The <EM>I</EM>-adic completion and local homology for Artinian modules

Details The <EM>I</EM>-adic completion and local homology for Artinian modules The <EM>I</EM>-adic completion and local homology for Artinian modules

Jul 2001

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001mpcps.131...61c&link_type=abstract

Mathematical Proceedings of the Cambridge Philosophical Society, vol. 131, Issue 1, p.61-72

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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 [Lambda]I defined by [Lambda]I(M) = lim[leftward arrow]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, [Lambda]I 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 [Lambda]I, in which LI0 is right exact, but in general LI0 [not equal] [Lambda]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 [fraktur m] and I is generated by a R-regular sequence, Matlis proved in [9, 10] thatwhere D([minus sign]) = HomR([minus sign]; E(R/[fraktur m])) is the Matlis dual functor, and thatIn [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) bywhere 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•(M) [congruent with] LI•(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.

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