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Properties of cofinite modules and applications to local cohomology. (English) Zbl 0921.13009
The paper contains some nice results about the so-called $$I$$-cofinite modules. Let $$A$$ be a noetherian ring, $$M$$ an $$A$$-module, $$I$$ an ideal of $$A$$. $$M$$ is called $$I$$-cofinite if $$\text{Supp}(M)\subseteq V(I)$$ and $$\text{Ext}^i_A(A/I,M)$$ is finitely generated over $$A$$ for all $$i$$.
The first result is that $$M$$ is $$I$$-cofinite iff $$\text{Supp}(M)\subseteq V(I)$$ and $$H^i(x_1,\dots,x_n;M)$$ is a finite $$A$$-module for all $$i$$, where $$x_1,\dots,x_n$$ is a system of generators of $$I$$. The second one is that if $$M$$ is $$I$$-cofinite for some ideal $$I$$, then all Bass numbers $$\mu_i(P,M)$$ and all Betti numbers $$\beta_i(P,M)$$ are finite for all prime ideals $$P$$ and all $$i$$.
As a corollary, if $$I$$ is an ideal such that $$\dim(A/I)=1$$ and $$M$$ is an $$A$$-module then the Betti numbers of $$H^i_I(M)$$ are finite for all $$i$$. If $$A$$ is local, $$\dim(A)\leq 2,I$$ is an ideal of $$A$$, $$M$$ is a finitely generated $$A$$-module then all $$H^i_I(M)$$ are $$I$$-cofinite. Finally, it is shown that for every local 3-dimensional local ring $$A$$ there is an ideal $$I$$ such that $$H^2_I(A)$$ is not $$I$$-cofinite.
Editorial remark: N. Abazari and K. Bahmanpour [Commun. Algebra 42, No. 3, 1270–1275 (2014; Zbl 1291.13028)] gave a counterexample to Theorem 1.6, which seems to be only true if one assumes $$R$$ is complete.

MSC:
 13D45 Local cohomology and commutative rings 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
Zbl 1291.13028
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