Let be a unital ring.
Consider also a finite sequence of elements in .
Given any central element , one can define a two term complex
concentrated in degrees and , where the map is the left multiplication by . Given a sequence of central elements in one can define the tensor product
of complexes of left -modules. Degree part of equals the exterior power . Consider the usual bases elements of , where . Then the differential is given by
The differential can be obtained from the faces of the obvious Koszul semi-simplicial -module and the chain complex above is obtained by the usual alternating sum rule.
Now let be a finitely generated left -module. Then the abelian groups
together with connecting homomorphisms, form a homological and cohomological delta functors (in the sense of Tohoku) respectively, deriving the zero parts
where is the left -submodule generated by . A Poincare-like duality holds: .
The sequence is called -regular (or regular on ) if for all the image of in annihilates only zero. This terminology is in accord with calling a non-zero divisor in a ring a “regular element” (and is in accord with the terminology regular local rings).
If is a regular sequence on/in then is a free resolution of the module and the cohomology while Koszul homology is .
The resolution of can be written
and the -linear map is given by the row vector .
The original reference is
- Jean-Louis Koszul, Homologie et cohomologie des algèbres de Lie , Bulletin de la Société Mathématique de France, 78, 1950, pp 65-127.
A standard textbook reference is
A generalization of Koszul complexes to (appropriate resolutions of) algebras over operads is in
- Joan Millès, The Koszul complex is the cotangent complex, MPIM2010-32, pdf