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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.)
##### Citations:
Zbl 0196.24301; Zbl 0921.13009; Zbl 1042.13010; Zbl 0893.13005
Full Text:
##### References:
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