Complete Cohomology and Gorensteinness of Schemes ✩
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 319 (2008) 2626–2651 www.elsevier.com/locate/jalgebra Complete cohomology and Gorensteinness of schemes ✩ J. Asadollahi a,b,∗, F. Jahanshahi c,Sh.Salarianc,b a Department of Mathematics, Shahre-Kord University, PO Box 115, Shahre-Kord, Iran b School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5746, Tehran, Iran c Department of Mathematics, University of Isfahan, PO Box 81746-73441, Isfahan, Iran Received 20 April 2007 Available online 21 December 2007 Communicated by Steven Dale Cutkosky Abstract We develop and study Tate and complete cohomology theory in the category of sheaves of OX-modules. Different approaches are included. We study the properties of these theories and show their power in reflect- ing the Gorensteinness of the underlying scheme. The connection of these two theories will be discussed. © 2007 Elsevier Inc. All rights reserved. Keywords: Gorenstein scheme; Locally free sheaf; Cohomology of sheaves; Homological dimension; Complete cohomology Contents 1. Introduction . 2627 2. Totally reflexive sheaves . 2628 2.1. Examples and descriptions . 2629 2.2. Gorenstein homological dimension . 2632 3. Tate cohomology sheaves over noetherian schemes . 2634 ✩ This research was in part supported by a grant from IPM (Nos. 86130019 and 86130024). The first author thanks the Center of Excellence of Algebraic Methods and Applications of Isfahan University of Technology (IUT-CEAMA). The second and the third authors thank the Center of Excellence for Mathematics (University of Isfahan). * Corresponding author at: Department of Mathematics, Shahre-Kord University, PO Box 115, Shahre-Kord, Iran.
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