Gorenstein weak global dimension is symmetric

Gorenstein derived functors from special rings to arbitrary rings

In this note, we mainly extend some Gorenstein homological properties from special rings (Noetherian or coherent rings ) to arbitrary rings by introducing the notions of Gorenstein weak injective and weak projective modules respectively.

弱Hopf Galois扩张的Hochschild上同调与AS-Gorenstein性质

Hochschild cohomology and AS-Gorenstein property of weak Hopf Galois extensions

本文针对具有双射反极的弱Hopf代数H，研究受限忠实平坦弱H-Galois扩张A/B的AS-Gorenstein性质，建立联系A与B的Hochschild上同调的谱序列，证明当A为诺特仿射PI代数、B为AS-Gorenstein时，A也具有该性质。

A Note on Gorenstein Projective Conjecture II

In this paper, we prove that Gorenstein projective conjecture is left and right symmetric and the co-homology vanishing condition can not be reduced in general. Moreover, the Gorenstein projective conjecture is proved to be true for CM-finite algebras.

Gorenstein $\mathrm{FP}_n$-平坦模与弱整体维数

Gorenstein $\mathrm{FP}_n$-flat modules and weak global dimensions

本文研究广义 Gorenstein $\mathrm{FP}_n$-平坦模的相对弱整体维数，证明其在 Gorenstein $n$-凝聚环上有限并等于右 $\mathrm{FP}_n$-内射模的平坦维数，且获得平衡对。

相对弱全局Gorenstein维数、AB-语境与模型结构

Relative weak global Gorenstein dimension, AB-contexts and model structures

本文研究环关于模的弱Gorenstein全局维数，给出有界性刻画，并用半对偶双模统一计算左右模的G_C-平坦维数，从而为Bennis猜想提供新证明，同时刻画了G_C-平坦模型结构中的弱等价。

具有有限 Gorenstein 弱整体维数的环上的同伦等价与 Grothendieck 对偶

Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension

本文证明具有有限 Gorenstein 弱整体维数的环满足同伦范畴的三角等价，并将其应用于 Ding-Chen 环和 Gorenstein n-相干环的 Grothendieck 对偶。

Gorenstein modifications and $\mathbb{Q}$-Gorenstein rings

Let $R$ be a Cohen--Macaulay normal domain with a canonical module $ω_R$. It is proved that if $R$ admits a noncommutative crepant resolution (NCCR), then necessarily it is $\mathbb{Q}$-Gorenstein. Writing $S$ for a Zariski local canonical cover of $R$, then a tight relationship between the existence of noncommutative (crepant) resolutions on $R$ and $S$ is given. A weaker notion of Gorenstein modification is developed, and a similar tight relationship is given. There are three applications: non-Gorenstein quotient singularities by connected reductive groups cannot admit an NCCR, the centre of…

Gorenstein projective modules over rings of Morita contexts

Under semi-weak and weak compatibility of bimodules, we establish sufficient and necessary conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This generalises and extends results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita con…

环的经典与Gorenstein同调不变量

On classical and Gorenstein homological invariants of rings

本文证明任意环R上平坦左R-模的投射维数上确界等于Gorenstein平坦左R-模的Gorenstein投射维数上确界，由此得到左n-完全环的新刻画，并引入经典同调不变量的Gorenstein类比，研究其关系，最后给出左弱n-Σ-余挠环的Gorenstein类刻画。

Cotorsion对、厚子范畴与有限生成Gorenstein投射模

Cotorsion pairs, thick subcategories, and finitely generated Gorenstein projective modules

在Cohen-Macaulay环上的诺特代数上，证明有限生成Gorenstein投射模构成遗传cotorsion对的左半部分，并回答Takahashi的问题，刻画左弱Gorenstein性质。

$(\mathcal{F},\mathcal{A})$-Gorenstein flat homological dimensions

In this paper we develop the homological properties of the $(\mathcal{L}, \mathcal{A})$-Gorenstein flat $R$-modules $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ proposed by Gillespie. Where the class $\mathcal{A} \subseteq \mathrm{Mod} (R^{op})$ sometimes corresponds to a duality pair $(\mathcal{L}, \mathcal{A})$. We study the weak global and finitistic dimensions that comes with $\mathcal{GF}_{(\mathcal{F}(R), \mathcal{A})}$ and show that over a $(\mathcal{L}, \mathcal{A})$-Gorenstein ring, the functor $-\otimes _R -$ is left balanced over $\mathrm{Mod} (R^{op}) \times \mathrm{Mod} (R)$ by …

Auslander条件下自内射维数的对称性

Symmetry of self-injective dimensions under the Auslander condition

针对交换Noether环上的模有限代数A，给出n-Gorenstein条件的局部判据，并证明满足Auslander条件时A的自内射维数具有左右对称性。

Relative Gorenstein dimensions over triangular matrix rings

Let $A$ and $B$ be rings, $U$ a $(B,A)$-bimodule and $T=\begin{pmatrix} A&0\\U&B \end{pmatrix}$ 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 $G_C$-projective modules over $T$. As an application, we study when a morphism in $T$-Mod has a special $G_CP(T)$-precover and when the class $G_CP(…

cleft扩张下的同调不变性质

Homological invariant properties under cleft extensions

本文研究环的cleft扩张下Gorenstein弱整体维数的有限性不变性，并比较相对奇点范畴，应用于θ-扩张和Morita context环。

Sally类型数半群环的结构与对称性

Structure and Symmetry of Sally Type Semigroup Rings

本文研究Sally类型半群的结构与对称性，证明了特定条件下半群为Gorenstein，并构造了极小自由解析及Betti数计算。

The stable module category of a general ring

For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each other and with the usual stable module category. Our stable module categories are homotopy categories of Quillen model structures on the category of R-modules. These model categories involve generalizations of Gorenstein projective and injective modules that we derive by replacing finitely presented modules by modules of type FP-infinity. Along the way, we …

The Existence of Maximal $n$-Orthogonal Subcategories

For an $(n-1)$-Auslander algebra $Λ$ with global dimension $n$, we give some necessary conditions for $Λ$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $Λ$-modules with projective dimension $n-1$ or $n$. For an almost hereditary algebra $Λ$ with global dimension 2, we prove that $Λ$ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable $Λ$-module $M$, $M$ is injective is equivalent to that the reduced grade of $M$ is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence o…

Invariants of (-1)-Skew Polynomial Rings under Permutation Representations

The symmetric group S_n acts on the skew polynomial ring A=k_{-1}[x_1,..., x_n] as permutations. For subgroups G of S_n we consider properties of the ring of invariants A^G. We show that A^G is always Artin-Schelter Gorenstein, and we investigate when A^G is a "complete intersection", in a sense we define. We obtain bounds on the degrees of the algebra generators. The properties of A^G are compared with those in the classical case, k[x_1, ..., x_n]^G.

Weakly associative and symmetric Leibniz algebras

We study a special class of weakly associative algebras: the symmetric Leibniz algebras. We describe the structure of the commutative and skew symmetric algebras associated with the polarization-depolarization principle. We also give a structure theorem for the symmetric Leibniz algebras and we study formal deformations in the context of deformation quantization.