nLab homological resolution

Idea

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

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

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

For instance if $V_\bullet$ consists of projective modules, then it is called a projective resolution of $N$, 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 $N$ itself may be far from being projective or injective or free, etc. and so the corresponding resolution, if it exists, allows to nevertheless regard $N$ 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 $N$ via any isomorphism $H_0(V) \simeq N$, but that there exists a chain map $f_\bullet$ from $V$ to $N$ or from $N$ to $V$, 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 $A$-module by free modules ove or of an $A$-algebra by free $A$-algebras.

For example, consider $A$ is a commutative augmented algebra over a field $k$ and $I$ an ideal in $A$. Resolve the quotient $A/I$ by a free $A$-algebra $R$ with a derivation differential so that $H(R) = A/I$ is concentrated in degree $0$ and all other homology vanishes. Iff $I$ is a regular ideal?, the Koszul complex will do; if $I$ 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 $C^\bullet$ of abelian groups whose (co)homology $H^\bullet(C^\bullet)$ in some degree, usually in degree 0, is the desired quotient, $H^0(C) = desired quotient$. Depending on the situation one may want to demand that the (co)homology in all other degrees vanish, in which case $C^\bullet$ would be weakly equivalent to the desired quotient.

Related concepts

• simplicial resolution

References

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

• Marc Henneaux, Claudio Teitelboim, section 8.3 of Quantization of Gauge Systems, Princeton University Press, 1992