Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Rings over which Invertible Modulesare Free

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Rings over which all modules are semiregular

For a ring A, it is proved that all A-modules are semiregular if and only if A is an Artinian serial ring and J ²(A) = 0. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 185–194, 2007.     

Quasi-socle ideals in a Gorenstein local ring

This paper explores the structure of quasi-socle ideals I=Q:m² in a Gorenstein local ring A, where Q is a parameter ideal and m is the maximal ideal in A. The purpose is to answer the problems as to when Q is a reduction of I and when the associated graded ring is Cohen-Macaulay. Wild examples are explored.     

Rings over which all torsion theories cogenerated by simple modules are jansian

Summary It is given a caracterization of (associative) rings (with unit element) over which all torsion theories cogenerated by simple modules are jansian. In the commutative case it is shown that such rings are finite products of local rings.

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Riassunto Vengono caratterizzati gli anelli (associativi con unità) per i quali tutte le teorie di torsione cogenerate da moduli semplici sono Jansiane. Nel caso commutativo si prova che tali anelli sono prodotti finiti di anelli locali.

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This work was written while the second author was visiting Università dell'Aquila, supported by a grant from the Consiglio Nazionale delle Ricerche.     

Skew inverse power series rings over a ring with projective socle

A ring R is called a right PS-ring if its socle, Soc(R _R ), is projective. Nicholson and Watters have shown that if R is a right PS-ring, then so are the polynomial ring R[x] and power series ring R[[x]]. In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R[[x ^?1; α, δ]] and the skew polynomial ring R[x; α, δ], where R is an associative ring equipped with an automorphism α and an α-derivation δ. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.     

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

In this paper we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.     

On the higher delta invariants of a Gorenstein local ring

Let be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by for each module and for each integer . We propose a conjecture asking if for any positive integers and . We prove that this is true provided the associated graded ring of has depth not less than . Furthermore we show that there are only finitely many possibilities for a pair of positive integers for which .

     

Transfer of Gorenstein dimensions along ring homomorphisms

A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.     

Rings over which all modules are I 0-modules. II

All right R-modules are I ₀-modules if and only if either R is a right SV-ring or R/I ⁽²⁾ (R) is an Artinian serial ring such that the square of the Jacobson radical of R/I ⁽²⁾ (R) is equal to zero.     

Homology algebra of the Koszul complex of a local Gorenstein ring

It is proved that local Gorenstein rings have the property that the homology algebra of their Koszul complex is a Poincaré algebra.Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 53–58, January, 1971.     

The Lichtenbaum-Hartshorne theorem for modules which are finite over a ring homomorphism

Let φ:(R,m)→S be a flat ring homomorphism such that mS≠S. Assume that M is a finitely generated S-module with dim_R(M)=d. If the set of support of M has a special property, then it is shown that if and only if for each prime ideal satisfying , we have . This gives a generalization of the Lichtenbaum-Hartshorne vanishing theorem for modules which are finite over a ring homomorphism. Furthermore, we provide two extensions of Grothendieck’s non-vanishing theorem. Applications to connectedness properties of the support are given.