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 AA-module by free AA-modules or of an AA-algebra by free AA-algebras.

For example, let AA be a commutative augmented algebra over a field kk and II an ideal in AA. Resolve 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.

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 GG acts non-freely on a manifold XX, the action groupoid X//GX//G exists as a Lie groupoid, an internal groupoid in manifolds, whereas the “true” quotient X/GX/G cannot be equipped with the structure of a manifold anymore.

Revised on February 13, 2013 01:59:33 by Urs Schreiber (82.169.65.155)