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The I-adic completion and local homology for Artinian modules

Published online by Cambridge University Press:  26 October 2001

Institute of Mathematics, P.O. Box 631, BoHo, 10.000 Hanoi, Vietnam. e-mail:
Institute of Mathematics, P.O. Box 631, BoHo, 10.000 Hanoi, Vietnam. e-mail:


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) = limtM/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.

Research Article
2001 Cambridge Philosophical Society

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This work is supported in part by the National Basis Research Programme in Natural Science of Vietnam.