Complete Cohomology for Extriangulated Categories

Let [Formula: see text] be an extriangulated category with a proper class [Formula: see text] of [Formula: see text]-triangles. We study complete cohomology of objects in [Formula: see text] by applying [Formula: see text]-projective resolutions and [Formula: see text]-injective coresolutions constructed in [Formula: see text]. Vanishing of complete cohomology detects objects with finite [Formula: see text]-projective dimension and finite [Formula: see text]-injective dimension. As a consequence, we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein. Moreover, we give a general technique for computing complete cohomology of objects with finite [Formula: see text]-[Formula: see text]projective dimension. As an application, the relations between [Formula: see text]-projective dimension and [Formula: see text]-[Formula: see text]projective dimension for objects in [Formula: see text] are given.

Relative Gorenstein Dimensions over Triangular Matrix Rings

Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.

Gorenstein-Projective Modules over Morita Rings

Let [Formula: see text] be a Morita ring which is an Artin algebra. In this paper we investigate the relations between the Gorenstein-projective modules over a Morita ring [Formula: see text] and the algebras [Formula: see text] and [Formula: see text]. We prove that if [Formula: see text] is a Gorenstein algebra and both [Formula: see text] and [Formula: see text] (resp., both [Formula: see text] and [Formula: see text]) have finite projective dimension, then [Formula: see text] (resp., [Formula: see text]) is a Gorenstein algebra. We also discuss when the CM-freeness and the CM-finiteness of a Morita ring [Formula: see text] is inherited by the algebras [Formula: see text] and [Formula: see text].

Igusa-Todorov function on path rings

The aim of this paper is to study the relation between the Igusa-Todorov functions for $A$, a finite dimensional algebra, and the algebra $AQ$. In particular it is proved that $\fidim (AQ) = \fidim (A) + 1$ when $A$ is a Gorenstein algebra. As a consequence of the previous result, it is exhibited an example of a family of algebras $\{A_n\}_{n \in \mathbb{N}}$ such that $\fidim (A_n) = n$ and each $A_n$ is of $\Omega^{\infty}$-infinite representation type.

An upper bound for the first Hilbert coefficient of Gorenstein algebras and modules

Let R R be a polynomial ring over a field and M = ⨁ n M n M= \bigoplus _n M_n be a finitely generated graded R R -module, minimally generated by homogeneous elements of degree zero with a graded R R -minimal free resolution F \mathbf {F} . A Cohen-Macaulay module M M is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, e 1 e_1 in terms of the shifts in the graded resolution of M M . When M = R / I M = R/I , a Gorenstein algebra, this bound agrees with the bound obtained in [ES09] in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound for the higher coefficients.

LOCAL DUALITY FOR THE SINGULARITY CATEGORY OF A FINITE DIMENSIONAL GORENSTEIN ALGEBRA

A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$ -local and $\mathfrak{p}$ -torsion subcategories of the derived category, for each homogeneous prime ideal $\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.

Endomorphism algebras of Gorenstein projective modules

Let [Formula: see text] be an Artin algebra, [Formula: see text] be a Gorenstein projective [Formula: see text]-module and [Formula: see text]. We give a characterization of modules on [Formula: see text] and show that if [Formula: see text] is [Formula: see text]-representation-finite, then [Formula: see text] is also [Formula: see text]-representation-finite. As an application, we prove if [Formula: see text] is a CM-finite [Formula: see text]-Gorenstein algebra, then [Formula: see text] is a [Formula: see text]-Igusa-Todorov algebra.

Hilbert regularity of Stanley–Reisner rings

In this note, we characterize the Hilbert regularity of the Stanley–Reisner ring [Formula: see text] in terms of the [Formula: see text]-vector and the [Formula: see text]-vector of a simplicial complex [Formula: see text]. We also compute the Hilbert regularity of a Gorenstein algebra.