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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.
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)
Full Text: DOI
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