Notes:Chain complex of modules

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Notes

Chain complex of modules

A chain complex of modules is an infinite sequence:

• [ilmath]\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 } [/ilmath]
  • Caution:Sequences.... start from [ilmath]0[/ilmath]? Or something, so not sure what's going on here - Will resolve later

of modules, [ilmath]C_i[/ilmath] and boundary homomorphisms, [ilmath]\partial_i[/ilmath] such that:

• [ilmath]\partial_n\circ\partial_{n+1}=0[/ilmath]

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

• [ilmath]C_0\leftarrow C_1 \leftarrow \cdots[/ilmath]

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

• [ilmath]C^0\rightarrow C^1\rightarrow \cdots[/ilmath]

With [ilmath]C^n:=C_{-n}[/ilmath] and module homomorphisms [ilmath]\delta^n:=\partial_{-n}:C^n\rightarrow C^{n+1}[/ilmath]

Homology module

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

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

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

• Caution:What on Earth is this [ilmath]\text{cls} [/ilmath] business...?

Chain transformation

Let [ilmath]\mathcal{A} [/ilmath] and [ilmath]\mathcal{B} [/ilmath] be chain complexes of modules. A chain transformation:

• [ilmath]\varphi:\mathcal{A}\rightarrow\mathcal{B} [/ilmath]

is a family of module homomorphisms:

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

Diagramatically:

[ilmath]\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 }[/ilmath]

For example every continuous map [ilmath]f:X\rightarrow Y[/ilmath] induces a chain transformation [ilmath]\mathcal{C}(f):\mathcal{C}(X)\rightarrow\mathcal{C}(Y)[/ilmath] 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 [ilmath]\varphi:\mathcal{A}\rightarrow\mathcal{B} [/ilmath] be a chain transformation.