi> <mn>3</mn></msub><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g' : A_2 / im(f) \to A_3 / im(g \circ f)</annotation></semantics></math>. By the existence of the retract <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> this has itself a retract. Moreover it factors as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo><mo>:</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>A</mi> <mn>3</mn></msub><mo stretchy="false">/</mo><mi>im</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g' : A_2/im(f) \to im(g)/im(g \circ f) \hookrightarrow A_3/ im(g \circ f) \,. </annotation></semantics></math></div> <p>Therefore the first morphism here on the left has to be an isomorphism, too.</p> </div> <div class="proof"> <h6 id="proof_of_theorem_">Proof (of theorem <a class="maruku-ref" href="#OrdinaryStatementInCohomology"></a>)</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to B_n \to Z_n \to H_n \to 0 </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a>, <a class="existingWikiWord" href="/nlab/show/cycles">cycles</a>, and <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">C_n</annotation></semantics></math> is assumed to be a <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">B_n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Z_n</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a>, it follows that these are also free abelian, by the abelian <a class="existingWikiWord" href="/nlab/show/Nielsen-Schreier+theorem">Nielsen-Schreier theorem</a>. Therefore this sequence exhibits a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a> of the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">H_n</annotation></semantics></math>. It follows that the <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a>-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^1(H_n,A)</annotation></semantics></math> is characterized by the short exact sequence</p> <div class="maruku-equation" id="eq:ExtES"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom(Z_n, A) \to Hom(B_n,A) \to Ext^1(H_n,A) \to 0 \,. </annotation></semantics></math></div> <p>Notice also that the short exact sequence</p> <div class="maruku-equation" id="eq:ChainCycleBoundarySES"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to Z_n \to C_n \stackrel{\partial}{\to} B_{n-1} \to 0 </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split</a> because, as before, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">B_{n-1}</annotation></semantics></math> is free abelian. Using these two exact sequences on the left and right of the short exact sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to Z_n/B_n \to C_n/B_n \to C_n/Z_n \to 0 </annotation></semantics></math></div> <p>shows that this is equivalent to</p> <div class="maruku-equation" id="eq:HomologyBoundariesSES"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mover><mo>→</mo><mo>∂</mo></mover><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to H_n \to C_n/B_n \stackrel{\partial}{\to} B_{n-1} \,. </annotation></semantics></math></div> <p>Again this splits as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">B_{n-1}</annotation></semantics></math> is free abelian.</p> <p>In addition to these exact sequence consider the decomposition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo>:</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>Z</mi> <mi>n</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \partial : C_n \to C_n/B_n \to C_n/Z_n \stackrel{\simeq}{\to} B_{n-1} \hookrightarrow Z_{n-1} \hookrightarrow C_{n-1} </annotation></semantics></math></div> <p>and apply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(-,A)</annotation></semantics></math> to obtain the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>i</mi></mover></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mn>0</mn></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>←</mo></mtd> <mtd><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mo>←</mo></mtd> <mtd></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>0</mn></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && && 0 \\ && && \uparrow \\ && && Hom(H_n,A) \\ && && \uparrow \\ Hom(B_n,A) &\leftarrow& Hom(C_n,A) &\stackrel{i}{\leftarrow}& Hom(C_n/B_n,A) &\leftarrow& 0 && 0 \\ && && \uparrow^{\mathrlap{Hom(\bar \partial,A)}} && && \uparrow \\ 0 &\leftarrow& Ext^1(H_n,A) &\leftarrow& Hom(B_{n-1},A) && \leftarrow && Hom(Z_{n-1},A) \\ && && \uparrow && \nwarrow && \uparrow \\ && && 0 && && Hom(C_{n-1},A) } </annotation></semantics></math></div> <p>Here the right vertical sequence is exact, because <a class="maruku-eqref" href="#eq:ChainCycleBoundarySES">(2)</a> splits, and the left vertical sequence is exact because <a class="maruku-eqref" href="#eq:HomologyBoundariesSES">(3)</a> splits. The upper horizontal sequence is exact because the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> takes <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a> to <a class="existingWikiWord" href="/nlab/show/kernels">kernels</a> and finally the lower horizontal sequence is the exact sequence <a class="maruku-eqref" href="#eq:ExtES">(1)</a>.</p> <p>Since therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mover><mo>∂</mo><mo stretchy="false">¯</mo></mover><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(\bar \partial,A)</annotation></semantics></math> are monomorphisms, it follows that the degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>≔</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ker</mi><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Z^{n-1} \coloneqq ker( Hom(C_{n-1},A) \to Hom(C_n,a) ) \simeq ker( Hom(C_{n-1},A) \to Hom(B_{n-1}, A) ) \,. </annotation></semantics></math></div> <p>Using this for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math> replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> shows by the upper horizontal exact sequence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Z</mi> <mi>n</mi></msup><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Z^n = Hom( C_n/B_n, A) \,. </annotation></semantics></math></div> <p>Similarly the <a class="existingWikiWord" href="/nlab/show/coboundaries">coboundaries</a> are seen to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mi>n</mi></msup><mo>≔</mo><mi>im</mi><mi>Hom</mi><mo stretchy="false">(</mo><mo>∂</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>im</mi><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> B^n \coloneqq im Hom(\partial,A) \simeq im ( Hom(Z_{n-1}, A) \to Hom(C_n/B_n), A ) \,. </annotation></semantics></math></div> <p>Together this gives the cochain cohomology as</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> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>Z</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><msup><mi>B</mi> <mi>n</mi></msup><mo>≃</mo><mi>coker</mi><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^n(C,A) \coloneqq Z^n / B^n \simeq coker ( Hom(Z_n, A) \to Hom( C_n/B_n, A ) ) \,. </annotation></semantics></math></div> <p>Now the universal coefficient theorem follows by going into lemma <a class="maruku-ref" href="#LemmaForUCTInOrdinaryCohomology"></a> with the identifications <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_1 = Hom(Z_{n-1}, A)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_2 = Hom(B_{n-1}, A)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>3</mn></msub><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_3 = Hom(C_n/B_n,A)</annotation></semantics></math>.</p> </div> <h4 id="UCTForOrdinaryCohomologyInTermsOfCohomology">In terms of cohomology</h4> <p>There is also a UCT relating cohomology to cohomology:</p> <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> and <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> be chain complexes of <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> over a <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>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> which is a <a class="existingWikiWord" href="/nlab/show/principal+ideal+domain">principal ideal domain</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_2</annotation></semantics></math> be <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>-modules. Assume that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Tor</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Tor_R(N_1,N_2) = 0</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a> group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N_1</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_2</annotation></semantics></math> vanishes);</p> </li> <li> <p>at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>N</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(A,N_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(B,N_2)</annotation></semantics></math> is of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a></p> </li> </ol> <p>then there are <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><munder><mo>⊕</mo><mrow><mi>p</mi><mo>+</mo><mi>q</mi><mo>=</mo><mi>n</mi></mrow></munder><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>N</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo>,</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Tor</mi><mo stretchy="false">(</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>N</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \underset{p+q = n}{\oplus} H^p(A,N_1) \otimes H^q(B,N_2) \longrightarrow H^n(A \otimes B, N_1 \otimes N_2) \longrightarrow Tor(H^\bullet(A,N_1), H^\bullet(B,N_2)) \to 0 </annotation></semantics></math></div> <p>(<a href="#Spanier66">Spanier 66, section 5.5, theorem 11</a>)</p> <h3 id="InHomology">For ordinary homology</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet \in Ch_\bullet(Ab)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/free+abelian+groups">free abelian groups</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> be any abelian group. 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>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C_\bullet \otimes A</annotation></semantics></math> etc. for the degreewise <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a>.</p> <p>More generally, let <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> be a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> which is a <a class="existingWikiWord" href="/nlab/show/principal+ideal+domain">principal ideal domain</a> (in the above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">R = \mathbb{Z}</annotation></semantics></math> is the ring of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>), let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet \in Ch_\bullet(R Mod)</annotation></semantics></math> be a chain complex of <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> 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>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A \in R Mod</annotation></semantics></math> be any <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>-<a class="existingWikiWord" href="/nlab/show/module">module</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>k</mi></msub><msub><mo>⊗</mo> <mi>R</mi></msub><mi>A</mi></mrow><annotation encoding="application/x-tex">C_k \otimes_R A</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> 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>.</p> <div class="num_theorem" id="TheoremInOrdinaryHomology"> <h6 id="theorem_2">Theorem</h6> <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> there is a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>R</mi></msub><mi>A</mi><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><msub><mo>⊗</mo> <mi>R</mi></msub><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>Tor</mi> <mn>1</mn> <mrow><mi>R</mi><mi>Mod</mi></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> 0 \to H_n(C_\bullet) \otimes_R A \to H_n(C_\bullet \otimes_R A) \to Tor_1^{R Mod}(H_{n-1}(C_\bullet), A) \to 0 \, </annotation></semantics></math></div> <p>where on the right we have the first <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>-module of the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{n-1}(C_\bullet)</annotation></semantics></math> with <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="num_remark"> <h6 id="remark_2">Remark</h6> <p>A fairly direct generalization of this statement and its proof is the <a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a> in ordinary homology, see at <em><a href="http://ncatlab.org/nlab/show/K%C3%BCnneth+theorem#InOrdinaryHomology">Künneth theorem - In ordinary homology</a></em>.</p> </div> <p>We spell out a proof along the lines for instance given in (<a href="#Hatcher">Hatcher, 3.A</a>) or (<a href="#Chen">Chen, section 3</a>).</p> <div class="num_lemma" id="LongSequenceForHomologyWithCoefficients"> <h6 id="lemma_2">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/free+abelian+groups">free abelian groups</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">A \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> any <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, there is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi></mrow></mover><msub><mi>Z</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⊗</mo><mi>A</mi><mover><mo>→</mo><mrow><msub><mi>i</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⊗</mo><mi>A</mi></mrow></mover><msub><mi>Z</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⊗</mo><mi>A</mi><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \to B_n \otimes A \stackrel{i_n \otimes A}{\to} Z_n \otimes A \to H_n(C_\bullet \otimes A) \to B_{n-1} \otimes A \stackrel{i_{n-1}\otimes A}{\to} Z_{n-1} \otimes A \to \cdots \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">B_n</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Z_n</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/cycles">cycles</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">C_\bullet</annotation></semantics></math> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>Z</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">i_n \colon B_n \hookrightarrow Z_n</annotation></semantics></math> is the canonical inclusion.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Since, by the <a class="existingWikiWord" href="/nlab/show/Dedekind-Nielsen-Schreier+theorem">Dedekind-Nielsen-Schreier theorem</a>, every <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of a <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> is itself free abelian, such as the subgroup of <a class="existingWikiWord" href="/nlab/show/cycles">cycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>n</mi></msub><mo>↪</mo><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Z_n \hookrightarrow C_n</annotation></semantics></math>, it follows that 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> we have a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">splitting</a> of the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">0 \to Z_n \to C_n \to B_{n-1}</annotation></semantics></math> and hence (as discussed at <em><a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></em>) a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> decomposition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub><mo>≃</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo>⊕</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_n \simeq Z_n \oplus B_{n-1} \,. </annotation></semantics></math></div> <p>Here the second direct summand on the right identifies under the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">\partial^C</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> in one degree lower, since by construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mi>C</mi></msup></mrow><annotation encoding="application/x-tex">\partial^C</annotation></semantics></math> is injective on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">/</mo><msub><mi>Z</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">C_n/Z_n</annotation></semantics></math>.</p> <p>Accordingly, if we regard the graded abelian groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">B_\bullet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">Z_\bullet</annotation></semantics></math> as chain complexes with vanishing <a class="existingWikiWord" href="/nlab/show/differential">differential</a>, then we have a sequence of <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>Z</mi> <mo>•</mo></msub><mo>↪</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to Z_\bullet \hookrightarrow C_\bullet \to B_{\bullet-1} \to 0 </annotation></semantics></math></div> <p>which is degreewise a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, hence is a short exact sequence of chain complexes. Now since the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a> distributes over <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>, the image of this sequence under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-)\otimes 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><mn>0</mn><mo>→</mo><msub><mi>Z</mi> <mo>•</mo></msub><mo>⊗</mo><mi>A</mi><mo>↪</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>⊗</mo><mi>A</mi><mo>→</mo><msub><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>⊗</mo><mi>A</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to Z_\bullet \otimes A \hookrightarrow C_\bullet \otimes A \to B_{\bullet-1} \otimes A \to 0 </annotation></semantics></math></div> <p>is still a short exact sequence. The induced <a class="existingWikiWord" href="/nlab/show/homology+long+exact+sequence">homology long exact sequence</a>, as discussed there, is the long exact sequence to be shown.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p><strong>of theorem <a class="maruku-ref" href="#TheoremInOrdinaryHomology"></a></strong></p> <p>By lemma <a class="maruku-ref" href="#LongSequenceForHomologyWithCoefficients"></a> we have <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to coker(i_n \otimes A) \to H_n(C_\bullet \otimes A) \to ker(i_n \otimes A) \to 0 </annotation></semantics></math></div> <p>Since the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a> is a <a class="existingWikiWord" href="/nlab/show/right+exact+functor">right exact functor</a> it preserves <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a> and hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi><mo>=</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> coker(i_n \otimes A) \simeq coker(i_n) \otimes A = H_n(C) \otimes A \,. </annotation></semantics></math></div> <p>The dual statement were true if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">(-)\otimes A</annotation></semantics></math> were also a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a>. In general it is not, and the failure is measure by the <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>-group:</p> <p>Notice that by assumption and by the <a class="existingWikiWord" href="/nlab/show/Dedekind-Nielsen-Schreier+theorem">Dedekind-Nielsen-Schreier theorem</a> the defining <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>B</mi> <mi>n</mi></msub><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msub><mi>Z</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to B_n \stackrel{i_n}{\to} Z_n \to H_n(C_\bullet) \to 0 </annotation></semantics></math></div> <p>exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>Z</mi> <mi>n</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>qi</mi></msub></mrow></mover><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\cdots \to 0 \to B_n \to Z_n] \stackrel{\simeq_{qi}}{\to} H_n(C)</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_n(C_\bullet)</annotation></semantics></math>. Therefore by definition of <a class="existingWikiWord" href="/nlab/show/Tor">Tor</a> the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Tor</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Tor_1(H_n(C_\bullet), A)</annotation></semantics></math> is the chain homology in degree 1 of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><msub><mi>B</mi> <mi>n</mi></msub><mo>⊗</mo><mi>G</mi><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi></mrow></mover><msub><mi>Z</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [\cdots \to 0 \to B_n \otimes G \stackrel{i_n \otimes A}{\to} Z_n \otimes A] \,, </annotation></semantics></math></div> <p>which is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Tor</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ker</mi><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>n</mi></msub><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Tor_1(H_n(C_\bullet), A) \simeq ker(i_n \otimes A) \,. </annotation></semantics></math></div></div> <div class="num_section" id="sectiona"></div> <h3 id="ForGeneralizedCohomology">For generalised cohomology theories</h3> <p>The situation for <a class="existingWikiWord" href="/nlab/show/generalised+cohomology+theories">generalised cohomology theories</a> is much more complicated than that for ordinary cohomology due to the fact that it is harder (or impossible!) to use the tools of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>. Nonetheless, it is possible to say something. The general case was studied by Adams in <a href="#Ada69">Ada69</a> (for use in the <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a>, see there for more) and the initial version of the rest of this section is heavily based on that treatment. This was also considered in the slightly later work, <a href="#Ada74">Ada74, III.13</a>. Adams’ opening paragraph in <a href="#Ada69">Ada69</a> is worth quoting in its entirety as motivation for this study.</p> <blockquote> <p>It is an established practice to take old theorems about ordinary homology, and generalise them so as to obtain theorems about generalised homology theories. For example, this works very well for duality theorems about manifolds. We may ask the following question. Take all those theorems about ordinary homology which are standard results in every day use. Which are the ones which still lack a fully satisfactory generalisation to generalised homology theories? I want to devote this lecture to such problems.</p> <p><em style="text-align: right">J. F. Adams</em></p> </blockquote> <p>The lecture concentrates on the Universal Coefficient Theorem and, as a by-product, the <a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">F^*</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theories">generalized cohomology theories</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">E_*</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">F_*</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/generalized+homology+theories">generalized homology theories</a>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative</a> and <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 a <a class="existingWikiWord" href="/nlab/show/module">module</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. Then the general problems that a Universal Coefficient Theorem should apply to are the following:</p> <ol> <li><p></p> <div class="num_enuma" id="enumiAa"></div> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{*}(X)</annotation></semantics></math>, calculate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_{*}(X)</annotation></semantics></math>.</p> </li> <li><p></p> <div class="num_enuma" id="enumiAb"></div> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{*}(X)</annotation></semantics></math>, calculate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F^{*}(X)</annotation></semantics></math>.</p> </li> <li><p></p> <div class="num_enuma" id="enumiAc"></div> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^{*}(X)</annotation></semantics></math>, calculate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_{*}(X)</annotation></semantics></math>.</p> </li> <li><p></p> <div class="num_enuma" id="enumiAd"></div> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^{*}(X)</annotation></semantics></math>, calculate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F^{*}(X)</annotation></semantics></math>.</p> </li> </ol> <p>In <a href="#Ada69">Ada69</a>, Adams works in a very general setting. On this page, we shall work in a more restricted situation (as spelled out in Note 2 in Adams’ lectures). We assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^{*}(-)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/generalised+cohomology+theory">generalised cohomology theory</a> associated to a <a class="existingWikiWord" href="/nlab/show/commutative+ring+spectrum">commutative ring spectrum</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. The cohomology theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F^{*}(-)</annotation></semantics></math> is assumed to come from a left <a class="existingWikiWord" href="/nlab/show/module-spectrum">module-spectrum</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, which we shall denote by <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>. We do not assume that <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 itself a ring spectrum. Following Adams, we shall also assume that all our cohomology and homology theories are <a class="existingWikiWord" href="/nlab/show/reduced+cohomology">reduced</a>.</p> <p>There are two statements that one would like to hold. These are not themselves theorems, rather the theorem would say “Under certain conditions, these statements hold”. The statements are the following.</p> <p> <div class='num_theorem' id='ucta'> <h6>Statement</h6> <p><em>(UCT1)</em> </p> <p>There is a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Tor</mo> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Tor_{p,*}^{E_{*}} (E_{*}(X), F_{*}) \xRightarrow[p]{} F_{*}(X) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow></msub><msub><mi>F</mi> <mo>*</mo></msub><mo>→</mo><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{*}(X) \otimes _{E_{*}} F_{*} \to F_{*}(X). </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_theorem' id='uctb'> <h6>Statement</h6> <p><em>(UCT2)</em> </p> <p>There is a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Ext</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Ext_{E_{*}}^{p,*} (E_{*}(X), F^{*}) \xRightarrow[p]{} F^{*}(X) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mo lspace="0em" rspace="thinmathspace">Hom</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> F^{*}(X) \to \Hom_{E_{*}} (E_{*}(X), F^{*}). </annotation></semantics></math></div> <p></p> </div> </p> <p>For finite <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> then we can derive two further statements from the above by <a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>. We use the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">D X</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+dual">Spanier-Whitehead dual</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>For a finite CW-complex <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>, we can apply <a href="#ucta">UCT1</a> and <a href="#uctb">UCT2</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">D X</annotation></semantics></math> in place 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> and then use the various isomorphisms relating the cohomologies 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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">D X</annotation></semantics></math> to reformulate them in terms 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>. We thus get the following statements.</p> <p> <div class='num_theorem' id='uctc'> <h6>Statement</h6> <p><em>(UCT3)</em> </p> <p>For <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 finite CW-complex, there is a spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Tor</mo> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Tor_{p,*}^{E^{*}}(E^{*}(X), F^{*}) \xRightarrow[p]{} F^{*}(X) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow></msub><msup><mi>F</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{*}(X) \otimes _{E^{*}} F^{*} \to F^{*}(X). </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_theorem' id='utcd'> <h6>Statement</h6> <p><em>(UCT4)</em> </p> <p>For <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 finite CW-complex, there is a spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Ext</mo> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Ext^{p,*}_{E^{*}} (E^{*}(X), F_{*}) \xRightarrow[p]{} F_{*}(X) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mo lspace="0em" rspace="thinmathspace">Hom</mo> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> F_{*}(X) \to \Hom^{*}_{E^{*}}(E^{*}(X), F_{*}). </annotation></semantics></math></div> <p></p> </div> </p> <p>(This is a generalization of the <a class="existingWikiWord" href="/nlab/show/Kronecker+pairing">Kronecker pairing</a>, see also e.g. <a href="#Schwede12">Schwede 12, prop. 6.20</a>).</p> <p>A particularly important special case of these statements is when we have a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, and a <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^{*}(-)</annotation></semantics></math>. Then we define a new homology theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_{*}(-)</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_{*}(X) = E_{*}(X \wedge Y)</annotation></semantics></math> and a new cohomology theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G^{*}(-)</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G^{*}(X) = E^{*}(X \wedge Y)</annotation></semantics></math>. These are representable, the homology theory by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∧</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">Y \wedge E</annotation></semantics></math> and the cohomology theory by the function spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(Y,E)</annotation></semantics></math>. Putting these into the statements of the universal coefficient theorem, we obtain similar statements for the <a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a>.</p> <p> <div class='num_theorem' id='kta'> <h6>Statement</h6> <p><em>(KT1)</em> </p> <p>There is a spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Tor</mo> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Tor_{p,*}^{E_{*}} (E_{*}(X), E_{*}(Y)) \xRightarrow[p]{} E_{*}(X \wedge Y) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow></msub><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{*}(X) \otimes _{E_{*}} E_{*}(Y) \to E_{*}(X \wedge Y). </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_theorem' id='ktb'> <h6>Statement</h6> <p><em>(KT2)</em> </p> <p>There is a spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Ext</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Ext_{E_{*}}^{p,*}(E_{*}(X), E^{*}(Y)) \xRightarrow[p]{} E^{*}(X \wedge Y) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mo lspace="0em" rspace="thinmathspace">Hom</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{*}(X \wedge Y) \to \Hom_{E_{*}}^{*} (E_{*}(X), E^{*}(Y)). </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_theorem' id='ktc'> <h6>Statement</h6> <p><em>(KT3)</em> </p> <p>For <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 finite CW-complex, there is a spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Tor</mo> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Tor_{p,*}^{E^{*}} (E^{*}(X), E^{*}(Y)) \xRightarrow[p]{} E^{*}(X \wedge Y) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow></msub><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{*}(X) \otimes _{E^{*}} E^{*}(Y) \to E^{*}(X \wedge Y). </annotation></semantics></math></div> <p></p> </div> </p> <p> <div class='num_theorem' id='ktd'> <h6>Statement</h6> <p><em>(KT4)</em> </p> <p>For <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 finite CW-complex, there is a spectral sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="0em" rspace="thinmathspace">Ext</mo> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow> <mrow><mi>p</mi><mo>,</mo><mo>*</mo></mrow></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munder><mo>⇒</mo><mrow><mi>p</mi></mrow></munder><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Ext^{p,*}_{E^{*}} (E^{*}(X), E_{*}(Y)) \xRightarrow[p]{} E_{*}(X \wedge Y) </annotation></semantics></math></div> <p>with edge homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mo lspace="0em" rspace="thinmathspace">Hom</mo> <mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_{*}(X \wedge Y) \to \Hom_{E^{*}}^{*} (E^{*}(X), E_{*}(Y)). </annotation></semantics></math></div> <p></p> </div> </p> <p>The key question is, thus: when do these statements hold? Adams gives some answers in <a href="#Ada69">Ada69</a>.</p> <ul> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">F_{*}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/flat+module">flat</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">E_{*}</annotation></semantics></math> then <a href="#ucta">UCT1</a> holds, whence <a href="#kta">KT1</a> holds if either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{*}(X)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_{*}(Y)</annotation></semantics></math> is flat.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">E = S</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>, then all the results are true.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a strict <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a> then <a href="#kta">KT1</a> holds, if also <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 a strict <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> then <a href="#ucta">UCT1</a> holds.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah">Atiyah</a> gives a method in <a href="#Ati62">Ati62</a> for <a href="#ktc">KT3</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">E =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/KU">KU</a> being complex <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> finite complexes.</p> </li> </ul> <h4 id="a_special_case">A Special Case</h4> <p>In both <a href="#Ada69">Ada69</a> and <a href="#Ada74">Ada74</a>, there is a particular focus on the universal coefficient theorem coming from its applications to the <a class="existingWikiWord" href="/nlab/show/Adams+spectral+sequence">Adams spectral sequence</a>. With that aim in mind, he studies the universal coefficient theorems with considerably strong assumptions. These assumptions are designed to allow Atiyah’s method (from <a href="#Ati62">Ati62</a>) to work.</p> <p> <div class='num_theorem' id='adams_assumption'> <h6>Assumption</h6> <p><em>(Condition 13.3 in <a href="#Ada74">Ada74</a>, see also Assumption 20 in <a href="#Ada69">Ada69</a>)</em> </p> <p>The spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/direct+limit">direct limit</a> of finite spectra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">E_\alpha</annotation></semantics></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>D</mi><msub><mi>E</mi> <mi>α</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(D E_\alpha)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/projective+object">projective</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">E_*</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>D</mi><msub><mi>E</mi> <mi>α</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msubsup><mo lspace="0em" rspace="thinmathspace">Hom</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>D</mi><msub><mi>E</mi> <mi>α</mi></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F^*(D E_\alpha) \to \Hom^*_{E_*}(E_*(D E_\alpha), F_*) </annotation></semantics></math></div> <p>is an isomorphism for all <a class="existingWikiWord" href="/nlab/show/module-spectra">module-spectra</a> <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> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. Here, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><msub><mi>E</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">D E_\alpha</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/S-dual">S-dual</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">E_\alpha</annotation></semantics></math>.</p> <p></p> </div> </p> <p>The main difference between the two treatments is that in <a href="#Ada69">Ada69</a>, the condition involving <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 stated for a <em>single</em> module-spectrum, not for <em>all</em> module-spectra, and there are alternatives for homology (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_*(-)</annotation></semantics></math>) and cohomology (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F^*(-)</annotation></semantics></math>).</p> <p>In the comments following Assumption 20 in <a href="#Ada69">Ada69</a>, Adams remarks that this is implied by a stronger condition (Proposition 17) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> which makes no reference to <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>. As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a ring spectrum, this reduces to:</p> <ol> <li>The spectral sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>α</mi></msub><mo>;</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⇒</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>α</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^*(E_\alpha; E^*) \Rightarrow E^*(E_\alpha)</annotation></semantics></math> is trivial, and</li> <li>For each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>α</mi></msub><mo>;</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^p(E_\alpha; E^*)</annotation></semantics></math> is projective as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math>-module.</li> </ol> <p>With this assumption, Adams shows the following result:</p> <div class="num_theorem" id="uctholds"> <h6 id="theorem_3">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> be a ring spectrum satisfying Assumption <a class="maruku-ref" href="#adams_assumption"></a>. Let <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> be a module-spectrum over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. Then <a class="maruku-ref" href="#uctb"></a> holds, and the spectral sequence is convergent.</p> </div> <p>What “convergent” means here is spelled out in <a href="#Ada74">Ada74, Theorem 8.2</a>.</p> <p> <div class='num_cor' id='uctproj'> <h6>Corollary</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> be a ring spectrum satisfying Assumption <a class="maruku-ref" href="#adams_assumption"></a>. Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X)</annotation></semantics></math> is projective over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">E_*</annotation></semantics></math>. Then the spectral sequence from <a class="maruku-ref" href="#uctb"></a> collapses at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E^2</annotation></semantics></math> term. That is,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>F</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mo lspace="0em" rspace="thinmathspace">Hom</mo> <mrow><msub><mi>E</mi> <mo>*</mo></msub></mrow> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>F</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F^*(X) \to \Hom^*_{E_*}(E_*(X),F_*) </annotation></semantics></math></div> <p>is an isomorphism.</p> </div> </p> <p>In <a href="#Ada74">Ada74</a>, Adams lists several cohomology theories (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(-)</annotation></semantics></math>) where the assumption holds. These are: <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>H</mi><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">H\mathbb{Z}_p</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MO</mi></mrow><annotation encoding="application/x-tex">MO</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MSp</mi></mrow><annotation encoding="application/x-tex">MSp</annotation></semantics></math>, <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KO</mi></mrow><annotation encoding="application/x-tex">KO</annotation></semantics></math>.</p> <h4 id="freeness_and_flatness">Freeness and Flatness</h4> <p>In <a href="#BJW95">BJW95</a> and <a href="#Boa95">Boa95</a> there are various versions of the universal coefficient theorems and Künneth theorems which are stated and proved (or indications given on how to prove) under assumptions of either <a class="existingWikiWord" href="/nlab/show/free+module">freeness</a> or <a class="existingWikiWord" href="/nlab/show/flat+module">flatness</a>.</p> <p>Here, we shall gather together all the statements made. In all the following, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(-)</annotation></semantics></math> is a multiplicative generalised cohomology theory with representing ring spectrum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. We use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(-)</annotation></semantics></math> for the associated homology theory. Following <a href="#Boa95">Boa95</a> and <a href="#BJW95">BJW95</a>, cohomology and homology are <em>not</em> reduced in this section.</p> <div class="num_theorem" id="homkun"> <h6 id="theorem_4">Theorem</h6> <p>(<a href="#Boa95">Boa95, 4.2</a>) Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(Y)</annotation></semantics></math> is a free or flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math>-module. Then the pairing:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>×</mo><mo lspace="verythinmathspace">:</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex"> \times \colon E_*(X) \otimes E_*(Y) \to E_*(X \times Y), </annotation></semantics></math></div> <p>induces the Künneth isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X \times Y) \cong E_*(X) \otimes E_*(Y)</annotation></semantics></math> in homology.</p> </div> <p>The next result relates homology and cohomology.</p> <div class="num_theorem" id="strdual"> <h6 id="theorem_5">Theorem</h6> <p>(<a href="#Boa95">Boa95, 4.14</a>) Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X)</annotation></semantics></math> is a free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math>-module. Then <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> has strong duality, i.e. the duality map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo lspace="verythinmathspace">:</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \colon E^*(X) \to D E_*(X)</annotation></semantics></math> is a homeomorphism between the profinite topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(X)</annotation></semantics></math> and the dual-finite topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D E_*(X)</annotation></semantics></math>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(X)</annotation></semantics></math> is complete Hausdorff.</p> </div> <p>Combining these two gives the Künneth theorem for cohomology.</p> <div class="num_theorem" id="cokun"> <h6 id="theorem_6">Theorem</h6> <p>(<a href="#Boa95">Boa95, 4.19</a>) Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(Y)</annotation></semantics></math> are free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math>-modules. Then we have the Künneth homeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≅</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⊗</mo><mo>^</mo></mover><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(X \times Y) \cong E^*(X) \widehat{\otimes} E^*(Y)</annotation></semantics></math> in cohomology.</p> </div> <p>There are similar results for spectra. Boardman, Johnson, and Wilson write reduced homology and cohomology as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X,o)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(X,o)</annotation></semantics></math>, even when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a spectrum (and so the reduced theories are all that there are).</p> <div class="num_theorem" id="homkunsp"> <h6 id="theorem_7">Theorem</h6> <p>(<a href="#Boa95">Boa95, 9.20</a>) Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X,o)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(Y,o)</annotation></semantics></math> is a free or flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math>-module. Then the pairing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo><mo lspace="verythinmathspace">:</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\times \colon E_*(X,o) \otimes E_*(Y,o) \to E_*(X \wedge Y,o)</annotation></semantics></math> is an isomorphism in homology.</p> </div> <div class="num_theorem" id="strdusp"> <h6 id="theorem_8">Theorem</h6> <p>(<a href="#Boa95">Boa95, 9.25</a>) Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X,o)</annotation></semantics></math> is a free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math>-module. Then <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> has strong duality, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo lspace="verythinmathspace">:</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \colon E^*(X,o) \to D E_*(X,o)</annotation></semantics></math> is a homeomorphism between the profinite topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(X,o)</annotation></semantics></math> and the dual-finite topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D E_*(X,o)</annotation></semantics></math>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^*(X,o)</annotation></semantics></math> is complete Hausdorff.</p> </div> <div class="num_theorem" id="cokunsp"> <h6 id="theorem_9">Theorem</h6> <p>(<a href="#Boa95">Boa95, 9.31</a>) Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(X,o)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_*(Y,o)</annotation></semantics></math> are free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">E^*</annotation></semantics></math>-modules. Then the pairing</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>×</mo><mo lspace="verythinmathspace">:</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo><mover><mo>⊗</mo><mo>^</mo></mover><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>E</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo>,</mo><mi>o</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \times \colon E^*(X,o) \widehat{\otimes} E^*(Y,o) \to E^*(X \wedge Y,o) </annotation></semantics></math></div> <p>induces the cohomology Künneth homeomorphism.</p> </div> <h2 id="examples">Examples</h2> <h3 id="InTopology">For singular cohomology</h3> <p>For <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 <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Sing X</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>N</mi><mi>ℤ</mi><mo stretchy="false">[</mo><mi>Sing</mi><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> C_\bullet(X) \coloneqq N \mathbb{Z}[Sing X] </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/normalized+chain+complex">normalized chain complex</a> of the <a class="existingWikiWord" href="/nlab/show/simplicial+abelian+group">simplicial abelian group</a> obtained by forming degreewise the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a>.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></em> <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>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_\bullet(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> of <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>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet(X)</annotation></semantics></math>, and for <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> some <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, the <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular 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><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(X,A)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a>, of <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>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet(X)</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>Comparison with the ordinary universal coefficient theorem <a class="maruku-ref" href="#OrdinaryStatementInCohomology"></a> shows that:</p> <div class="num_theorem" id="OrdinaryStatementInTopology"> <h6 id="theorem_10">Theorem</h6> <p><strong>(universal coefficient theorem in topology)</strong></p> <p>For <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 <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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \geq 1 \in \mathbb{N}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> and <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> 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> fit into a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split</a> <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to Ext^1(H_{n-1}(X), A) \longrightarrow H^n(X,A) \longrightarrow Hom_{Ab}(H_n(X), A) \to 0 \,. </annotation></semantics></math></div></div> <div class="num_example" id="UniversalCoefficientTheoremInRationalCohomology"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a> in <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a>)</strong></p> <p>For <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>-<a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Ext+groups">Ext groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>;</mo><mi>ℚ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext^1(-;\mathbb{Q})</annotation></semantics></math> vanish, and hence the <a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a> identifies <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> <a class="existingWikiWord" href="/nlab/show/cohomology+groups">groups</a> with the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a> of the <a class="existingWikiWord" href="/nlab/show/rational+vector+space">rational vector space</a> of rational <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a>:</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 maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℚ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>ℤ</mi></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>H</mi> <mo>•</mo></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℚ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℚ</mi><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^\bullet \big( -; \, \mathbb{Q} \big) \;\; \simeq \;\; Hom_{\mathbb{Z}} \Big( H_\bullet\big(-;\,\mathbb{Q} \big); \, \mathbb{Q} \Big) \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="rational+homotopy+theory#Moerman15">Moerman 15, Cor. 1.2.1</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <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/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bootstrap+category">bootstrap category</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="for_ordinary_cohomology_2">For ordinary (co)homology</h3> <ul> <li id="Spanier66"><a class="existingWikiWord" href="/nlab/show/Edwin+Spanier">Edwin Spanier</a>, section 5.5 of <em>Algebraic topology</em>, 1966</li> </ul> <p>An exposition of the universal coefficient theorem for ordinary cohomology and homology is in section 3.1 of</p> <ul> <li id="Hatcher"><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Algebraic topology</em> (<a href="http://www.math.cornell.edu/~hatcher/AT/AT.pdf">pdf</a>); also <a href="http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf">section 3.A</a>.</li> </ul> <p>The note</p> <ul> <li>Adam Clay, <em>The universal coefficient theorems and Künneth formulas</em> (<a href="http://www.math.ubc.ca/~aclay/Algtopology.pdf">pdf</a>)</li> </ul> <p>surveys and spells out statement and proof of the theorem. A detailed proof of the theorem in cohomology is also in</p> <ul> <li id="Boardman"><a class="existingWikiWord" href="/nlab/show/Michael+Boardman">Michael Boardman</a>, <em>The universal coefficient theorem</em> (<a href="http://www.math.jhu.edu/~jmb/note/uctcoh.pdf">pdf</a>)</li> </ul> <p>and a detailed proof of the statement in homology is in section 3 of</p> <ul> <li id="Chen">Chen, <em>Universal coefficient theorem for homology</em> (<a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/ChenJ.pdf">pdf</a>)</li> </ul> <h3 id="ForGeneralizedCoHomology">For generalized (co)homology</h3> <p>The universal coefficient theorem in <a class="existingWikiWord" href="/nlab/show/symmetric+smash+product+of+spectra">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/model+categories+of+spectra">model categories of spectra</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, section 8 of <em><a class="existingWikiWord" href="/nlab/show/Modern+foundations+for+stable+homotopy+theory">Modern foundations for stable homotopy theory</a></em> (<a href="http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf">pdf</a>)</li> </ul> <p>Universal coefficient theorems for <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> are discussed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Friedrich+Bauer">Friedrich Bauer</a>, <em>Remarks on universal coefficient theorems for generalized homology theories</em> Quaestiones Mathematicae <strong>9</strong> 1, 4 (1986) 29-54</li> </ul> <p>More on the universal coefficient theorem in generalized homology is in:</p> <ul> <li id="Ada69"> <p><strong>Ada69</strong> <a class="existingWikiWord" href="/nlab/show/John+Adams">J. F. Adams</a>. Lectures on generalised cohomology. pages 1–138, Berlin, 1969. Springer.</p> </li> <li id="Ada74"> <p><strong>Ada74</strong> <a class="existingWikiWord" href="/nlab/show/John+Adams">J. F. Adams</a>, (1974). <a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a>. Chicago, Ill.: University of Chicago Press.</p> </li> <li id="Ati62"> <p><strong>Ati62</strong> <a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">M. F. Atiyah</a>. Vector bundles and the Künneth formula. Topology, 1:245–248, 1962.</p> </li> <li id="Boa95"> <p><strong>Boa95</strong> <a class="existingWikiWord" href="/nlab/show/J.+M.+Boardman">J. M. Boardman</a>, <em>Stable Operations in Generalized Cohomology</em> [<a href="https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Boardman-StableOperations.pdf" title="pdf">pdf</a>] in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em> Oxford 1995 (<a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li> <p><strong>BJW95</strong> <a class="existingWikiWord" href="/nlab/show/J.+M.+Boardman">J. M. Boardman</a>, and Johnson, David Copeland and Wilson, W. Stephen. (1995). Unstable operations in generalized cohomology. (pp. 687–828). Amsterdam: North-Holland.</p> </li> </ul> <p>See also</p> <ul> <li id="Schwede12"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, prop. 6.20 of <em><a class="existingWikiWord" href="/nlab/show/Symmetric+spectra">Symmetric spectra</a></em>, 2012 (<a href="http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf">pdf</a>)</li> </ul> <p>Further discussion along these lines includes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrew+Baker">Andrew Baker</a>, Andrey Lazarev, <em>On the Adams Spectral Sequence for <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>-modules</em>, Algebr. Geom. Topol. 1 (2001) 173-199 (<a href="http://arxiv.org/abs/math/0105079">arXiv.0105079</a>)</li> </ul> <h3 id="ReferencesForKK">For KK-theory</h3> <p>Discussion for <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, Claude Schochet, <em>The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor</em>, Duke Math. J. Volume 55, Number 2 (1987), 431-474. (<a href="http://projecteuclid.org/euclid.dmj/1077306030">EUCLID</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 5, 2023 at 19:37:11. 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