Complexes in Local Ring Theory

Abstract

In this article we shall discuss a certain homological tool, the Koszul complex, which relates two concepts important in local ring theory, namely depth and multiplicity.

We recall that a local ring R is a commutative, notherian ring with identity, having a unique maximal ideal, m,. The dimension of the local ring R is the longest integer d for which a strictly descending chain of prime ideals, m = J₀ ⊃ J₁⊃…⊃ J₁, of length d exists. Since R is noetherian, all ideals of R are finitely generated. In particular, m, is finitely generated, and according to Krull's principal ideal theorem, the number of elements required to generate m, is always greater than or equal to. dim R (the dimension of R). If m can be generated by precisely d= dim R elements, R is said to be a regular local ring. An ideal r of R is said to be an ideal of definition or m-primary if r contains some power of m This is equivalent to saying that R/r is an R-module of finite length. A set of elements x₁,…, x_d of R (where d = dim R) is said to be a system of parameters if the elements generate an ideal of definition.