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A characterization of cofinite complexes over complete Gorenstein domains. (English) Zbl 1252.13010
Let \(R\) be a Noetherian ring and \(J\) and ideal of \(R\). An \(R\)-module is said to be \(J\)-cofinite if \(\text{Supp} N \subset V(J)\) and \(\text{Ext}^i(R/J, N)\) is finitely generated for each \(j \geq 0\). On the other hand, the definition of \(J\)-cofiniteness of complexes is quite different. That is, a complex \(N^\bullet\) of \(R\)-modules is said to be \(J\)-cofinite if there is a complex \(M^\bullet\) with finitely generated cohomologies such that \[ N^\bullet \cong \text{RHom}(M^\bullet, {R \Gamma_J}(D^\bullet)), \] where \(D^\bullet\) is a dualizing complex of \(R\).
In [Math. Z. 269, No. 1-2, 587–608 (2011; Zbl 1228.13020)], the second author proved that if \(R\) is a regular local ring, it is \(J\)-adic compete and \(\dim R/J = 1\), then a bounded-below complex \(N^\bullet\) is \(J\)-cofinite if and only if all the cohomology modules of \(N^\bullet\) are \(J\)-cofinite. It is a partial answer to R. Hartshorne’s question [Invent. Math. 9, 145–164 (1970; Zbl 0196.24301)]. In the present paper, the authors replace the assumption “\(R\) is a regular local ring and \(J\)-adic complete” with “\(R\) is a Gorenstein local domain.”

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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