The I-adic completion and local homology for Artinian modules

Abstract

Let I be an ideal of a commutative ring R and M an R-module. It is well known that the I-adic completion functor Λ_I defined by Λ_I(M) = lim_←tM/I^tM is an additive exact covariant functor on the category of finitely generated R-modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, Λ_I is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence of left derived functors {L^I_i} of Λ_I, in which L^I₀ is right exact, but in general L^I₀ ≠ Λ_I. Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R-regular sequence, Matlis proved in [9, 10] that

where D(−) = Hom_R(−; E(R/[mfr ])) is the Matlis dual functor, and that

In [18, 5] A.-M. Simon shows that L^I_o(M) = M and L^I_i(M) = 0 for i > 0, provided that M is complete with respect to the I-adic topology.

Later, Greenlees and May [3] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R) by

where x is a finitely generated system of I. Then they showed, under some conditions on x which are satisfied when R is Noetherian, that H^I_{[bull ]}(M) ≅ L^I_{[bull ]}(M). Recently, Tarrío, López and Lipman [1] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.