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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.

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