Notes:Chain complex of modules

These come from.

Notes

Chain complex of modules

A chain complex of modules is an infinite sequence:

• $\mathcal{C}: \xymatrix { \cdots \ar@{<-}[r] & C_{n-1} \ar@{<-}[r]^{\partial_n} & C_n \ar@{<-}[r]^{\partial_{n+1} } & C_{n+1} \ar@{<-}[r] & \cdots }$
  • Caution:Sequences.... start from $0$? Or something, so not sure what's going on here - Will resolve later

of modules, $C_i$ and boundary homomorphisms, $\partial_i$ such that:

• $\partial_n\circ\partial_{n+1}=0$

A positive complex $\mathcal{C}$ has $C_n=0$ (the trivial module) for all $n<0$ and is usually written:

• $C_0\leftarrow C_1 \leftarrow \cdots$

A negative complex $\mathcal{C}$ has $C_n=0$ for $n>0$ and is usually re-written for convenience as a positive complex:

• $C^0\rightarrow C^1\rightarrow \cdots$

With $C^n:=C_{-n}$ and module homomorphisms $\delta^n:=\partial_{-n}:C^n\rightarrow C^{n+1}$

Homology module

Let $\mathcal{C}$ be a chain complex of modules. The $n$^th homology module of $\mathcal{C}$ is:

• $H_n(\mathcal{C}):=\text{Ker}(\partial_n)/\text{Im}(\partial_{n+1})$

We denote the homology class of $x\in\text{Ker}(\partial_n)$ by $\text{cls }z:=z+\text{Im}(\partial_{n+1})$

• Caution:What on Earth is this $\text{cls}$ business...?

Chain transformation

Let $\mathcal{A}$ and $\mathcal{B}$ be chain complexes of modules. A chain transformation:

• $\varphi:\mathcal{A}\rightarrow\mathcal{B}$

is a family of module homomorphisms:

• $\varphi_n:A_n\rightarrow B_n$ such that:
  • $\forall n\in\text{SOMETHING? Z? N?}[\partial^\mathcal{B}_n\circ\varphi_n=\varphi_{n-1}\circ\partial_n^\mathcal{A}]$

Diagramatically:

$\xymatrix{ \mathcal{A}:\cdots \ar@<-2ex>[d]_\varphi \ar@{<-}[r] & A_{n-1} \ar[d]_{\varphi_{n-1} } \ar@{<-}[r] & A_n \ar@{<-}[r] \ar[d]_{\varphi_{n} } & A_{n+1} \ar@{<-}[r] \ar[d]_{\varphi_{n+1} } & \cdots \\ \mathcal{B}: \cdots \ar@{<-}[r] & B_{n-1} \ar@{<-}[r] & B_n \ar@{<-}[r] & B_{n+1} \ar@{<-}[r] & \cdots }$

For example every continuous map $f:X\rightarrow Y$ induces a chain transformation $\mathcal{C}(f):\mathcal{C}(X)\rightarrow\mathcal{C}(Y)$ of their singular chain complexes.

In general chain transformations can be added and composed componentwise. The results of which are also chain transformations.

Every chain transformation induces a homomorphism between homology modules

Let $\varphi:\mathcal{A}\rightarrow\mathcal{B}$ be a chain transformation.