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On a category of cofinite modules which is abelian. (English) Zbl 1228.13020
Math. Z. 269, No. 1-2, 587-608 (2011); addendum ibid. 275, No. 1-2, 641-646 (2013).
Let $$I$$ be an ideal of a commutative noetherian local ring $$(R,\mathfrak{m})$$ and $$M$$ a finitely generated $$R$$-module. In 1968, A. Grothendieck [Séminaire de géométrie algébrique: Cohomologie locale des faisceaux cohérents et théoremes de Lefschetz locaux et globaux (1962; Zbl 0159.50402)] has conjectured that $$\operatorname{Hom}_R(R/I,H_{I}^i(M))$$ is finitely generated for all $$i\geq 0$$. R. Hartshorne gave a counter-example to this conjecture [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)]. In the same paper, Hartshorne introduced the notion of $$I$$-cofinite modules and asked the following question: For which local rings $$R$$ and ideals $$I$$ is $$H_{I}^i(N)$$ $$I$$-cofinite for all $$i\geq 0$$ and any finitely generated $$R$$-module $$N$$? An $$R$$-module $$X$$ is said to be $$I$$-cofinite if $$\mathrm{Supp}_RX\subseteq V(I)$$ and $$\mathrm{Ext}^i_R(R/I,X)$$ is finitely generated for all $$i\geq 0$$. Now, it is known that if either $$I$$ is principal or $$\dim R/I=1$$, then $$H_{I}^i(N)$$ is $$I$$-cofinite for all $$i\geq 0$$ and any finitely generated $$R$$-module $$N$$.
Let $$\mathcal{M}(R,I)_{cof}$$ denote the full subcategory of $$I$$-cofinite $$R$$-modules. When $$I$$ is principal, Hartshorne showed that $$\mathcal{M}(R,I)_{cof}$$ is abelian. He asked whether $$\mathcal{M}(R,I)_{cof}$$ is abelian when $$\dim R/I=1$$. In the paper under review, the author provides an affirmative answer to this questions.

##### MSC:
 13D45 Local cohomology and commutative rings 13D09 Derived categories and commutative rings
##### Citations:
Zbl 0196.24301; Zbl 0159.50402
Full Text:
##### References:
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