Koszul complex

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts about the depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension. Moreover, in certain circumstances, the complex is the complex of syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.

Let R be a commutative ring and E a free module of finite rank r over R. We write ${\displaystyle \wedge ^{i}E}$ for the i-th exterior power of E. Then, given an R-linear map ${\displaystyle s:E\to R}$, the Koszul complex associated to s is the chain complex of R-modules:

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