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Universal coefficient theorem
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<h2 id="statement-of-the-homology-case">Statement of the homology case</h2> <p>Consider the <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product of modules</a> H i ( X , Z ) ⊗ A {\displaystyle H_{i}(X,\mathbb {Z} )\otimes A} . The theorem states there is a <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequence</a> involving the <a href="/facts/Tor_functor/oFn2L3gd">Tor functor</a> </p> 0 → H i ( X , Z ) ⊗ A → μ H i ( X , A ) → Tor 1 ( H i − 1 ( X , Z ) , A ) → 0. {\displaystyle 0\to H_{i}(X,\mathbb {Z} )\otimes A\,{\overset {\mu }{\to }}\,H_{i}(X,A)\to \operatorname {Tor} _{1}(H_{i-1}(X,\mathbb {Z} ),A)\to 0.} <p>Furthermore, this sequence <a href="/facts/Splitting_lemma/L81DDNaA">splits</a>, though not naturally. Here μ {\displaystyle \mu } is the map induced by the bilinear map H i ( X , Z ) × A → H i ( X , A ) {\displaystyle H_{i}(X,\mathbb {Z} )\times A\to H_{i}(X,A)} . </p><p>If the coefficient ring A {\displaystyle A} is Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} } , this is a special case of the <a href="/facts/Bockstein_spectral_sequence/3dGougYy">Bockstein spectral sequence</a>. </p> <h2 id="universal-coefficient-theorem-for-cohomology">Universal coefficient theorem for cohomology</h2> <p>Let G {\displaystyle G} be a module over a <a href="/facts/Principal_ideal/Cga66qgM">principal ideal</a> domain R {\displaystyle R} (for example Z {\displaystyle \mathbb {Z} } , or any field.) </p><p>There is a universal coefficient theorem for <a href="/facts/Cohomology/J7FJzTNh">cohomology</a> involving the <a href="/facts/Ext_functor/ktggDk7I">Ext functor</a>, which asserts that there is a natural short exact sequence </p> 0 → Ext R 1 ( H i − 1 ( X ; R ) , G ) → H i ( X ; G ) → h Hom R ( H i ( X ; R ) , G ) → 0. {\displaystyle 0\to \operatorname {Ext} _{R}^{1}(H_{i-1}(X;R),G)\to H^{i}(X;G)\,{\overset {h}{\to }}\,\operatorname {Hom} _{R}(H_{i}(X;R),G)\to 0.} <p>As in the homology case, the sequence splits, though not naturally. In fact, suppose </p> H i ( X ; G ) = ker ∂ i ⊗ G / im ∂ i + 1 ⊗ G , {\displaystyle H_{i}(X;G)=\ker \partial _{i}\otimes G/\operatorname {im} \partial _{i+1}\otimes G,} <p>and define </p> H ∗ ( X ; G ) = ker ( Hom ( ∂ , G ) ) / im ( Hom ( ∂ , G ) ) . {\displaystyle H^{*}(X;G)=\ker(\operatorname {Hom} (\partial ,G))/\operatorname {im} (\operatorname {Hom} (\partial ,G)).} <p>Then h {\displaystyle h} above is the canonical map: </p> h ( [ f ] ) ( [ x ] ) = f ( x ) . {\displaystyle h([f])([x])=f(x).} <p>An alternative point of view can be based on representing cohomology via <a href="/facts/Eilenberg%25E2%2580%2593MacLane_space/mrux6oMJ">Eilenberg–MacLane space</a>, where the map h {\displaystyle h} takes a <a href="/facts/Homotopy/1nWrPxdj">homotopy</a> class of maps X → K ( G , i ) {\displaystyle X\to K(G,i)} to the corresponding homomorphism induced in homology. Thus, the Eilenberg–MacLane space is a <i>weak right <a href="/facts/Adjoint_functor/bJ1P7AN2">adjoint</a></i> to the homology <a href="/facts/Functor/l2u3rIBE">functor</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="example-mod-2-cohomology-of-the-real-projective-space">Example: mod 2 cohomology of the real projective space</h2> <p>Let X = R P n {\displaystyle X=\mathbb {RP} ^{n}} , the <a href="/facts/Real_projective_space/svbABDtd">real projective space</a>. We compute the singular cohomology of X {\displaystyle X} with coefficients in G = Z / 2 Z {\displaystyle G=\mathbb {Z} /2\mathbb {Z} } using integral homology, i.e., R = Z {\displaystyle R=\mathbb {Z} } . </p><p>Knowing that the integer homology is given by: </p> H i ( X ; Z ) = { Z i = 0 or i = n odd, Z / 2 Z 0 < i < n , i odd, 0 otherwise. {\displaystyle H_{i}(X;\mathbb {Z} )={\begin{cases}\mathbb {Z} &i=0{\text{ or }}i=n{\text{ odd,}}\\\mathbb {Z} /2\mathbb {Z} &0<i<n,\ i\ {\text{odd,}}\\0&{\text{otherwise.}}\end{cases}}} <p>We have Ext ( G , G ) = G {\displaystyle \operatorname {Ext} (G,G)=G} and Ext ( R , G ) = 0 {\displaystyle \operatorname {Ext} (R,G)=0} , so that the above exact sequences yield </p> H i ( X ; G ) = G {\displaystyle H^{i}(X;G)=G} <p>for all i = 0 , … , n {\displaystyle i=0,\dots ,n} . In fact the total <a href="/facts/Cohomology_ring/eyUR2o1d">cohomology ring</a> structure is </p> H ∗ ( X ; G ) = G [ w ] / ⟨ w n + 1 ⟩ . {\displaystyle H^{*}(X;G)=G[w]/\left\langle w^{n+1}\right\rangle .} <h2 id="corollaries">Corollaries</h2> <p>A special case of the theorem is computing integral cohomology. For a finite <a href="/facts/CW_complex/rN7buLMo">CW complex</a> X {\displaystyle X} , H i ( X , Z ) {\displaystyle H_{i}(X,\mathbb {Z} )} is finitely generated, and so we have the following <a href="/facts/Fundamental_theorem_of_finitely_generated_abelian_groups/dwtl7mp4">decomposition</a>. </p> H i ( X ; Z ) ≅ Z β i ( X ) ⊕ T i , {\displaystyle H_{i}(X;\mathbb {Z} )\cong \mathbb {Z} ^{\beta _{i}(X)}\oplus T_{i},} <p>where β i ( X ) {\displaystyle \beta _{i}(X)} are the <a href="/facts/Betti_number/r42Y8k6f">Betti numbers</a> of X {\displaystyle X} and T i {\displaystyle T_{i}} is the torsion part of H i {\displaystyle H_{i}} . One may check that </p> Hom ( H i ( X ) , Z ) ≅ Hom ( Z β i ( X ) , Z ) ⊕ Hom ( T i , Z ) ≅ Z β i ( X ) , {\displaystyle \operatorname {Hom} (H_{i}(X),\mathbb {Z} )\cong \operatorname {Hom} (\mathbb {Z} ^{\beta _{i}(X)},\mathbb {Z} )\oplus \operatorname {Hom} (T_{i},\mathbb {Z} )\cong \mathbb {Z} ^{\beta _{i}(X)},} <p>and </p> Ext ( H i ( X ) , Z ) ≅ Ext ( Z β i ( X ) , Z ) ⊕ Ext ( T i , Z ) ≅ T i . {\displaystyle \operatorname {Ext} (H_{i}(X),\mathbb {Z} )\cong \operatorname {Ext} (\mathbb {Z} ^{\beta _{i}(X)},\mathbb {Z} )\oplus \operatorname {Ext} (T_{i},\mathbb {Z} )\cong T_{i}.} <p>This gives the following statement for integral cohomology: </p> H i ( X ; Z ) ≅ Z β i ( X ) ⊕ T i − 1 . {\displaystyle H^{i}(X;\mathbb {Z} )\cong \mathbb {Z} ^{\beta _{i}(X)}\oplus T_{i-1}.} <p>For X {\displaystyle X} an <a href="/facts/Orientability/1SEQM2Pt">orientable</a>, <a href="/facts/Closed_manifold/jMG6G6Bl">closed</a>, and <a href="/facts/Connected_space/5CUTxfoA">connected</a> n {\displaystyle n} -<a href="/facts/Manifold/rXPERJtu">manifold</a>, this corollary coupled with <a href="/facts/Poincar%25C3%25A9_duality/lwNjUCIa">Poincaré duality</a> gives that β i ( X ) = β n − i ( X ) {\displaystyle \beta _{i}(X)=\beta _{n-i}(X)} . </p> <h2 id="universal-coefficient-spectral-sequence">Universal coefficient spectral sequence</h2> <p>There is a generalization of the universal coefficient theorem for (co)homology with <a href="/facts/Twisted_Poincar%25C3%25A9_duality/XhvLatq7">twisted coefficients</a>. </p><p>For cohomology we have </p> E 2 p , q = Ext R q ( H p ( C ∗ ) , G ) ⇒ H p + q ( C ∗ ; G ) , {\displaystyle E_{2}^{p,q}=\operatorname {Ext} _{R}^{q}(H_{p}(C_{*}),G)\Rightarrow H^{p+q}(C_{*};G),} <p>where R {\displaystyle R} is a ring with unit, C ∗ {\displaystyle C_{*}} is a chain complex of free modules over R {\displaystyle R} , G {\displaystyle G} is any ( R , S ) {\displaystyle (R,S)} -bimodule for some ring with a unit S {\displaystyle S} , and Ext {\displaystyle \operatorname {Ext} } is the <a href="/facts/Ext_functor/ktggDk7I">Ext group</a>. The differential d r {\displaystyle d^{r}} has degree ( 1 − r , r ) {\displaystyle (1-r,r)} . </p><p>Similarly for homology, </p> E p , q 2 = Tor q R ( H p ( C ∗ ) , G ) ⇒ H ∗ ( C ∗ ; G ) , {\displaystyle E_{p,q}^{2}=\operatorname {Tor} _{q}^{R}(H_{p}(C_{*}),G)\Rightarrow H_{*}(C_{*};G),} <p>for Tor {\displaystyle \operatorname {Tor} } the <a href="/facts/Tor_functor/oFn2L3gd">Tor group</a> and the differential d r {\displaystyle d_{r}} having degree ( r − 1 , − r ) {\displaystyle (r-1,-r)} . </p> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Allen_Hatcher/dWXuqkXA">Allen Hatcher</a>, <i>Algebraic Topology</i>, Cambridge University Press, Cambridge, 2002. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the <a href="http://pi.math.cornell.edu/~hatcher/AT/ATpage.html">author's homepage</a>.</li> <li><a href="/facts/Paul_Chester_Kainen/ajToeKL6">Kainen, P. C.</a> (1971). "Weak Adjoint Functors". <i>Mathematische Zeitschrift</i>. 122: 1–9. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fbf01113560">10.1007/bf01113560</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:122894881">122894881</a>.</li> <li><a href="/facts/Jerome_Levine/McPKaq79">Jerome Levine</a>. “Knot Modules. I.” Transactions of the American Mathematical Society 229 (1977): 1–50. <a href="https://doi.org/10.2307/1998498">https://doi.org/10.2307/1998498</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://math.stackexchange.com/q/768481">Universal coefficient theorem with ring coefficients</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>(Kainen 1971) - Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1–9. doi:10.1007/bf01113560. S2CID 122894881. <a href="https://doi.org/10.1007%2Fbf01113560" target="_blank">https://doi.org/10.1007%2Fbf01113560</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>