Foxby equivalence, local duality and Gorenstein homological dimensions

Let \((R,\fm)\) be a local ring and \((-)^{\vee}\) denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that \(R\) possesses a normalized dualizing complex \(D\) and \(X\) and \(Y\) are two...

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Bibliographic Details
Published in:arXiv.org 2012-01
Main Authors: Fatemeh Mohammadi Aghjeh Mashhad, Divaani-Aazar, Kamran
Format: Article
Language:English
Subjects:
Equivalence
Homology
Modules
Rings (mathematics)
Online Access:Get full text
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Summary:Let \((R,\fm)\) be a local ring and \((-)^{\vee}\) denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that \(R\) possesses a normalized dualizing complex \(D\) and \(X\) and \(Y\) are two homologically bounded complexes of \(R\)-modules with finitely generated homology modules. We present several duality results for \(\fm\)-section complex \({\bf R}\Gamma_{\fm}({\bf R}\Hom_R(X,Y))\). In particular, if G-dimension of \(X\) and injective dimension of \(Y\) are finite, then we show that $${\bf R}\Gamma_{\fm}({\bf R}\Hom_R(X,Y))\simeq ({\bf R}\Hom_R(Y,D\otimes_ R^{{\bf L}}X))^{\vee}.$$ We deduce several applications of these duality results. In particular, we establish Grothendieck's non-vanishing Theorem in the context of generalized local cohomology modules.
ISSN:2331-8422