Strongly Gorenstein graded modules

Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.     

Gorenstein gr-injective modules over graded isolated singularities

Graded isolated singularities appear very naturally in algebraic pro-jcctive geometry (cf. [18]) or invariant theory of the binary polyhedral groups (cf. [15], [17]). We define mock finitely generated modules in the category of graded modules and we use such graded modules which are also Gorenstein injective (cf. [1], [2]) in this category to characterize graded isolated Gorenstein singularities.     

Gorenstein injective complexes of modules over Noetherian rings

A complex C is called Gorenstein injective if there exists an exact sequence of complexes $?\to I_{\mathrm{?1}}\to I_0\to I_1\to ?$ such that each $I_i$ is injective, $C=\mathrm{Ker}(I_0\to I_1)$ and the sequence remains exact when $\mathrm{Hom}(E,?)$ is applied to it for any injective complex E. We show that over a left Noetherian ring R, a complex C of left R-modules is Gorenstein injective if and only if $C^m$ is Gorenstein injective in R-Mod for all $m\in \mathbb{Z}$. Also Gorenstein injective dimensions of complexes are considered.     

Poincaré series of modules over compressed Gorenstein local rings

Given two positive integers e and s we consider Gorenstein Artinian local rings R   whose maximal ideal m$\mathfrak{m}$ satisfies m^s≠0=m^s+1${\mathfrak{m}}^s\ne 0={\mathfrak{m}}^{s+1}$ and _rankR/m(m/m²)=e${\mathrm{rank}}_{R/\mathfrak{m}}(\mathfrak{m}/{\mathfrak{m}}^2)=e$. We say that R is a compressed Gorenstein local ring   when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3$s\ne 3$, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s  . Note that for s=3$s=3$ examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad.     

Gorenstein fp-平坦和强Gorenstein fp-平坦模

引入了Gorenstein fp-平坦模和强Gorenstein fp-平坦模的概念,讨论了这两类模的一些性质、联系以及稳定性.     

Anti-isomorphisms of graded endomorphism rings of graded modules close to free ones

We obtain criteria that answer the question of when an anti-isomorphism of graded endomorphism rings of the strict gr-generators is induced by a graded Morita anti-equivalence or a graded anti-semilinear isomorphism.     

Isomorphisms of graded endomorphism rings of graded modules close to free ones

We obtain criteria that answer the question of when an isomorphism of graded endomorphism rings of strong gr-generators is induced by a gr-generator, graded Morita equivalence, or semi-linear isomorphism. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 3–18, 2007.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.