The derived tensor product and derived internal hom now give concrete computational consequences for ordinary homology and cohomology. We first record the algebraic input over principal ideal domains, then apply it to the chain complexes of animae constructed in Section 6.2.
6.7.1 Tor and Ext over principal ideal domains
For complexes of modules over a principal ideal domain, the homology groups of the derived tensor product and derived hom admit particularly explicit descriptions in terms of Tor and Ext groups of the homology of the individual complexes. The key simplification is that projective resolutions over a PID have length at most one, which allows for elementary proofs of the Künneth formula and universal coefficient theorem.
Proposition 6.7.1. Let \(R\) be a principal ideal domain. Then every \(R\)-module \(M\) admits a projective resolution of length at most one: \[ 0 \to P_1 \to P_0 \to M \to 0. \] In particular, \(\Tor _n^R(M,N) = 0\) and \(\Ext ^n_R(M,N) = 0\) for all \(R\)-modules \(M, N\) and all \(n \geq 2\).
Proof. Every \(R\)-module \(M\) admits a surjection from a free module: take \(P_0 := R^{(M)}\), the free module on the underlying set of \(M\), with the canonical surjection \(\varepsilon \colon P_0 \twoheadrightarrow M\). The kernel \(P_1 := \ker (\varepsilon )\) is a submodule of the free module \(P_0\). Over a PID, every submodule of a free module is itself free, hence projective. This gives the desired resolution.
The vanishing of higher Tor and Ext groups follows from the formulas \(\Tor _n^R(M,N) \cong H_n(P_{\bullet } \otimes _R N)\) and \(\Ext ^n_R(M,N) \cong H^n(\Hom _R(P_{\bullet }, N))\) of Corollary 6.6.28 and Proposition 6.4.14, which are both zero for \(n \geq 2\) since \(P_k = 0\) for \(k \geq 2\). □
Theorem 6.7.2 (Algebraic Künneth formula). Let \(R\) be a principal ideal domain and let \(A, B \in \D ^-(R)\) be bounded below complexes of \(R\)-modules. Then for each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to \bigoplus _{i+j=n} H_i(A) \otimes _R H_j(B) \to H_n(A \otimes ^{\bL }_R B) \to \bigoplus _{i+j=n-1} \Tor _1^R(H_i(A), H_j(B)) \to 0, \] and this sequence splits (non-naturally).
Proof. We may represent the complexes \(A\) and \(B\) up to quasi-isomorphism by bounded below chain complexes \(C\) and \(C'\), respectively, such that the \(R\)-modules \(C_i\) and \(C'_i\) are free for all \(i \in \Z \). Then the derived tensor product \(A \otimes ^{\bL }_R B\) is represented by the algebraic tensor product \(C \otimes _R C'\).
We first treat the special case where all the differentials in \(C\) are zero, so that \(H_i(C) = C_i\) for all \(i\). In this case, the differential on \(C \otimes _R C'\) is given by \(d(c \otimes c') = (-1)^i c \otimes dc'\) for \(c \in C_i\), and the chain complex \(C \otimes _R C'\) decomposes as the direct sum of the complexes \(C_i[i] \otimes _R C'\). Since each \(C_i\) is free, the complex \(C_i \otimes _R C'\) is a direct sum of copies of \(C'\), and hence \[ H_n(C_i[i] \otimes _R C') \;\cong \; C_i \otimes _R H_{n-i}(C') \;=\; H_i(C) \otimes _R H_{n-i}(C'). \] Summing over \(i\) yields an isomorphism \(H_n(C \otimes _R C') \cong \bigoplus _{i+j=n} H_i(C) \otimes _R H_j(C')\), which is the statement of the theorem since the Tor terms vanish (\(H_i(C) = C_i\) being free, hence flat).
For the general case, let \(Z_i \subseteq C_i\) and \(B_i \subseteq C_i\) denote the cycles and boundaries in degree \(i\), respectively. These give subchain complexes \(Z\) and \(B\) of \(C\) with trivial differentials. We have a short exact sequence of chain complexes \[ 0 \to Z \to C \to B[1] \to 0 \] arising from the short exact sequences \(0 \to Z_i \to C_i \to B_{i-1} \to 0\) in each degree. Each of these splits since \(B_{i-1}\) is free, being a submodule of the free module \(C_{i-1}\). Because of this splitting, tensoring with \(C'\) preserves exactness, and we obtain a long exact sequence in homology: \[ \cdots \to H_n(Z \otimes _R C') \to H_n(C \otimes _R C') \to H_{n-1}(B \otimes _R C') \xrightarrow {i_{n-1}} H_{n-1}(Z \otimes _R C') \to \cdots \] where \(i_k\colon H_k(B\otimes _R C')\to H_k(Z\otimes _R C')\) is induced by the inclusion \(B \hookrightarrow Z\).
Since \(Z\) and \(B\) are chain complexes with trivial differentials (and have free terms, being submodules of the free modules \(C_i\)), the special case treated before converts this to: \[ \cdots \xrightarrow {i_n} \bigoplus _{i + j = n} Z_i \otimes _R H_{j}(C') \to H_n(C \otimes _R C') \to \bigoplus _{i+j = n-1} B_i \otimes _R H_{j}(C') \xrightarrow {i_{n-1}} \bigoplus _{i+j = n-1} Z_i \otimes _R H_{j}(C') \to \cdots \] The long exact sequence thus provides short exact sequences \[ 0 \to \coker (i_{n}) \to H_n(C \otimes _R C') \to \ker (i_{n-1}) \to 0. \] To identify \(\coker (i_n)\) and \(\ker (i_{n-1})\), observe that the short exact sequence \(0 \to B_i \to Z_i \to H_i(C) \to 0\) functions as a free resolution of \(H_i(C)\). The long exact sequence on Tor-groups between \(H_i(C)\) and \(H_j(C')\) takes the form \[ 0 \to \Tor _1^R(H_i(C), H_{j}(C')) \to B_i \otimes _R H_{j}(C') \to Z_i \otimes _R H_{j}(C') \to H_i(C) \otimes _R H_{j}(C') \to 0. \] By taking the direct sum over all \(j\), this results in isomorphisms \[ \coker (i_{n}) \;\cong \; \bigoplus _{i+j = n} H_i(C) \otimes _R H_j(C') \qquadtext { and } \ker (i_{n-1}) \;\cong \; \bigoplus _{i + j = n-1} \Tor _1^R(H_i(C), H_j(C')). \] All maps used to construct this short exact sequence are functorial in chain maps. Through the projective-complex model of \(\D ^-(R)\) from Proposition 6.6.16, the sequence is therefore natural in \(A\) and \(B\).
Finally, we show that the short exact sequence splits. Since each \(C_i\) is free, the short exact sequence \(0 \to Z_i \to C_i \to B_{i-1} \to 0\) splits, and the quotient maps \(Z_i \to H_i(C)\) extend to homomorphisms \(C_i \to H_i(C)\). We similarly obtain homomorphisms \(C'_j \to H_j(C')\). Viewing the sequences of homology groups \(H_i(C)\) and \(H_j(C')\) as chain complexes with trivial differentials, we obtain chain maps \(C \to H(C)\) and \(C' \to H(C')\). Their tensor product is a chain map \(C \otimes _R C' \to H(C) \otimes _R H(C')\). Since the differentials on the target are trivial, the induced map on homology \[ H_n(C \otimes _R C') \to H_n(H(C) \otimes _R H(C')) \;=\; \bigoplus _{i+j=n} H_i(C) \otimes _R H_j(C') \] provides the desired splitting. □
An analogous argument yields the universal coefficient theorem for the derived hom:
Theorem 6.7.3 (Algebraic universal coefficient theorem). Let \(R\) be a principal ideal domain, let \(A \in \D ^-(R)\), and let \(M\) be an \(R\)-module, regarded as a complex concentrated in degree zero. For each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to \Ext ^1_R(H_{n-1}(A),M) \to H_{-n}(\bR \uHom _R(A,M[0])) \to \Hom _R(H_n(A),M) \to 0, \] and this sequence splits non-naturally.
Proof. We refer to [Weibel (1994), Theorem 3.6.5], applied to \(\uHom _R(C,M)\), which represents \(\bR \uHom _R(A,M[0])\). □
Example 6.7.4. For \(R = \Z \), the Künneth formula specializes to the classical result for abelian groups. Given chain complexes \(A\) and \(B\) of abelian groups, the homology of the derived tensor product fits into the short exact sequence \[ 0 \to \bigoplus _{i+j=n} H_i(A) \otimes _{\Z } H_j(B) \to H_n(A \otimes ^{\bL }_{\Z } B) \to \bigoplus _{i+j=n-1} \Tor _1^{\Z }(H_i(A), H_j(B)) \to 0. \] Since \(\Tor _1^{\Z }(M,N)\) consists of the common torsion of \(M\) and \(N\) (see Exercise 6.5.8), this shows that the homology of the derived tensor product is determined by the homology groups of \(A\) and \(B\), up to an extension involving their torsion.
Example 6.7.5. Let \(R\) be a PID and let \(A, B \in \D ^-(R)\). If all homology groups \(H_i(A)\) are flat (equivalently, torsion-free), then \[ H_n(A \otimes ^{\bL }_R B) \;\cong \; \bigoplus _{i+j=n} H_i(A) \otimes _R H_j(B). \]
Example 6.7.6. Let \(R\) be a PID, let \(A \in \D ^-(R)\), and let \(M\) be an \(R\)-module. If all homology groups \(H_i(A)\) are projective (equivalently, free), then \[ H_{-n}(\bR \uHom _R(A,M[0])) \;\cong \; \Hom _R(H_n(A),M). \]
6.7.2 Künneth theorem
The Künneth theorem describes the homology of a product of spaces in terms of the homology of the factors. The key observation is that the chain complex of a product is the derived tensor product of the chain complexes of the factors.
Proposition 6.7.7. Let \(R\) be a commutative ring. For animae \(X, Y \in \An \), there is a natural equivalence \[ C_*(X \times Y; R) \; \simeq \; C_*(X;R) \otimes ^{\bL }_R C_*(Y;R) \] in \(\D (R)\).
Proof. Chain complexes of animae are connective, so both sides are functors \(\An \times \An \to \D (R)_{\geq 0}\), and the derived tensor product of Definition 6.6.25 is defined on them. It preserves colimits in each variable: it is exact and preserves coproducts of uniformly bounded below families, and every colimit of connective objects is of this form. Similarly, the functor \(- \times -\colon \An \times \An \to \An \) preserves colimits in both variables, and \(C_*(-;R)\) preserves colimits. It follows that both sides define functors \(\An \times \An \to \D (R)_{\geq 0}\) preserving colimits in each variable. Both send \((\pt , \pt )\) to \(R[0]\), hence they are naturally equivalent by the universal property of \(\An \). □
Theorem 6.7.8 (Künneth theorem). Let \(R\) be a principal ideal domain and let \(X, Y \in \An \) be animae. For each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to \bigoplus _{i+j=n} H_i(X;R) \otimes _R H_j(Y;R) \to H_n(X \times Y;R) \to \bigoplus _{i+j=n-1} \Tor _1^R(H_i(X;R), H_j(Y;R)) \to 0, \] and this sequence splits, though not naturally.
Proof. By Proposition 6.7.7, we have \(H_n(X \times Y;R) \cong H_n(C_*(X;R) \otimes ^{\bL }_R C_*(Y;R))\). The result now follows immediately from the algebraic Künneth formula (Theorem 6.7.2). □
Corollary 6.7.9. If \(R\) is a principal ideal domain and \(H_i(X;R)\) is a free \(R\)-module for all \(i\), then \[ H_n(X \times Y;R) \,\cong \, \bigoplus _{i+j=n} H_i(X;R) \otimes _R H_j(Y;R) \] for all \(n \in \Z \) and all animae \(Y\). □
6.7.3 Universal coefficient theorems
The universal coefficient theorems express the (co)homology with coefficients in a module in terms of the integral (co)homology. We begin with the homological version.
Proposition 6.7.10. For an anima \(X \in \An \) and a bounded below chain complex \(A \in \D ^-(\Z )\), there is a natural equivalence \[ C_*(X;A) \,\simeq \, C_*(X;\Z ) \otimes ^{\bL }_{\Z } A \] in \(\D ^-(\Z )\).
Proof. Say \(A\) is concentrated in degrees \(\geq m\). Both \(C_*(-;A)\) and \(C_*(-;\Z ) \otimes ^{\bL }_{\Z } A\) then take values in \(\D (\Z )_{\geq m}\), which is closed under colimits in \(\D (\Z )\). Both preserve colimits: for the first this is the defining property from Construction 6.2.12, and for the second it follows as in Proposition 6.7.7, since \(C_*(-;\Z )\) is colimit-preserving with connective values and \(-\otimes ^{\bL }_{\Z }A\) is exact and preserves coproducts of uniformly bounded below families. Finally, both send \(\pt \) to \(A\). By the universal property of \(\An \), they are naturally equivalent. □
Theorem 6.7.11 (Universal coefficient theorem for homology). Let \(X \in \An \) be an anima and let \(A\) be an abelian group. For each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to H_n(X) \otimes _{\Z } A \to H_n(X;A) \to \Tor _1^{\Z }(H_{n-1}(X), A) \to 0, \] and this sequence splits, though not naturally.
Proof. By Proposition 6.7.10, we have \(H_n(X;A) \cong H_n(C_*(X;\Z ) \otimes ^{\bL } A)\). Viewing \(A\) as a chain complex concentrated in degree \(0\), the statement is thus an instance of the algebraic Künneth formula (Theorem 6.7.2), using that \(H_j(A) = 0\) for \(j \neq 0\) and \(H_0(A) = A\). □
For cohomology, we use the internal hom in \(\D (\Z )\) instead of the derived tensor product.
Proposition 6.7.12. For an anima \(X \in \An \) and a chain complex \(A \in \D (\Z )\), there is a natural equivalence \[ C^*(X;A) \,\simeq \, \bR \uHom _{\Z }(C_*(X;\Z ), A) \] in \(\D (\Z )\).
Proof. The chain complexes \(C_*(X;\Z )\) are connective. Thus every diagram in the image of the colimit-preserving functor \(C_*(-;\Z )\colon \An \to \D (\Z )_{\geq 0}\) is uniformly bounded below, and Corollary 6.6.36 shows that \(\bR \uHom _{\Z }(C_*(-;\Z ),A)\) preserves limits as a functor on \(\An \catop \). It sends \(\pt \) to \(\bR \uHom _{\Z }(\Z ,A) \simeq A\), so the universal property of \(\An \) identifies it with \(C^*(-;A)\). □
Theorem 6.7.13 (Universal coefficient theorem for cohomology). Let \(X \in \An \) be an anima and let \(A\) be an abelian group. For each \(n \in \Z \), there is a natural short exact sequence \[ 0 \to \Ext ^1_{\Z }(H_{n-1}(X), A) \to H^n(X;A) \to \Hom _{\Z }(H_n(X), A) \to 0, \] and this sequence splits, though not naturally.
Proof. By Proposition 6.7.12, we have \(H^n(X;A)\cong H_{-n}(\bR \uHom _{\Z }(C_*(X;\Z ),A))\). The result is therefore the case \(R=\Z \), \(M=A\) of Theorem 6.7.3. □
Corollary 6.7.14. If \(H_{n-1}(X)\) is a free abelian group, then \[ H^n(X;A) \,\cong \, \Hom _{\Z }(H_n(X), A) \] for all abelian groups \(A\). □
Exercises
Exercise 6.1 (Two-term resolutions, Tor, and Ext). For an integer \(n \geq 1\), let \(C(n)_{\bullet }\) be the chain complex of abelian groups with \(\Z \) in degrees \(1\) and \(0\), differential \[ d_1\colon C(n)_1 = \Z \xrightarrow {\cdot n} C(n)_0 = \Z , \] and zero in all other degrees.
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Compute the homology groups \(H_k(C(n))\).
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Show that the evident map \(C(n)_{\bullet } \to \Z /n\Z [0]\) is a quasi-isomorphism.
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Regard \(C(n)_{\bullet }\) as a projective resolution of \(\Z /n\Z \) and compute \(\Tor ^{\Z }_i(\Z /n\Z ,A)\) and \(\Ext ^i_{\Z }(\Z /n\Z ,A)\) for every abelian group \(A\) and every \(i\geq 0\).
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Specialize to \(A=\Z /m\Z \) and express the answer in terms of \(\gcd (m,n)\).
Exercise 6.2. Let \(C\) and \(D\) be stable \(\infty \)-categories with t-structures, let \(F\colon C \to D\) be an exact functor, and let \[ 0 \to A \hookrightarrow B \twoheadrightarrow C' \to 0 \] be a short exact sequence in the heart \(C^{\heartsuit }\). Construct a long exact sequence in \(D^{\heartsuit }\) of the form \[ \dots \to \pi _{n+1}F(C') \to \pi _nF(A) \to \pi _nF(B) \to \pi _nF(C') \to \pi _{n-1}F(A) \to \dots . \] Show that if \(F\) is t-exact, this long exact sequence collapses to the original short exact sequence in the heart.
Exercise 6.3 (Truncating a two-term complex). Let \(\Aa \) be an abelian category and let \(C_{\bullet }\) be the complex \[ \cdots \longrightarrow 0\longrightarrow A_1\xrightarrow {d}A_0\longrightarrow 0\longrightarrow \cdots \] concentrated in degrees \(1\) and \(0\). Describe \(\tau _{\geq 1}C_{\bullet }\) and \(\tau _{\leq 0}C_{\bullet }\) explicitly, and identify the canonical exact sequence \[ \tau _{\geq 1}C_{\bullet }\longrightarrow C_{\bullet }\longrightarrow \tau _{\leq 0}C_{\bullet } \] in terms of \(\ker (d)\) and \(\coker (d)\).
Exercise 6.4 (Chains on spheres). Use the construction of ordinary chains from Construction 6.2.12.
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Show that the reduced chain functor \(\widetilde {C}_*(-;\Z )\colon \An _* \to \D (\Z )\) sends cofiber sequences of pointed animae to exact sequences in \(\D (\Z )\).
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Deduce that \(\widetilde {C}_*(\Sigma X;\Z ) \cong \widetilde {C}_*(X;\Z )[1]\).
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Compute \(\widetilde {C}_*(S^n;\Z )\) and \(C_*(S^n;\Z )\) in \(\D (\Z )\).
Exercise 6.5 (Extensions and addition in \(\Ext ^1\)). Let \(C\) be a stable \(\infty \)-category with a t-structure, and let \(M,N \in C^{\heartsuit }\).
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Starting from a morphism \(\phi \colon M \to N[1]\), form its fiber and show that this gives an extension \[ 0 \to N \to E_{\phi } \to M \to 0 \] in \(C^{\heartsuit }\).
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Starting from an extension of \(M\) by \(N\), construct the corresponding morphism \(M \to N[1]\).
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Check that these two constructions are inverse to each other on isomorphism classes.
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Describe the addition law on extensions which corresponds to the abelian group structure on \(\Ext ^1_C(M,N)\).
Exercise 6.6 (A derived functor from a homotopy invariant functor). Let \(F\colon \Top \to D\) be a functor to an \(\infty \)-category \(D\) which sends homotopy equivalences of topological spaces to isomorphisms.
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Using CW-complexes as cofibrant objects, explain how Proposition 6.6.3 produces a total left derived functor \[ \bL F\colon \An \to D. \]
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Show that if \(X\) is a CW-complex, then \(\bL F(\Pi _{\infty }X) \cong F(X)\).
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Explain why, if \(F\) already sends weak homotopy equivalences to isomorphisms, then \(\bL F\) is just the functor induced by the universal property of the localization \(\Top \to \An \).
For the next two exercises, you may use that \(H_k(\CP ^m)\cong \Z \) for \(k=0,2,\ldots ,2m\) and vanishes otherwise, while \(H_0(\RP ^2)\cong \Z \), \(H_1(\RP ^2)\cong \Z /2\Z \), and all other homology groups of \(\RP ^2\) vanish.
Exercise 6.7 (Künneth computations). Use Theorem 6.7.8 to compute the integral homology groups of the following products:
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\(S^m \times S^n\).
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\(\CP ^m \times S^n\).
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\(\RP ^2 \times \RP ^2\).
In each case, indicate whether the \(\Tor _1^{\Z }\) term vanishes and why.
Exercise 6.8 (Universal coefficient computations). Use Theorem 6.7.11 and Theorem 6.7.13.
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Compute \(H_*(\RP ^2;\Z /2\Z )\) from the integral homology of \(\RP ^2\).
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Compute \(H^*(\RP ^2;\Z )\).
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Compute \(H^*(\RP ^2;\Z /2\Z )\) and identify the contribution of the \(\Ext ^1_{\Z }\) term.
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