nLab
homological resolution

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

In homological algebra a homological resolution of a vector space or more generally of a module NN is a chain complex V V_\bullet whose chain homology H (V)H_\bullet(V) reproduces NN, in that

H n(V){N | n=0 0 | otherwise H_n(V) \;\simeq\; \left\{ \array{ N &\vert& n = 0 \\ 0 &\vert& \text{otherwise} } \right.

Of course if one regards NN itself as a chain complex N N_\bullet which is concentrated in degree 0, then N N_\bullet has this property, trivially. Therefore, typically when asking for a homological resolution V V_\bullet of NN, it is understood that the entries of V V_\bullet have certain nice properties that NN is lacking.

For instance if V V_\bullet consists of projective modules, then it is called a projective resolution of NN, or if it consists of injective modules then it is called an injective resolution, or if it consists of free modules, then it is called a free resolution, etc. The module NN itself may be far from being projective or injective or free, etc. and so the corresponding resolution, if it exists, allows to nevertheless regard NN with tools applicable to these particularly nice classes of modules, up to chain homology

Typically in these constructions one demands not just that the chain homology of the resolution reproduces NN via any isomorphism H 0(V)NH_0(V) \simeq N, but that there exists a chain map f f_\bullet from VV to NN or from NN to VV, such that it is this chain map which under passage to chain homology induces this isomorphism. A chain map which induces isomorphisms on chain homology groups is called a quasi-isomorphism, and so typically a homological resolution means a choice of quasi-isomorphism from a module to a (particularly nice) chain complex.

Famous classes of examples of such resolutions are the injective and projective resolutions that are used to construct derived functors in homological algebra, see there for more.

Phrased this way, there is nothing special about starting with a single module, and more generally one may speak of resolutions of one chain complex by another chain complex with some better properties. The theory of homotopical categories such as model categories or fibration categories/cofibration categories is used to handle homological resolutions in this generality, see at model structure on chain complexes for more on this.

Viewed from this general perspective of homotopical categories, homological resolutions are a special case of the general concept of resolutions in homotopy theory. (In fact homological algebra may be understood as a fragment of stable homotopy theory, see at Dold-Kan correspondence and stable Dold-Kan correspondence for more on this.)

Examples

The ur examples of homological resolutions are the Koszul complexes or more generally Koszul-Tate resolutions of an AA-module by free modules ove or of an AA-algebra by free AA-algebras.

For example, consider AA is a commutative augmented algebra over a field kk and II an ideal in AA. Resolve the quotient A/IA/I by a free AA-algebra RR with a derivation differential so that H(R)=A/IH(R) = A/I is concentrated in degree 00 and all other homology vanishes. Iff II is a regular ideal?, the Koszul complex will do; if II is not regular, continue the process forming the Koszul-Tate resolution, the algebraic analog of a Moore-Postnikov system, which was indeed Tate’s inspiration.

If the original object is itself graded or differential graded, the resolution will be bigraded by resolution degree and internal degree.

By homological resolution of a quotient, one means a weak quotient or homotopy quotient in some \infty-categorical homotopy theory context which is equivalent to a category of (co)chain complexes. This means: a homological resolution of a quotient is a (co)chain complex C C^\bullet of abelian groups whose (co)homology H (C )H^\bullet(C^\bullet) in some degree, usually in degree 0, is the desired quotient, H 0(C)=desiredquotientH^0(C) = desired quotient. Depending on the situation one may want to demand that the (co)homology in all other degrees vanish, in which case C C^\bullet would be weakly equivalent to the desired quotient.

References

Discussion for an audience of physicists in the context of BV-BRST formalism is in

Revised on September 29, 2017 12:06:37 by Urs Schreiber (188.105.77.106)