Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories


Algebraic topology


Of chain complexes


Under the Dold-Kan correspondence, ∞-groupoids with strict abelian group structure (modeled by Kan complexes that are simplicial abelian groups) are identified with non-negatively graded chain complexes of abelian groups

N :SimpAbCh +. N_\bullet : SimpAb \stackrel{\simeq}{\to} Ch_+ \,.

The homology groups of a chain complex of abelian groups are the image under this identification of the homotopy groups of the corresponding ∞-groupoids. More details on this are at chain homology and cohomology.

So at least for the case of chain complexes of abelian groups we have the slogan

Of course historically the development of concepts was precisely the opposite: chain homology is an old fundamental concept in homological algebra that is simpler to deal with than simplicial homotopy groups. The computational simplification for chain complexes is what makes the Dold-Kan correspondence useful after all.

But conceptually it is useful to understand homology as a special kind of homotopy. This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts. But later generalizations of cohomology to generalized (Eilenberg-Steenrod) cohomology and further to nonabelian cohomology showed that the restricted notion of homology is an insufficient dual model for cohomology: what cohomology is really dual to is the more general concept of homotopy. More on this is at cohomotopy and Eckmann-Hilton duality.


The category of abelian groups is in particular an abelian category. We can define chain complexes and their homology in any abelian category CC.

Let CC be an abelian category and let

V =(V n+1 nV n n1V n1) V_\bullet = ( \cdots \to V_{n+1} \stackrel{\partial_n}{\to} V_n \stackrel{\partial_{n-1}}{\to} V_{n-1} \to \cdots )

be a chain complex in CC. For each integer nn \in \mathbb{N} this induces the following diagram of kernels, cokernels and images

imδ n kerδ n1 V n+1 n V n n1 V n1 cokerδ n imδ n1 \array{ && im \delta_n &&\to&& ker \delta_{n-1} \\ & \nearrow && \searrow && \swarrow \\ V_{n+1} &&\stackrel{\partial_n}{\to}&& V_n &&\stackrel{\partial_{n-1}}{\to}&& V_{n-1} \\ & && \swarrow && \searrow && \nearrow \\ && coker \delta_n &&\stackrel{}{\to}&& im \delta_{n-1} }

the homology H n(V)H_n(V) of VV in degree nn is the object

im(kerδ n1V ncokerδ) coker(imδ nkerδ n1) coker(V n+1kerδ n1) ker(cokerδ nimδ n1) ker(cokerδ nV n1) \begin{aligned} im(ker \delta_{n-1} \to V_n \to coker \delta) & \simeq coker(im \delta_n \to ker \delta_{n-1}) \\ & \simeq coker(V_{n+1} \to ker \delta_{n-1}) \\ & \simeq ker(coker \delta_n \to im \delta_{n-1}) \\ & \simeq ker(coker \delta_n \to V_{n-1}) \end{aligned}
  • If H n(V)0H_n(V) \simeq 0 then one says that the complex VV is exact in degree nn.


In the special case that CC is the category of abelian groups, or of vector spaces, this definition reduces to the more familiar simpler statement:

the nn-th homology group of the chain complex V V_\bullet is the quotient group

H n(V)=ker( n)/im( n+1). H_n(V) = ker(\partial_n) / im(\partial_{n+1}) \,.

Generalized homology

By the Brown representability theorem every spectrum AA induces a generalized (Eilenberg-Steenrod) cohomology theory, and dually a generalized homology theory.

For XX a topological space and AA a spectrum, the generalized homology of spectrum of XX with coefficients in AA is

XA:=Σ (X)A, X \wedge A := \Sigma^\infty(X)\wedge A \,,

where on the right we have the smash product of spectra with the suspension spectrum of XX and on the left we abbreviate this to the (∞,1)-tensoring of Spec over Top.

The corresponding homology groups are the homotopy groups of this spectrum:

E n(X,A):=π n(XA):=[Σ n𝕊,XA]. E_n(X,A) := \pi_n(X \wedge A) := [\Sigma^n \mathbb{S}, X \wedge A ] \,.

where 𝕊\mathbb{S} is the sphere spectrum. For more see generalized homology.


The relation between homology, cohomology and homotopy:

[S n,][S^n,-][,A][-,A]()A(-) \otimes A
category theorycovariant homcontravariant homtensor product
homological algebraExtExtTor
enriched category theoryendendcoend
homotopy theoryderived hom space Hom(S n,)\mathbb{R}Hom(S^n,-)cocycles Hom(,A)\mathbb{R}Hom(-,A)derived tensor product () 𝕃A(-) \otimes^{\mathbb{L}} A

The ingredients of homology and cohomology:

H n=Z n/B nH_n = Z_n/B_n(chain-)homology(cochain-)cohomologyH n=Z n/B nH^n = Z^n/B^n
C nC_nchaincochainC nC^n
Z nC nZ_n \subset C_ncyclecocycleZ nC nZ^n \subset C^n
B nC nB_n \subset C_nboundarycoboundaryB nC nB^n \subset C^n

Revised on April 20, 2017 04:34:46 by Anonymous (