Existence of Gorenstein Projective Resolutions
نویسنده
چکیده
Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of Gorenstein rings has led to the advent of a whole branch of homological algebra, known as Gorenstein homological algebra. This paper solves one of the open problems of Gorenstein homological algebra by showing that so-called Gorenstein projective resolutions exist over quite general rings, thereby enabling the definition of a Gorenstein version of derived functors. An application is given to the theory of Tate cohomology.
منابع مشابه
Existence of Gorenstein Projective Resolutions and Tate Cohomology
Existence of proper Gorenstein projective resolutions and Tate cohomology is proved over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.
متن کاملA Brief Introduction to Gorenstein Projective Modules
Since Eilenberg and Moore [EM], the relative homological algebra, especially the Gorenstein homological algebra ([EJ2]), has been developed to an advanced level. The analogues for the basic notion, such as projective, injective, flat, and free modules, are respectively the Gorenstein projective, the Gorenstein injective, the Gorenstein flat, and the strongly Gorenstein projective modules. One c...
متن کاملThe existence totally reflexive covers
Let $R$ be a commutative Noetherian ring. We prove that over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover$varphi:C rightarrow M$ such that $C$ is finitely generated and the projective dimension of $Kervarphi$ is finite and $varphi$ is surjective.
متن کاملAll Abelian Quotient C.i.-singularities Admit Projective Crepant Resolutions in All Dimensions All Abelian Quotient C.i.-singularities Admit Projective Crepant Resolutions in All Dimensions
For Gorenstein quotient spaces C d =G, a direct generalization of the classical McKay correspondence in dimensions d 4 would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstei...
متن کاملOn crepant resolutions of 2-parameter series of Gorenstein cyclic quotient singularities
An immediate generalization of the classical McKay correspondence for Gorenstein quotient spaces C/G in dimensions r ≥ 4 would primarily demand the existence of projective, crepant, full desingularizations. Since this is not always possible, it is natural to ask about special classes of such quotient spaces which would satisfy the above property. In this paper we give explicit necessary and suf...
متن کامل