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Concept
Koszul complex
Summary
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 for the i-th exterior power of E. Then, given an R-linear map ,
the Koszul complex associated to s is the chain complex of R-modules:
where the differential is given by: for any in E,
The superscript means the term is omitted. To show that , use the self-duality of a Koszul complex.
Note that and . Note also that ; this isomorphism is not canonical (for example, a choice of a volume form in differential geometry provides an example of such an isomorphism.)
If (i.e., an ordered basis is chosen), then, giving an R-linear map amounts to giving a finite sequence of elements in R (namely, a row vector) and then one sets
If M is a finitely generated R-module, then one sets:
which is again a chain complex with the induced differential .
The i-th homology of the Koszul complex
is called the i-th Koszul homology. For example, if and is a row vector with entries in R, then is
and so
Similarly,
Given a commutative ring R, an element x in R, and an R-module M, the multiplication by x yields a homomorphism of R-modules,
Considering this as a chain complex (by putting them in degree 1 and 0, and adding zeros elsewhere), it is denoted by . By construction, the homologies are
the annihilator of x in M.
Official source
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