Koszul complex

Let be a commutative ring with unit element and a sequence of elements of . The Koszul complex defined by these data then consists of the modules , where is the canonical basis for the -module , and the differentials

where, as usual, a over a symbol means deletion. More generally one also considers the chain and cochain complexes and . If , consists of just two non-zero modules and in dimensions 0 and 1 and the only non-zero differential is multiplication by in . The general Koszul complex can be viewed as built up from these elementary constituents as

For , define a morphism of chain complexes by taking multiplication by in dimension zero and the identity in dimension 1. Taking tensor products and iterates one thus defines morphisms of chain complexes

where . Let denote the -th cohomology group of the cochain complex . For a Noetherian ring , the local cohomology of an -module with respect to an ideal , can then be calculated as:

where is a set of generators for .

An element is called an -regular element (where is an -module) if is not a zero-divisor on , i.e. if is injective. A sequence of elements is called an -regular sequence of elements or an -sequence if is not a zero-divisor on , i.e. if is -regular. Let be an ideal of . Then an -regular sequence is called an -regular sequence in if for . A maximal -regular sequence in is an -regular sequence in such that there is no for which is an -regular sequence.

Let be Noetherian, a finitely-generated -module and an ideal. Then the following are equivalent: i) for all and for all finitely-generated -modules with support in (cf. Support of a module); ii) for ; and iii) there exists an -regular sequence in .

The -depth of a module is the length of the longest -regular sequence in . It is also called the grade of on . The depth of a module is the -depth.

The homology of the Koszul complex associated with an -regular sequence satisfies for and . This (and the above) makes Koszul complexes an important tool in commutative and homological algebra, for instance in dimension theory and the theory of multiplicities (and intersection theory), cf. [a1], [a2], [a3], [a4]; cf. also Depth of a module and Cohen–Macaulay ring.

References

[a1]	A. Grothendieck, "Local cohomology" , Lect. notes in math. , 41 , Springer (1967)
[a2]	J. Herzog (ed.) E. Kunz (ed.) , Der kanonische Modul eines Cohen–Macaulay-Rings , Lect. notes in math. , 238 , Springer (1971)
[a3]	H. Matsumura, "Commutative algebra" , Benjamin (1970)
[a4]	D.G. Northcott, "Lessons on rings, modules, and multiplicities" , Cambridge Univ. Press (1968)