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Modules cofinite with respect to an ideal. (English) Zbl 1093.13012
Let \(A\) be a commutative noetherian ring. An \(A\)-module \(M\) is \(\mathbf{a}\)-cofinite (\(\mathbf{a}\) an ideal of \(A\)) if \(\text{Supp}_{A} M \subset V ( \mathbf{a})\) and \(\text{Ext}_{A}^{i} (A/{\mathbf{a}},M)\) is a finite module for all \(i \in R\) [R. Hartshorne, Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)]. In a previous paper [Math. Proc. Cambridge Philos. Soc. 125, 417–423 (1999; Zbl 0921.13009)], the author showed that one could require the finiteness of the Koszul cohomology modules \(H^{i}(x_{1}, \dots , x_{n} ; M)\) where \(x_{1}, \dots x_{n}\) are generators for \(\mathbf{a}\), instead of the finiteness of the modules \(\text{Ext}_{A}^{i}(A/{\mathbf{a}}, M)\) in the definition of \(\mathbf{a}\)-cofiniteness. In the proof of this, the change of ring principle involving a spectral sequence argument was used T. Marley and J. Vassilev [J. Algebra 256, 180–193 (2002; Zbl 1042.13010)]. An alternative proof of the change of ring principle which avoids spectral sequences is supplied in this paper.
Various conditions for cofiniteness are given, for example if \(x \in \mathbf{a},\) \(\text{Supp}_{A} M \subseteq V(\mathbf{a})\) and both \(0:_{M}x\) and \(M/xm\) are \(\mathbf{a}\)-cofinite, then so is \(M\).
Minimax modules (i.e. modules that have a finite submodule such that the quotient by it is an artinian module) are also studied. A necessary and sufficient condition for an artinian module \(M\) with support in \(V(\mathbf{a})\) to be \(\mathbf{a}\)-cofinite is found. It is shown that if \(\dim(A)=1\), then every \(\mathbf{a}\)-cofinite module is a minimax module and the class of \(\mathbf{a}\)-cofinite modules is closed with respect to submodules and quotients.
For \(M\) a module over a ring of finite Krull dimension \(d\), the top cohomology module \(H^{d}_{a}(M)\) is studied. Conditions for \(H^{d}_{a}(M)\) to be \(\mathbf{a}\)-cofinite are found.
In the final part, the question of when the kernel of a homomorphism between \(\mathbf{a}\)-cofinite modules is again \(\mathbf{a}\)-cofinite, is considered. This is known for a complete local ring [D. Delfino and T. Marley, J. Pure Appl. Algebra 121, 45–52 (1997; Zbl 0893.13005)]. Though the author does not succeed in proving this for a non-complete ring, the question is reduced to the study of certain local cohomology modules. Results on the case where \(\dim(A) \leq 2\) are also obtained.

MSC:
13D45 Local cohomology and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
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