Homology

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 29 September 2012

Complexes

Let $A$ be a ring and let $𝒜$ be the category of $A$-modules. By a theorem of Freyd (see [Benson I p. 22]) this setup is equivalent to letting $𝒜$ be an abelian category.

A complex of $A$-modules is a $\mathbb{Z}$-graded $A$-module $C$, with a morphism $d:C⟶C$ such that $$\cdots ⟶C_{n+1}\stackrel{d_{n+1}}{⟶}{C}_n\stackrel{d_n}{⟶}{C}_{n–1}⟶\cdots \text{such that}\mathrm{deg}d=-1\text{and}d\circ d=0.$$

A morphism is a graded $A$-module homomorphism $u:C⟶C\prime$ such that $\mathrm{deg}u=0$ and $d\prime \circ u=u\circ d$.

The homology of a complex $C$ is $$H\left(C\right)=\frac{Z\left(C\right)}{B\left(C\right)}\text{,}\text{where}Z\left(C\right)=\ker d\text{and}B\left(C\right)=\mathrm{im}d.$$

A quasiisomorphism is a morphism $f:C\to C\prime$ such that $H\left(f\right):H\left(C\right)\to H\left(C\prime \right)$ is an isomorphism.

A complex is exact if $H\left(C\right)=0$.

The Grothendieck group of $𝒜$ is the abelian group generated by symbols $$\left[M\right],\text{for}M\in 𝒜,\text{with relations}\left[M_1\right]=\left[M_2\right],\text{if}{M}_1\cong M_2$$ and $$\left[M\right]=\left[M_1\right]+\left[M_2\right],\text{if there exists an exact sequence}0⟶M_1⟶M⟶M_2⟶0.$$

The Euler characteristic of a graded $A$-module $M$ is $$\chi \left(M\right)=\sum _{n\in \mathbb{Z}}{(-1)}^n\left[M_n\right]\text{and}{P}_M\left(t\right)=\sum _{n\in \mathbb{Z}}{t}^n\left[M_n\right]$$ is the Poincaré polynomial of $M$.

Let $$0⟶C\prime \stackrel{u}{⟶}C\stackrel{v}{⟶}C\prime \prime ⟶0$$ be an exact sequence of complexes. The long exact sequence in homology is the exact triangle $$\begin{array}{c}H\left(C\right)\\ H\left(u\right)\nearrow \searrow H\left(v\right)\\ H\left(C\right)\phantom{,}\stackrel{\partial (u,v)}{⟵}\phantom{,}H\left(C\prime \prime \right)\\ \left[t^{\prime }\right]\phantom{mm}⟵\phantom{mm}\alpha \end{array}$$ $$\cdots ⟶H_{n+1}\left(C\prime \prime \right)\stackrel{{\partial }_{n+1}(u,v)}{⟶}{H}_n\left(C\prime \prime \right)\stackrel{H_n\left(u\right)}{⟶}{H}_n\left(C\right)\stackrel{H_n\left(v\right)}{⟶}{H}_n\left(C\prime \prime \right)\stackrel{{\partial }_n(u,v)}{⟶}{H}_{n-1}\left(C\prime \right)\stackrel{H_{n-1}\left(u\right)}{⟶}{H}_{n-1}\left(C\right)\stackrel{H_{n-1}\left(v\right)}{⟶}{H}_{n-1}\left(C\prime \prime \right)⟶\cdots$$

where

if $z\prime \prime )\in Z(C\prime \prime )$ such that $\left[z\prime \prime \right)]=\mathrm{\alpha \alpha }$, and

$x\in C_n$ such that $v\left(x\right)=z\prime \prime$, then

$t\prime \in {C\prime }_{n-1}$ such that $u\left(t\prime \right)=dx$.

Derived functors

A complex is exact if $H\left(C\right)=0$. An exact functor is a functor $F:𝒞⟶𝒟$ such that if $$\text{if}0⟶X\stackrel{f}{⟶}Y\stackrel{g}{⟶}Z⟶0\text{is exact}\text{then}0⟶F\left(X\right)\stackrel{F\left(f\right)}{⟶}F\left(Y\right)\stackrel{F\left(g\right)}{⟶}F\left(Z\right)⟶0\text{is exact.}$$

A left exact functor is a functor $F:𝒞⟶𝒟$ such that if $$\text{if}0⟶X\stackrel{f}{⟶}Y\stackrel{g}{⟶}Z\text{is exact}\text{then}0⟶F\left(X\right)\stackrel{F\left(f\right)}{⟶}F\left(Y\right)\stackrel{F\left(g\right)}{⟶}F\left(Z\right)\text{is exact.}$$

• A projective object is an object $Y\in 𝒞$ such that $\mathrm{Hom}(Y,\cdot )$ is an exact functor.
• An injective object is an object $Y\in 𝒞$ such that $\mathrm{Hom}(\cdot ,Y)$ is an exact functor.
• A flat $A$-module is an $A$-module $Y$ such that $\cdot {\otimes }_{A}Y$ is an exact functor.
• A free $A$-module is an ???.
• A torsion free $A$-module is an ???

A presentation of an $A$-module $M$ is an exact sequence $$R⟶X⟶M⟶0,$$ where $R$ and $X$ are free modules.

An injective resolution of $M$ is an exact sequence $$0⟶M⟶I^0⟶I^1⟶I^2⟶\cdots ,\text{with all}{I}^k\text{injective.}$$

A projective resolution of $M$ is an exact sequence $$\cdots ⟶P^{-2}⟶P^{-1}⟶P^0⟶M⟶0\text{,}\text{with all}{P}^{-k}\text{projective.}$$

Let $F:𝒞⟶𝒟$ be a left exact functor. The right derived functors of $F$ are $$R^{i}F:𝒞⟶𝒟\text{given by}{R}^{i}F\left(M\right)=H^i\left(F\right(I\left)\right),$$ where $I$ is an injective resolution of $M$.

Let $F:𝒞⟶𝒟$ be a right exact functor. The left derived functors of $F$ are $$L_{i}F:𝒞⟶𝒟\text{given by}{L}_{i}F\left(M\right)=H^{-i}\left(F\right(P\left)\right),$$ where $P$ is a projective resolution of $M$.

Let $$0⟶N⟶M⟶P⟶0$$ be an exact sequence. The long exact sequence is $$0⟶F\left(N\right)⟶F\left(M\right)⟶F\left(P\right)\stackrel{{\delta }^0}{⟶}{R}^{1}F\left(N\right)⟶R^{1}F\left(M\right)⟶R^{1}F\left(P\right)\stackrel{{\delta }^1}{⟶}{R}^{2}F\left(N\right)⟶R^{2}F\left(M\right)⟶R^{2}F\left(P\right)⟶\cdots$$

A quasiisomorphism is a morphism of complexes $f:C\to D$ such that $H\left(f\right):H\left(C\right)\to H\left(D\right)$ is an isomorphism.

The derived category of $𝒜$ is the category $D\left(𝒜\right)$ with a functor $Q:\mathrm{Kom}\left(𝒜\right)\to D\left(𝒜\right)$ such that

• if $f$ is a quasiisomorphism then $Q\left(f\right)$ is an isomorphism, and
• if $F:\mathrm{Kom}\left(𝒜\right)\to 𝒞$ is a functor that takes quasiisomorphisms to isomorphisms then there exists a unique functor $\stackrel{\sim }{F}:D\left(𝒜\right)\to 𝒟$ such that

$$\begin{array}{ccc}\mathrm{Kom}\left(𝒜\right)& \stackrel{Q}{⟶}& D\left(𝒜\right)\\ & F\searrow & \downarrow \stackrel{\sim }{F}\\ & & 𝒞\end{array}$$

Let $D^{+}\left(𝒞\right)$ denote the left bounded derived category of $𝒞$. Let $F:𝒞⟶𝒟$ be a left exact functor. The derived functor of $F$ is the functor $$\mathrm{RF}:D^{+}\left(𝒞\right)⟶D^{+}\left(𝒟\right)\text{determined by}\mathrm{RF}(K{)}^i=F(K^i),\text{for}K\in \mathrm{Kom}(𝒞).$$

Then $$R^{i}F=H^i\left(\mathrm{RF}\right).$$

Examples

Ext: ${\mathrm{Ext}}_{𝒞}^i(M,N)={\mathrm{Hom}}_{D\left(𝒞\right)}\left(M\right[0],N[i\left]\right)$, ${\mathrm{Ext}}^i(M,\cdot )=R^i\mathrm{Hom}(M,\cdot )$, ${\mathrm{Ext}}^i(\cdot ,N)=R^i\mathrm{Hom}(\cdot ,N)$.

Tor: Let $\otimes$ be the left adjoint functor to $\mathrm{Hom}$ so that $$\mathrm{Hom}(M\otimes V,N)\cong \mathrm{Hom}(M,\mathrm{Hom}(N,V\left)\right),\text{for all}N.$$

Then ${\mathrm{Tor}}_i(\cdot ,N)=R^i\left(\cdot {\otimes }_{A}N\right)$, $\text{and}{\mathrm{Tor}}_i(M,\cdot )=R^i\left(M{\otimes }_A\cdot \right)$.

Sheaf cohomology: Let $X$ be a topological space. The sheaf cohomology of $X$ is $$H^i(X;\cdot )=R^i\Gamma (X;\cdot ),\text{where}\begin{array}{cccc}\Gamma :& \left\{\text{sheaves on}X\right\}& ⟶& \left\{\text{abelian groups}\right\}\\ & ℱ& ⟼& ℱ\left(X\right)\end{array}$$ is the global sections functor.

Group cohomology: Let $G$ be a group and let $M$ be a $G$-module. The cohomology of $G$ is $$H^i(G;\cdot )=R^i\left({\cdot }^G\right),\text{where}{M}^G=\{m\in M|gm=m\text{for all}g\in G\}$$ is the invariants of $M$.

Lie algebra cohomology: Let $𝔤$ be a Lie algebra and let $M$ be a $𝔤$-module. The cohomology of $𝔤$ is $$H^i(𝔤;\cdot )=R^i\left({\cdot }^{𝔤}\right),\text{where}{M}^{𝔤}=\{m\in M|xm=m\text{for all}x\in 𝔤\}$$ is the invariants of $M$.

Notes and References

Stuff

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