Peter Jorgensen
A vanishing theorem is proved for Ext groups over non-commutative graded algebras. Along the way, an "infinite" version is proved of the non-commutative Auslander-Buchsbaum theorem.
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An Auslander-Buchsbaum formula for higher Auslander algebras and applications
本文建立高阶 Auslander 代数的非交换 Auslander–Buchsbaum 公式，据此证明倾斜 Auslander 代数与整体维数不超过2的 QF-1 代数等同，并给出分类及向更高同调维数的推广。
Quasi-projective dimensions of complexes over rings
本文将关联环上模的准射影维数推广到复形模上，研究其基本性质，包括与射影维数的比较及有限准射影维数复形的导出Auslander-Buchsbaum公式，并给出若干条件使局部诺特环为完全交集环。
In studying Nakayama's 1958 conjecture on rings of infinite dominant dimension, Auslander and Reiten proposed the following generalization: Let Lambda be an Artin algebra and M a Lambda-generator such that Ext^i_Lambda(M,M)=0 for all i \geq 1; then M is projective. This conjecture makes sense for any ring. We establish Auslander and Reiten's conjecture for excellent Cohen--Macaulay normal domains containing the rational numbers, and slightly more generally.
Let $R$ be a commutative artinian ring. We extend higher Auslander correspondence from Artin $R$-algebras of finite representation type to dualizing $R$-varieties. More precisely, for a positive integer $d$, we show that a dualizing $R$-variety is $d$-abelian if and only if it is a $d$-Auslander dualizing $R$-variety if and only if it is equivalent to a $d$-cluster-tilting subcategory of the category of finitely presented modules over a dualizing $R$-variety.
Infinitely Many Components in Auslander--Reiten Quivers of Representation-Infinite Algebras over Perfect Fields
研究完美域上表示无限的有限维 \(k -\) 代数的奥斯特兰德 - 赖滕箭图，通过结合局部化多项式表示嵌入、半线性扭转及可分基变换等方法，证明其有无限多个连通分支，确立相关猜想。
The Auslander--Reiten conjecture for quantum complete intersections
本文证明任意域上有限维量子完全交满足 Auslander--Reiten 猜想与 Tachikawa 第二猜想，通过刚性模分级扭转与自扩张判据实现。
An approach to Martsinkovsky's invariant via Auslander's approximation theory
本文利用Auslander逼近理论构造收敛于Martsinkovsky $ξ$-不变量的非降整数序列，并给出计算方法。
In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $Λ$, we show that the right self-injective dimension of $Λ$ is at most $k$ if and only if each $M \in$mod $Λ$ satisfying the condition that Ext$_Λ^i(M, Λ)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $Λ$ are identical. In addition, we show that if $Λ$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $Λ$ with grade$M$ finite, then Ext$_…
We extend the definition of a semidualizing module to associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of C-flats, C-projectives, and C-injectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new in the commutative, noetherian setting.
A higher algebraic approach to liftings of modules over derived quotients
本文用高阶代数方法，在自Ext^2消没条件下证明导出商上模提升的存在性，并借助Auslander零因子定理，得出该条件等价于局部完全交环。
In this short paper we study for the skew PBW (Poincaré-Birkhoff-Witt) extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. For some key examples we present the …
In commutative invariant theory, a classical result due to Auslander says that if $R = \Bbbk[x_1, \dots, x_n]$ and $G$ is a finite subgroup of $\text{Aut}_{\text{gr}}(R) \cong \text{GL}(n,\Bbbk)$ which contains no reflections, then there is a natural graded isomorphism $R \hspace{1pt} \# \hspace{1pt} G \cong \text{End}_{R^G}(R)$. In this paper, we show that a version of Auslander's Theorem holds if we replace $R$ by an Artin-Schelter regular algebra $A$ of global dimension 2, and $G$ by a finite subgroup of $\text{Aut}_{\text{gr}}(A)$ which contains no quasi-reflections. This extends work of C…
A ring R satisfies the Generalized Auslander-Reiten Condition if any R-module M with no self-extensions in degrees higher than m must have projective dimension at most m. We prove that this condition is satisfied by all n-symmetric algebras of quasitilted type- a broad class of self-injective algebras where every module is periodic under the Nakayama automorphism.
Auslander-Reiten combinatorics and $q$-characters of representations of affine quantum groups
本文通过扩展同态 $\widetilde{D}_\xi$ 并证明标准模的 $q$-特征在其核中，揭示了 Auslander-Reiten 组合学与仿射量子群表示 $q$-特征之间的深层联系。
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