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On the characterizations of cofinite complexes over affine curves and hypersurfaces. (English) Zbl 1386.13042
In [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)] R. Hartshorne asked about four questions. The fourth one was as follows:
Question. Let $$R$$ be a regular ring of finite Krull dimension and $$I$$ an ideal of $$R$$. Suppose that $$R$$ is complete with respect to an $$I$$-adic topology. Then does there exist an abelian subcategory $$\mathcal{M}_{cof}$$ consisting of $$R$$-modules, such that $$I$$-cofinite complexes $$N^*$$ are characterized by the property $$\mathrm{H}^i (N^*) \in \mathcal{M}_{cof}$$ for all $$i$$?
Let $$A$$ be a homomorphic image of a Gorenstein ring of finite Krull dimension, $$J$$ an ideal of $$A$$ of dimension one, and $$N^*$$ a bounded below complex of $$A$$-modules. suppose that $$A$$ is complete with respect to a $$J$$-adic topology. In this paper, it is proved that $$N^*$$ is a $$J$$-cofinite complex if and only if $$\mathrm{H}^i(N^*)$$ is a $$J$$-cofinite module for all $$i$$. The same result is also proved for principal ideals $$J$$. Consequently, for the above question, one obtain an answer over the ring, on affine curves and hypersurfaces.
Note: The section 4 of the reviewer’s paper [J. Pure Appl. Algebra 95, No. 1, 103–119 (1994; Zbl 0843.13005)] gives some information about cofinite complexes.
##### MSC:
 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 14B15 Local cohomology and algebraic geometry 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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