Local homology theory for artinian modules

Abstract

where K.(x1 , . . . , x n r , M) is the Koszul complex of M with respect to x n 1 , . . . , x n r . Then H p (·) is an additive, A-linear covariant functor from the category of Artinian A-modules and A-homomorphisms to the category of A-modules and Ahomomorphisms and there is a long exact sequence of local homology modules for any short exact sequence of Artinian A-modules. For any ideal I of A, by [4,Lemma 3], there exist finitely many elements x1, . . . , xr ∈ I such that 0 :M I = 0 :M (x1, . . . , xr). It is proved that H x p (M) does not depend on the choice of x = (x1, . . . , xr), up to A-isomorphisms. We use H I p (M) to denote one of these H p (M). The two important concepts of Artinian modules, cograde and dimension, are connected with local homology modules. The paper is carried out when the author is visiting the Department of Pure Mathematics of Sheffield University. I am grateful to Prof.R.Y.Sharp for helpful discussions.