In this paper, we show the existence of attached comultiplication structure on $\operatorname{Hom}_{eAe}(eA,D(Ae))$ if an $(eAe,A)$-bimodule $eA$ has double centraliser property over algebra $A$ with idempotent $e$. Then apply it gendo-Gorenstein algebras. As applications, give a sufficient and necessary condition for to be Gorenstein, bocs-theoretic characterisation property.

It was proved by Avramov and Buchweitz that if A is a commutative local complete intersection ring with finitely generated modules M and N , then the Ext groups between M and N vanish from some step if and only if the Ext groups between N and M vanish from some step. This paper shows that the same is true under the weaker conditions that A is Gorenstein and that M and N have finite complete int...

LetR be a complete local ring of dimension d over a perfect field of prime characteristic p, and let M be an R-module of finite length and finite projective dimension. S. Dutta showed that the equality limn→∞ `(F n R(M)) pnd = `(M) holds when the ring R is a complete intersection or a Gorenstein ring of dimension at most 3. We construct a module over a Gorenstein ring R of dimension five for wh...

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...

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves over a semi-separated noetherian scheme, generalising complete flat resolutions over a ring. By studying these complexes as objects of the pure derived category of flat sheaves we extend several results about totally acyclic complexes of projective modules to schemes; for example, we prove that a scheme is Goren...

We give a precise formulation of the modularity principle for the log canonical models Mg(α) := Proj ⊕ d≥0 H (Mg , bd(KMg +αδ)c). We define a new invariant of Gorenstein curve singularities with Gm-action which can be used to predict the critical α-value at which a singularity first arises in the modular interpretation of Mg(α). We compute these critical α-values for large classes of singularit...

After the structure theorem of Buchsbaum and Eisenbud [1] on Gorenstein ideals of codimension 3, much progress was made in this area from the algebraic point of view; in particular some characterizations of these ideals using h−vectors (Stanley [9]) and minimal resolutions (Diesel [3]) were given. On the other hand, the Liaison theory gives some tools to exploit, but, at the same time, it requi...

We give some experimental data of Gorenstein liaison, working with points in P and curves in P , to see how far the familiar situation of liaison, biliaison, and Rao modules of curves in P will extend to subvarieties of codimension 3 in higher P . AMS classification numbers: 14H50, 14M07

Article history: Received 26 March 2009 Available online 23 October 2010 Communicated by Efim Zelmanov MSC: 16E05 16E10

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...