nLab
homological resolution

A homological resolution of a quotient is a special case of a more general homological resolution of an object in a category in which (co)chain complexes make sense. Typically, one is interested in resolutions of non-free objects by free ones. The ur examples are the Koszul or Koszul–Tate resolutions of an $A$-module by free $A$-modules or of an $A$-algebra by free $A$-algebras.

For example, let $A$ be a commutative augmented algebra over a field $k$ and $I$ an ideal in $A$. Resolve $A/I$ 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 $\mathrm{\infty }$-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 $C^{•}$ of abelian groups whose (co)homology $H^{•}(C^{•})$ in some degree, usually in degree 0, is the desired quotient, $H^0(C)=\mathrm{desired}\mathrm{quotient}$. Depending on the situation one may want to demand that the (co)homology in all other degrees vanish, in which case $C^{•}$ would be weakly equivalent to the desired quotient.

The term ‘resolution’ is usually reserved for the case in which (co)homology in all other resolution degrees vanishes. (Jim)

In general, i.e. not restricted to the context of (co)chain complexes, weak or homotopy quotients are a common theme throughout homotopy theory and higher category theory. It underlies for instance the idea of regarding Lie groupoids as orbifolds (see Orbifolds as Groupoids) and the idea of stacky quotients. The point of all these weak quotients is usually that the weak quotient in general exists only in a category of nicer objects than the true quotient: if a Lie group $G$ acts non-freely on a manifold $X$, the action groupoid $X//G$ exists as a Lie groupoid, an internal groupoid in manifolds, whereas the “true” quotient $X/G$ cannot be equipped with the structure of a manifold anymore.