Complexes and Homology

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 18 November 2009

Complexes

Let $A$ be a ring and let $𝒜$ be the category of $A$-modules. More generally, $𝒜$ could be any abelian category.

A complex of $A$-modules is a $\mathbb{Z}$-graded $A$-module $C$, with a morphism $d:C⟶C$ such that

$$\cdots ⟶C_{n+1}\stackrel{d_{n+1}}{⟶}{C}_n\stackrel{d_n}{⟶}{C}_{n–1}⟶\cdots \phantom{i},\text{such that}\mathrm{deg}d=–1\text{and}d·d=0.$$

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

The homology of a complex $C$ is

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 group

$$\left[M\right]\text{,}\text{for}M\in 𝒜\text{,}\text{with relations}\left[M_1\right]=\left[M_2\right]\text{,}\text{if}{M}_1\cong M_2$$

and

$$\left[M\right]=\left[M_1\right]+\left[M_2\right]\text{,}\text{if there exists an exact sequence}0⟶M_1⟶M⟶M_2⟶0$$.

The Euler characteristic of a graded $A$-module $M$ is

is the Poincare 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

$$\begin{array}{c}H\left(C\right)\\ H\left(u\right)\nearrow \searrow H\left(v\right)\\ H\left(C\right)\phantom{,}\stackrel{\partial (u,v)}{⟵}\phantom{,}H\left(C^{\prime \prime }\right)\\ \left[t^{\prime }\right]\phantom{mm}⟵\phantom{mm}\alpha \end{array}$$

$$···⟶H_{n+1}\left(C^{\prime \prime }\right)\stackrel{{\partial }_{n+1}(u,v)}{⟶}{H}_n\left(C^{\prime }\right)\stackrel{H_n\left(u\right)}{⟶}{H}_n\left(C\right)\stackrel{H_n\left(v\right)}{⟶}{H}_n\left(C^{\prime \prime }\right)\stackrel{{\partial }_n(u,v)}{⟶}{H}_{n–1}\left(C^{\prime }\right)\stackrel{H_{n–1}\left(u\right)}{⟶}{H}_{n–1}\left(C\right)\stackrel{H_{n–1}\left(v\right)}{⟶}{H}_{n–1}\left(C^{\prime \prime }\right)⟶···$$

where

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

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

$t^{\prime }\in C_{n–1}^{\prime }$ 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 $F:𝒞⟶{𝒞}^{\prime }$ 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 $F:𝒞⟶{𝒞}^{\prime }$ 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

$$R⟶X⟶M⟶0\text{,}$$ where $R$ and $X$ are free modules.

An injective resolution of $M$ is an exact sequence

$$0⟶M⟶I^0⟶I^1⟶I^2⟶\cdots \text{,}\text{with all}{I}^k\text{injective.}$$

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 $F:𝒞⟶{𝒞}^{\prime }$ be a left exact functor. The right derived functors of $F$ are

$$R^{i}F:𝒞⟶{𝒞}^{\prime }\text{given by}{R}^{i}F\left(M\right)=H^i\left(F\right(I\left)\right)\text{,}$$

where $I$ is an injective resolution of $M$.

Let $F:𝒞⟶{𝒞}^{\prime }$ be a right exact functor. The left derived functors of $F$ are

$$L_{i}F:𝒞⟶{𝒞}^{\prime }\text{given by}{L}_{i}F\left(M\right)=H^{–i}\left(F\right(P\left)\right)\text{,}$$

where $P$ is a projective resolution of $M$.

Let

$$0⟶C^{\prime }⟶C⟶C^{\prime \prime }⟶0$$

be an exact sequence. The long exact sequence is

$0⟶F\left(C^{\prime }\right)⟶F\left(C\right)⟶F\left(C^{\prime \prime }\right)\stackrel{{\delta }^0}{⟶}{R}^{1}F\left(C^{\prime }\right)⟶R^{1}F\left(C\right)⟶R^{1}F\left(C^{\prime \prime }\right)\stackrel{{\delta }^1}{⟶}{R}^{2}F\left(C^{\prime }\right)⟶R^{2}F\left(C\right)⟶R^{2}F\left(C^{\prime \prime }\right)⟶\cdots$

A quasiisomorphism is a morphism of complexes $f:C\to C^{\prime }$ such that $H\left(f\right):H\left(C\right)\to H\left(C^{\prime }\right)$ 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 $\stackrel{\sim }{F}:D\left(𝒜\right)\to 𝒞$ 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 $F:𝒞⟶{𝒞}^{\text{'}}$ be a left exact functor. The derived functor of $F$ is the functor

$$\mathrm{RF}:D^{+}\left(𝒞\right)⟶D^{+}\left(C^{\prime }\right)\text{determined by}\mathrm{RF}(K{)}^i=F(K^i),\text{for}K\in \mathrm{Kom}(𝒞)\text{.}$$

Then

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

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

$$\mathrm{Hom}(M\otimes V,N)\simeq \mathrm{Hom}(M,\mathrm{Hom}(N,V\left)\right)\text{,}\text{for all}\phantom{t}N\text{.}$$

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

Sheaf cohomology: Let $X$ be a topological space. The sheaf cohomology of $X$ is

is the global sections functor.

Group cohomology: Let $G$ be a group and let $M$ be a $G$-module. The cohomology of $G$ is

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

is the invariants of $M$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)