## Homology

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 $ℤ$-graded $A$-module $C$, with a morphism $d:C⟶C$ such that $⋯⟶ Cn+1 ⟶ dn+1 Cn ⟶dn Cn–1 ⟶⋯ such that deg⁡d=-1 and d∘d=0.$

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

The homology of a complex $C$ is $H(C) = Z(C) B(C) , where Z(C) =ker⁡d and B(C) =im⁡d.$

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 $[M], for M∈𝒜, with relations [M1] =[M2], if M1≅M2$ and $[M]= [M1] +[M2], if there exists an exact sequence 0 ⟶M1 ⟶M ⟶M2 ⟶0.$

The Euler characteristic of a graded $A$-module $M$ is $χ(M)= ∑n∈ℤ (-1)n [Mn] and PM(t) = ∑n∈ℤ tn[Mn]$ is the Poincaré polynomial of $M$.

Let $0⟶C′ ⟶u C⟶v C′′⟶0$ be an exact sequence of complexes. The long exact sequence in homology is the exact triangle $H(C) H(u) ↗↘ H(v) H(C), ⟵ ∂(u,v) ,H(C′′) [t′]mm ⟵mmα$ $⋯⟶ Hn+1( C′′) ⟶ ∂n+1 (u,v) Hn( C′′) ⟶ Hn(u) Hn(C) ⟶ Hn(v) Hn( C′′) ⟶ ∂n(u,v) Hn-1( C′) ⟶ Hn-1 (u) Hn-1(C) ⟶ Hn-1 (v) Hn-1 ( C′′) ⟶⋯$

where
if $z\prime \prime \right)\in Z\left(C\prime \prime \right)$ such that $\left[z\prime \prime \right)\right]=\mathrm{\alpha \alpha }$, 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 $if 0⟶X ⟶f Y⟶gZ ⟶0 is exact then 0 ⟶F(X) ⟶ F(f) F(Y) ⟶F(g) F(Z)⟶0 is exact.$

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

• A projective object is an object $Y\in 𝒞$ such that $\mathrm{Hom}\left(Y,\cdot \right)$ is an exact functor.
• An injective object is an object $Y\in 𝒞$ such that $\mathrm{Hom}\left(\cdot ,Y\right)$ 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 $0⟶M⟶ I0⟶ I1⟶ I2⟶ ⋯, with all Ik injective.$

A projective resolution of $M$ is an exact sequence $⋯⟶ P-2⟶ P-1⟶ P0⟶ M⟶0, with all P-k projective.$

Let $F:𝒞⟶𝒟$ be a left exact functor. The right derived functors of $F$ are $RiF: 𝒞⟶𝒟 given by RiF(M) =Hi(F(I)),$ where $I$ is an injective resolution of $M$.

Let $F:𝒞⟶𝒟$ be a right exact functor. The left derived functors of $F$ are $LiF:𝒞 ⟶𝒟 given by LiF(M) =H-i (F(P)),$ where $P$ is a projective resolution of $M$.

Let $0⟶N⟶M ⟶P⟶0$ be an exact sequence. The long exact sequence is $0⟶ F(N) ⟶F(M) ⟶F(P) ⟶δ0 R1F(N) ⟶ R1F(M) ⟶ R1F(P) ⟶δ1 R2F(N) ⟶R2F( M) ⟶R2F( P) ⟶⋯$

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
$Kom(𝒜) ⟶Q D(𝒜) F↘ ↓F∼ 𝒞$

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 $RF:D+ (𝒞) ⟶ D+(𝒟) determined by RF(K)i =F(Ki), for K∈Kom(𝒞).$

Then $RiF= Hi(RF).$

## Examples

Ext: $\phantom{\rule{1em}{0ex}}{\mathrm{Ext}}_{𝒞}^{i}\left(M,N\right)={\mathrm{Hom}}_{D\left(𝒞\right)}\left(M\left[0\right],N\left[i\right]\right)$, $\phantom{\rule{1em}{0ex}}{\mathrm{Ext}}^{i}\left(M,\cdot \right)={R}^{i}\mathrm{Hom}\left(M,\cdot \right)$, $\phantom{\rule{1em}{0ex}}{\mathrm{Ext}}^{i}\left(\cdot ,N\right)={R}^{i}\mathrm{Hom}\left(\cdot ,N\right)$.

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

Then $\phantom{\rule{1em}{0ex}}{\mathrm{Tor}}_{i}\left(\cdot ,N\right)={R}^{i}\left(\cdot {\otimes }_{A}N\right)$, $\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathrm{Tor}}_{i}\left(M,\cdot \right)={R}^{i}\left(M{\otimes }_{A}\cdot \right)$.

Sheaf cohomology: Let $X$ be a topological space. The sheaf cohomology of $X$ is $Hi (X;⋅) =RiΓ (X;⋅), where Γ: {sheaves on X} ⟶ {abelian groups} ℱ ⟼ ℱ(X)$ is the global sections functor.

Group cohomology: Let $G$ be a group and let $M$ be a $G$-module. The cohomology of $G$ is $Hi(G;⋅ ) =Ri( ⋅G), where MG={m ∈M | gm=m for all g∈G}$ 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 $Hi (𝔤;⋅) =Ri( ⋅𝔤), where M𝔤 ={m∈M | xm=m for all x∈𝔤}$ is the invariants of $M$.

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