The membership of the local cohomology modules H n a (M) of a module M in certain Serre subcategories of the category of modules is studied from below (i < n) and from above (i > n). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module.
The notion of associated prime ideal and the related one of primary decomposition are classical. In a dual way one defines attached prime ideals and secondary representation. This theory is developed in the appendix to §6 in Matsumura[5] and in Macdonald[3].
Definition [4]. Let A be a
noetherian ring, [afr ] an ideal of A and M an A-module.
M is said to be [afr ]-cofinite if M has support in V([afr ])
and
ExtiA(A/[afr ], M)
is a finite
A-module for each i.Remark. (a) If
0→M′→M→M″ →0 is exact
and two of the modules in the
sequence are [afr ]-cofinite, then so is the third one.This has the following consequence, which will be used several times.(b) If f[ratio ]M→N is a homomorphism
between two [afr ]-cofinite modules and one of
the three modules Ker f, Im f and Coker f is
[afr ]-cofinite, then all three of them are
[afr ]-cofinite.Example [5, remark 1·3]. If A
is local with maximal ideal [mfr ], then an A-module is
[mfr ]-cofinite if and only if it is an artinian A-module.
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