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 ΛI defined by
ΛI(M)
= lim←tM/ItM is an additive
exact covariant functor on the category of finitely generated R-modules, provided
R is Noetherian. Unfortunately, even if R is Noetherian, ΛI
is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence
of left derived functors {LIi} of ΛI,
in which LI0 is right exact, but in general
LI0 ≠ ΛI.
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(−) = HomR(−; E(R/[mfr ]))
is the Matlis dual functor, and that
In [18, 5] A.-M. Simon shows that
LIo(M)
= M and LIi(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
HI[bull ](M) ≅
LI[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.