On the annihilation of local cohomology modules
In this work a generalization of Faltings' annihilator theorem is proven. Here are some details: Not all local cohomology modules (LCM) are finitely generated (in some sense, most of them are not finite). Faltings realized that the question of finite generatedness of LCM is related to certain properties of the annihilators of LCM. \textit{G. Faltings'} famous annihilator theorem [Arch. Math. 30, 473--476 (1978; Zbl 0368.14004)] says under which conditions these annihilator properties are fulfilled. It holds for regular rings - its proof uses the Auslander-Buchsbaum formula, in particular projective dimension. \textit{K. Khashyarmanesh} and \textit{Sh. Salarian} [Proc. Am. Math. Soc. 132, No. 8, 2215--2220 (2004; Zbl 1099.13510)] generalized the annihilator theorem to Gorenstein rings, using Gorenstein dimension. Now in this paper an even more general version of the annihilator theorem is proven, namely a version for Cohen-Macaulay rings, using the so-called Gorenstein dimension with respect to a semi-dualizing module \(C\). In addition, a version of uniform annihilation is proven in this general context.
scientific article; zbMATH DE number 5128992