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On the category of confinite modules for principal ideals. (English) Zbl 1247.14003
Let \(R\) be a noetherian commutative ring and \(I\) be an ideal of \(R\). Let \({\mathcal M}(R,I)_{\mathrm{cof}}\) be the collection of all \(R\)-modules \(X\) satisfying the two conditions: i) \(\mathrm{Supp}_R(X)\subseteq V(I)\) and ii) \(\mathrm{Ext}_R^j(R/I,X)\) is of finite type, for all \(j\). In this paper, it is shown that if \(I\) is generated by an element up to radical, then \({\mathcal M}(R,I)_{\mathrm{cof}}\) is an abelian full subcategory of the category of all \(R\)-modules. A counterexample for ideals generated by two elements was given by R. Hartshorne [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)].

MSC:
14B15 Local cohomology and algebraic geometry
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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References:
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