Gorenstein Dimension of Modules over Homomorphisms Lars Winther Christensen and Srikanth Iyengar
نویسنده
چکیده
Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | TorRm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This result is analogous to a theorem of André on flat dimension.
منابع مشابه
G-dimension over Local Homomorphisms. Applications to the Frobenius Endomorphism
We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an R-modul...
متن کاملGorenstein Dimension of Modules over Homomorphisms
Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | Torm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This re...
متن کاملAscent Properties of Auslander Categories
Let R be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over R. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rin...
متن کاملBeyond Totally Reflexive Modules and Back A Survey on Gorenstein Dimensions
Lars Winther Christensen, Hans-Bjørn Foxby, and Henrik Holm 1 Department of Mathematics and Statistics, Texas Tech University, Mail Stop 1042, Lubbock, TX 79409, U.S.A. [email protected] 2 Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København Ø, Denmark [email protected] 3 Department of Basic Sciences and Environment, University of Copenh...
متن کاملGorenstein flat and Gorenstein injective dimensions of simple modules
Let R be a right GF-closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorensntein flat dimensnion of R/I as a right R-module and the Gorensntein injective dimensnnion of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorensntein ring R, ...
متن کامل