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2.5.1 Generalities on Chain Complexes

In this section, we provide a brief review of some of the homological algebra which will be needed throughout §2.5.

Definition 2.5.1.1. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$, and let $n$ be an integer. We will say that $C_{\ast }$ is concentrated in degrees $\geq n$ if objects $C_{m} \in \operatorname{\mathcal{A}}$ are zero for $m < n$. Similarly, we say that $C_{\ast }$ is concentrated in degrees $\leq n$ if the objects $C_ m$ are zero for $m > n$. We let $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\geq n}$ denote the full subcategory of $\operatorname{Ch}(\operatorname{\mathcal{A}})$ spanned by those chain complexes which are concentrated in degree $n$, and $\operatorname{Ch}(\operatorname{\mathcal{A}})_{\leq n}$ the full subcategory spanned by those chain complexes which are concentrated in degrees $\leq n$.

Example 2.5.1.2. Let $\operatorname{\mathcal{A}}$ be an additive category, let $C \in \operatorname{\mathcal{A}}$ be an object, and let $n$ be an integer. We will write $C[n]$ for the chain complex given by

\[ C[n]_{\ast } = \begin{cases} C & \text{ if } \ast = n \\ 0 & \text{ otherwise, } \end{cases} \]

where each differential is the zero morphism. Note that a chain complex $M_{\ast }$ is isomorphic to $C[n]$ (for some object $C \in \operatorname{\mathcal{A}}$) if and only if it both concentrated in degrees $\geq n$ and concentrated in degrees $\leq n$.

Notation 2.5.1.3 (Cycles and Boundaries). Let $\operatorname{\mathcal{A}}$ be an abelian category (Definition ) and let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$. For each integer $n$, we let $\mathrm{Z}_{n}(C)$ denote the kernel of the boundary operator $\partial : C_ n \rightarrow C_{n-1}$, and $\mathrm{B}_ n(C)$ the image of the boundary operator $\partial : C_{n+1} \rightarrow C_ n$. We regard $\mathrm{Z}_ n(C)$ and $\mathrm{B}_{n}(C)$ as subobjects of $C_ n$. Note that we have $\mathrm{B}_{n}(C) \subseteq \mathrm{Z}_{n}(C)$ (this is a reformulation of the identity $\partial ^2 = 0$).

In the special case where $\operatorname{\mathcal{A}}= \operatorname{ Ab }$ is the category of abelian groups, we will refer to the elements of $C_{n}$ as $n$-chains of $C_{\ast }$, to the elements of $\mathrm{Z}_{n}(C)$ as $n$-cycles of $C_{\ast }$, and to the elements of $\mathrm{B}_{n}(C)$ as $n$-boundaries of $C_{\ast }$.

Definition 2.5.1.4 (Homology). Let $\operatorname{\mathcal{A}}$ be an abelian category and let $C_{\ast }$ be a chain complex with values in $\operatorname{\mathcal{A}}$. For every integer $n$, we let $\mathrm{H}_{n}( C )$ denote the quotient $\mathrm{Z}_{n}(C) / \mathrm{B}_{n}(C)$. We will refer to $\mathrm{H}_{n}(C)$ as the $n$th homology of the chain complex $C_{\ast }$. We say that the chain complex $C_{\ast }$ is acyclic if the homology objects$\mathrm{H}_{n}(C)$ vanish for every integer $n$.

If $\operatorname{\mathcal{A}}= \operatorname{ Ab }$ is the category of abelian groups and If $x \in \mathrm{Z}_ n(C)$ is an $n$-cycle of $C_{\ast }$, we let $[ x ]$ denote its image in the homology group $\mathrm{H}_{n}(C)$: we refer to $[x]$ as the homology class of $x$. We say that a pair of $n$-cycles $x,x' \in \mathrm{Z}_{n}(C)$ are homologous if $[ x ] = [ x' ]$: that is, if there exists an $(n+1)$-chain $y$ satisfying $x' = x + \partial (y)$.

Definition 2.5.1.5 (Quasi-Isomorphisms). Let $\operatorname{\mathcal{A}}$ be an abelian category, let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in $\operatorname{\mathcal{A}}$, and let $f: C_{\ast } \rightarrow D_{\ast }$ be a chain map. We say that $f$ is a quasi-isomorphism if, for every integer $n$, the induced map of homology objects $\mathrm{H}_{n}(C) \rightarrow \mathrm{H}_{n}(D)$ is an isomorphism.

Remark 2.5.1.6. Let $C_{\ast }$ be a chain complex with values in an abelian category $\operatorname{\mathcal{A}}$. In practice, the homology objects $\mathrm{H}_{\ast }(C)$ are often primary objects of interest, while the chain complex $C_{\ast }$ itself plays an ancillary role. The terminology of Definition 2.5.1.5 emphasizes this perspective: a chain map $f: C_{\ast } \rightarrow D_{\ast }$ which induces an isomorphism on homology should allow us to view the chain complexes $C_{\ast }$ and $D_{\ast }$ as “the same” for many purposes (this idea is the starting point for Verdier's theory of derived categories, which we will discuss in §).

Remark 2.5.1.7 (Two-out-of-Three). Let $\operatorname{\mathcal{A}}$ be an abelian category and suppose we are given a commutative diagram of chain complexes

\[ \xymatrix { 0 \ar [r] & C'_{\ast } \ar [d]^{f'} \ar [r] & C_{\ast } \ar [d]^{f} \ar [r] & C''_{\ast } \ar [d]^{f''} \ar [r] & 0 \\ 0 \ar [r] & D'_{\ast } \ar [r] & D_{\ast } \ar [r] & D''_{\ast } \ar [r] & 0 } \]

in which the rows are exact. If any two of the chain maps $f$, $f'$, and $f''$ are quasi-isomorphisms, then so is the third. This follows by comparing the long exact homology sequences associated to the upper and lower rows (see ).

Proposition 2.5.1.8. Let $C_{\ast }$ and $D_{\ast }$ be chain complexes with values in an abelian category $\operatorname{\mathcal{A}}$, and let $f,f': C_{\ast } \rightarrow D_{\ast }$ be a pair of chain maps. If $f$ and $f'$ are chain homotopic, then they induce the same map from $\mathrm{H}_{n}(C)$ to $\mathrm{H}_{n}(D)$ for every integer $n$.

Proof. Let $h = \{ h_ m \} _{m \in \operatorname{\mathbf{Z}}}$ be a chain homotopy from $f$ to $f'$, so that $f'_{n} - f_ n = \partial _{D} \circ h_{n} + h_{n-1} \circ \partial _{C}$. It follows that, when restricted to the subobject $\mathrm{Z}_{n}(C) \subseteq C_ n$, the difference $f'_{n} -f_ n = \partial _{D} \circ h_ n$. factors through the subobject $\mathrm{B}_{n}(D) \subseteq \mathrm{Z}_{n}(D)$, so the induced maps $\mathrm{H}_{n}(f), \mathrm{H}_{n}(f'): \mathrm{H}_{n}(C) \rightarrow \mathrm{H}_{n}(D)$ are the same. $\square$

Corollary 2.5.1.9. Let $f: C_{\ast } \rightarrow D_{\ast }$ be a chain map between chain complexes with values in an abelian category $\operatorname{\mathcal{A}}$. If $f$ is a chain homotopy equivalence, then it is a quasi-isomorphism.

For later use, we record the following elementary fact:

Proposition 2.5.1.10. Let $P_{\ast }$ be a chain complex taking values in an abelian category $\operatorname{\mathcal{A}}$. Assume that $P_{\ast }$ is acyclic, concentrated in degrees $\geq 0$, and that each $P_{n}$ is a projective object of $\operatorname{\mathcal{A}}$. Then $P_{\ast }$ is a projective object of the category $\operatorname{Ch}(\operatorname{\mathcal{A}})$. In other words, every epimorphism of chain complexes $f: M_{\ast } \twoheadrightarrow P_{\ast }$ admits a section.

Proof. Our assumption that $P_{\ast }$ is acyclic guarantees that for every integer $n \geq 0$, we have a short exact sequence

\[ 0 \rightarrow \mathrm{Z}_{n}(P) \rightarrow P_{n} \xrightarrow {\partial } \mathrm{Z}_{n-1}(P) \rightarrow 0. \]

It follows by induction on $n$ that each of these exact sequences splits and that each $\mathrm{Z}_ n(P)$ is also a projective object of $\operatorname{\mathcal{A}}$. We can therefore choose a direct sum decomposition $P_{n} \simeq \mathrm{Z}_ n(P) \oplus Q_ n$, where the differential on $P_{\ast }$ restricts to isomorphisms $\partial : Q_{n} \simeq \mathrm{Z}_{n-1}(P)$. Since each $Q_{n}$ is projective and $f$ is an epimorphism in each degree, we can choose maps $u_{n}: Q_{n} \rightarrow M_ n$ for which the composition $f_ n \circ u_ n$ equal to the identity on $Q_{n}$. The maps $u_ n$ then extend uniquely to a map of chain complexes $s = \{ s_ n \} _{n \in \operatorname{\mathbf{Z}}}$, characterized by the requirement that each composition

\[ Q_{n+1} \oplus Q_{n} \xrightarrow { \partial \oplus \operatorname{id}} \mathrm{Z}_{n}(P) \oplus Q_ n = P_ n \xrightarrow {s_ n} M_ n \]

is the sum of the maps $u_{n+1}$ and $u_ n$. $\square$

We now specialize our attention to the category $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ of chain complexes of abelian groups, which we will endow with a monoidal structure.

Notation 2.5.1.11. Let $C_{\ast }$ and $D_{\ast }$ be graded abelian groups. We define a new graded abelian group $(C \boxtimes D)_{\ast } = C_{\ast } \boxtimes D_{\ast }$ by the formula

\[ (C \boxtimes D)_{n} = \bigoplus _{n = n' + n''} C_{n'} \otimes D_{n''}. \]

Here the direct sum is taken over the set $\{ (n', n'') \in \operatorname{\mathbf{Z}}\times \operatorname{\mathbf{Z}}: n = n' + n'' \} $ of all decompositions of $n$ as a sum of two integers $n'$ and $n''$, and $C_{n'} \boxtimes D_{n''}$ denotes the tensor product of $C_{n'}$ with $D_{n''}$ (formed in the category of abelian groups). For every pair of elements $x \in C_{m}$ and $y \in D_{n}$, we let $x \boxtimes y$ denote the image of the pair $(x,y)$ under the canonical map

\[ C_{m} \times D_{n} \rightarrow C_ m \otimes D_{n} \hookrightarrow (C \boxtimes D)_{m+n}. \]

Proposition 2.5.1.12. Let $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ be chain complexes. Then there is a unique homomorphism of graded abelian groups

\[ \partial : (C \boxtimes D)_{\ast } \rightarrow (C \boxtimes D)_{\ast - 1} \]

satisfying the identity

\[ \partial ( x \boxtimes y) = (\partial (x) \boxtimes y) + (-1)^{m} (x \boxtimes \partial (y)) \]

for $x \in C_{m}$ and $y \in D_{n}$. Moreover, this homomorphism satisfies $\partial ^2 = 0$, so we can regard the pair $( (C \boxtimes D)_{\ast }, \partial )$ as a chain complex.

Proof. For every pair of integers $m,n \in \operatorname{\mathbf{Z}}$, the construction

\[ (x,y) \mapsto (\partial x \boxtimes y) + (-1)^{m} (x \boxtimes \partial y) \]

determines a bilinear map $C_ m \times D_{n} \rightarrow (C \boxtimes D)_{m+n-1}$. Invoking the universal property of tensor products and direct sums, we deduce that there is a unique map $\partial : (C \boxtimes D)_{\ast } \rightarrow (C \boxtimes D)_{\ast -1}$ with the desired properties. The identity $\partial ^2 = 0$ follows from the calculation

\begin{eqnarray*} \partial ^2(x \boxtimes y) & = & \partial ( (\partial x \boxtimes y) + (-1)^{m} (x \boxtimes \partial y) ) \\ & = & (\partial ^2 x \boxtimes y) + (-1)^{m-1} (\partial x \boxtimes \partial y) + (-1)^{m} ( \partial x \boxtimes \partial y) + (-1)^{2m} ( x \boxtimes \partial ^2 y) \\ & = & 0. \end{eqnarray*}
$\square$

Notation 2.5.1.13. In the situation of Proposition 2.5.1.12, we will refer to $( (C \boxtimes D)_{\ast }, \partial )$ as the tensor product of the chain complexes $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$.

Warning 2.5.1.14 (The Koszul Sign Rule). Let $(C_{\ast }, \partial )$ and $( D_{\ast }, \partial )$ be chain complexes. There is a unique isomorphism of graded abelian groups $\tau : C_{\ast } \boxtimes D_{\ast } \rightarrow D_{\ast } \boxtimes C_{\ast }$ satisfying $\tau (x \boxtimes y) = y \boxtimes x$ for all $x \in C_{m}$, $y \in C_{n}$. Beware that $\tau $ is usually not a chain map: we have

\[ \partial \tau (x \boxtimes y) = \partial (y \boxtimes x) = (\partial y \boxtimes x) + (-1)^{n} (y \boxtimes \partial x) \]
\[ \tau ( \partial (x \boxtimes y) ) = \tau ( (\partial x \boxtimes y) + (-1)^{m} (x \boxtimes \partial y) ) = (-1)^{m} (\partial y \boxtimes x) + (\partial x \boxtimes y). \]

This can be remedied by modifying the isomorphism $\tau $: there is another isomorphism of graded abelian groups

\[ \sigma : C_{\ast } \boxtimes D_{\ast } \simeq D_{\ast } \boxtimes C_{\ast } \quad \quad \sigma (x \boxtimes y) = (-1)^{mn} (y \boxtimes x). \]

The isomorphism of $\sigma $ is a chain map (hence an isomorphism of chain complexes) by virtue of the calculation

\begin{eqnarray*} \partial \sigma (x \boxtimes y) & = & \partial ( (-1)^{mn} y \boxtimes x) \\ & = & (-1)^{mn} (\partial y \boxtimes x) + (-1)^{mn+n} (y \boxtimes \partial x) \\ & = & (-1)^{n} \sigma (x \boxtimes \partial y) + \sigma (\partial x \boxtimes y) \\ & = & \sigma ( \partial ( x \boxtimes y) ). \end{eqnarray*}

Exercise 2.5.1.15 (Universal Property of the Tensor Product). Let $(C_{\ast }, \partial )$, $(D_{\ast }, \partial )$, and $(E_{\ast }, \partial )$ be chain complexes. We will say that a collection of bilinear maps

\[ \{ f_{m,n}: C_{m} \times D_{n} \rightarrow E_{m+n} \} _{m,n \in \operatorname{\mathbf{Z}}} \]

satisfies the Leibniz rule if, for every pair of elements $x \in C_ m$ and $y \in D_{n}$, the identity

\[ \partial f_{m,n}(x,y)= f_{m-1,n}( \partial x, y ) + (-1)^{m} f_{m,n-1}( x, \partial y ) \]

holds in the abelian group $E_{m+n-1}$. Show that there is a canonical bijection from the collection of chain maps $f: C_{\ast } \boxtimes D_{\ast } \rightarrow E_{\ast }$ to the collection of systems of bilinear maps $\{ f_{m,n}: C_ m \times D_{n} \rightarrow E_{m+n} \} _{m,n \in \operatorname{\mathbf{Z}}}$ satisfying the Leibniz rule, given by the construction $f_{m,n}(x,y) = f( x \boxtimes y)$.

Remark 2.5.1.16 (Associativity Isomorphisms). Let $(C_{\ast }, \partial )$, $(D_{\ast }, \partial )$, and $(E_{\ast }, \partial )$ be chain complexes of abelian groups. Then there is a unique isomorphism of graded abelian groups

\[ \alpha : C_{\ast } \boxtimes (D_{\ast } \boxtimes E_{\ast } ) \rightarrow (C_{\ast } \boxtimes D_{\ast }) \boxtimes E_{\ast } \]

satisfying the identity $\alpha ( x \boxtimes (y \boxtimes z) ) = (x \boxtimes y) \boxtimes z$. Moreover, $\alpha $ is an isomorphism of chain complexes: this follows from observation that $\alpha ( \partial (x \boxtimes (y \boxtimes z) ) )$ and $\partial \alpha ( x \boxtimes (y \boxtimes z) )$ are both given by the sum

\[ (\partial x \boxtimes y) \boxtimes z + (-1)^{m} (x \boxtimes \partial y) \boxtimes z + (-1)^{m+n} ((x \boxtimes y) \boxtimes \partial z) \]

for $x \in C_{m}$, $y \in D_{n}$, $z \in E_{p}$.

Construction 2.5.1.17 (The Monoidal Structure on Chain Complexes). Let $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ denote the category of chain complexes of abelian groups (Definition 2.5.0.3). We define a monoidal structure on $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ as follows:

  • The tensor product functor $\boxtimes : \operatorname{Ch}(\operatorname{\mathbf{Z}}) \times \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$ carries each pair of chain complexes $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ to the tensor product chain complex $( C_{\ast } \boxtimes D_{\ast }, \partial )$ of Proposition 2.5.1.12, and carries a pair of chain maps $f: C_{\ast } \rightarrow C'_{\ast }$, $g: D_{\ast } \rightarrow D'_{\ast }$ to the tensor product map

    \[ (f \boxtimes g): C_{\ast } \boxtimes D_{\ast } \rightarrow C'_{\ast } \boxtimes D'_{\ast } \quad \quad (f \boxtimes g)(x \boxtimes y) = f(x) \boxtimes g(y). \]
  • For every triple of chain complexes $C = (C_{\ast }, \partial )$, $D = (D_{\ast }, \partial )$, and $E = (E_{\ast }, \partial )$, the associativity constraint

    \[ \alpha _{C,D,E}: C_{\ast } \boxtimes (D_{\ast } \boxtimes E_{\ast }) \simeq (C_{\ast } \boxtimes D_{\ast }) \boxtimes E_{\ast } \]

    is the isomorphism of Remark 2.5.1.16.

  • The unit object of $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ is the chain complex $\operatorname{\mathbf{Z}}[0]$ of Example 2.5.1.2, and the unit constraint $\upsilon : \operatorname{\mathbf{Z}}[0] \boxtimes \operatorname{\mathbf{Z}}[0] \simeq \operatorname{\mathbf{Z}}[0]$ is the isomorphism classified by the bilinear map

    \[ \operatorname{\mathbf{Z}}\times \operatorname{\mathbf{Z}}\rightarrow \operatorname{\mathbf{Z}}\quad \quad (m,n) \mapsto mn. \]

Remark 2.5.1.18. Let $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ be chain complexes. The tensor product chain complex $(C_{\ast } \boxtimes D_{\ast }, \partial )$ of Proposition 2.5.1.12 is characterized up to (unique) isomorphism by the universal property of Exercise 2.5.1.15. However, the construction of this tensor product complex (and, by extension, the monoidal structure on $\operatorname{Ch}(\operatorname{\mathbf{Z}})$) depends on auxiliary choices. These choices are ultimately irrelevant in the sense that they do not change the isomorphism class of the monoidal category $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ or, equivalently, of the classifying simplicial set $B_{\bullet } \operatorname{Ch}(\operatorname{\mathbf{Z}})$ of Example 2.3.1.18. This simplicial set can be described concretely (without auxiliary choices): its $n$-simplices can be identified with systems of chain complexes $\{ C(j,i)_{\ast } \} _{0 \leq i < j \leq n}$ together with bilinear maps

\[ C(k,j)_{q} \times C(j,i)_{p} \rightarrow C(k,i)_{q+p} \quad \quad (y,z) \mapsto yz \]

for $0 \leq i < j < k \leq n$ which satisfy the Leibniz rule $\partial (yz) = (\partial y)z + (-1)^{q} y(\partial z)$ together with the associative law $x(yz) = (xy)z$ for $x \in C(\ell ,k)_{r}$, $y \in C(k,j)_{q}$, $z \in C(j,i)_{p}$ with $0 \leq i < j < k < \ell \leq n$.