\( \usepackage{mathrsfs} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\cHom}{\underline{Hom}} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\inj}{inj} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\coim}{coim} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\eq}{eq} \DeclareMathOperator{\op}{op} \DeclareMathOperator{\ob}{ob} \DeclareMathOperator{\coeq}{coeq} \DeclareMathOperator{\cone}{cone} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\Ch}{Ch} \DeclareMathOperator{\Ab}{\text{Ab}} \DeclareMathOperator{\Top}{\text{Top}} \DeclareMathOperator{\Rmod}{R\text{mod}} \DeclareMathOperator{\SSet}{\text{SSet}} \DeclareMathOperator{\sAb}{\text{sAb}} \DeclareMathOperator{\cA}{\mathcal{A}} \DeclareMathOperator{\cB}{\mathcal{B}} \DeclareMathOperator{\cC}{\mathcal{C}} \DeclareMathOperator{\cO}{\mathcal{O}} \DeclareMathOperator{\cM}{\mathcal{M}} \DeclareMathOperator{\sC}{\mathscr{C}} \DeclareMathOperator{\sF}{\mathscr{F}} \DeclareMathOperator{\sW}{\mathscr{W}} \DeclareMathOperator{\dd}{\partial} \DeclareMathOperator{\rd}{\mathrm{d}} % Rings \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\CC}{\mathbf{C}} \DeclareMathOperator{\PP}{\mathbf{P}} \DeclareMathOperator{\QQ}{\mathbf{Q}} \DeclareMathOperator{\RR}{\mathbf{R}} \DeclareMathOperator{\ZZ}{\mathbf{Z}} \)

Section1.4Chain complexes

Definition1.4.1Chain complexes
A chain complex\(C_\bullet\) is a collection of objects \((C_i)_{i\in \ZZ}\) in \(\cA\) with maps \(d_i\colon C_i \to C_{i-1}\) such that \(d_{i-1}\circ d_i = 0\).
Definition1.4.2Cycles, boundaries, homology objects
We define the cycles\(Z_i = \ker d_i\) and boundaries\(B_i= \im d_{i+1}\) and the \(i\)th homology object\(H_i(C) = \coker(B_i\to Z_i)\). A complex is acyclic if it is exact i.e. \(H_\bullet(C) = 0\).
Definition1.4.3Cochain complexes
A cochain complex\(C^\bullet\) is a collection of objects \((C^i)_{i\in \ZZ}\) in \(\cA\) with maps \(d_i\colon C_i \to C_{i+1}\) such that \(d_{i+1}\circ d_i = 0\). We then have as above \(H^i\) the \(i\)th cohomology object.

We can switch between chain complexes and cochain complexes via \(C^i = C_{-i}\).

Example1.4.4
We have many such complexes:
  • Singular (co-)chain complex on a top space.
  • de Rahm complex.
  • Cellular chain complex.
  • Flabby resolution of a sheaf.
  • Bar resolution of a group.
  • Koszul complex.
Definition1.4.5Chain maps
Given \(B,C\) chain complexes, a chain map\(f\colon B\to C\) is a collection of maps \(f_i\colon B_i \to C_i\) such that \(df=fd\).

We now have formed the category of chain complexes \(\Ch(\cA)\) using these maps. We write \(\Ch(R)\) for \(\Ch(\Rmod)\). Note that \(\Ch(\cA)\) is an additive category moreover it is an abelian category, we can define and check everything level-wise. For example \(\ker(A\to B)_n = \ker(A_n\to B_n)\). Note that the \(H_n\) form a functor \(\Ch(\cA)\to \cA\). Define \(f_*\colon H_n A \to H_n B\) in the natural way and check it works. \(H_n\) is additive.

Naturality here means given two short exact sequences and compatible chain maps the induced maps on homology are compatible with \(\dd_n\). (The obvious diagram commutes.)

Recall that \(f\) is a chain map if \(\dd f - f\dd = 0\).

Definition1.4.8
Let \(\cHom_n(A,B)\) consist of functions \(\{f_i \colon A_i \to B_{i+n}\} \) and define \(df = d\cdot f - (-1)^n fd\) if \(f\in\cHom_n\).

Check that \[d^2 f = d\cdot (d\cdot f - (-1)^fd ) - (-1) (d\cdot f - (-1)^n f\cdot d)\cdot d = 0.\]

We use the ``Sign rule'' to help with definitions, this states that if \(a\) moves past \(b\) we pick a sign \((-1)^{\deg a\deg b}\).

\(\Ch(\cA)\) can be enriched over \(\Ch(\ZZ)\).

Definition1.4.9Shifted complexes
The shifted complex\(C[n]\) for \(C\in \Ch(\cA)\) is defined by \(C[n]_i = C_{n+i}\) and \(d_i^{C[n]} = (-1)^n d_{n+i}^C\).

Note that \(H_i(C) = H_0(C[i])\).

So a chain map \(f\colon A \to B[n]\) is exactly a cycle in \(\cHom_n(A,B)\).

Now \(\Hom(A,B) = Z_0(\cHom(A,B))\), so what is \(H_0(\cHom(A,B))\)?

Definition1.4.10Chain homotopies
A chain homotopy\(S\) between chain maps \(f,g\colon A \to B\) is a collection \(S_i \colon A_i \to B_i\) such that \(\dd S + S\dd = f-g\). Equivalently we could say a map \(A \to B[1]\) such that \(dS = g -f\) (note: not a chain map). We write \(f \simeq g\) to denote the fact that \(f\) is chain homotopic to \(g\).
Definition1.4.11Chain homotopy equivalences
Two chain complexes \(A\) and \(B\) are said to be chain homotopy equivalent if there are some \(f\colon A\to b\), \(g\colon B \to A\) such that \(gf \simeq 1_A\) and \(fg\simeq 1_B\).
Definition1.4.13Quasi-isomorphisms
A chain map \(f\) inducing isomorphisms on homology is called a quasi-isomorphism. Two chains \(A,B\) are quasi-isomorphic if there is a quasi-isomorphism \(A \to B\) and \(B\to A\).
Example1.4.14
\[ \xymatrix{ \dots\ar[r] &\ZZ \ar[r]\ar[d]_n &0 \ar[r]\ar[d] &\dots \\ \dots\ar[r] &\ZZ \ar[r]^{\text{pr}} &\ZZ \ar[r] &\dots } \]
is a quasi-isomorphism.

Any chain homotopy equivalence is a quasi-isomorphism, the converse is false however.

Definition1.4.16Cones
Given \(f\colon A \to B\) we define a chain complex called the cone of \(f\) by \(\cone(f)_n = A_{n-1} \oplus B_n\) with maps \[d = \begin{pmatrix}-d_A & 0 \\ -f & d_B\end{pmatrix}.\]

Note that there exists a short exact sequence \[ \xymatrix{B \ar[r]_{b\mapsto (0,b)}& \cone(f) \ar[r]_{(a,b)\mapsto -a}& A[-1].} \] Doing the diagram chase of the Snake lemma 1.4.6 we see that the boundary map is induced by \(f\) on homology i.e. \[f_*\colon H_{n-1}A \to H_{n-1}B.\] This proves the following.