Article
Mathematics, Applied
Ruipeng Zhu
Summary: We prove a version of Auslander's theorem for finite group actions or coactions on noetherian polynomial identity Artin-Schelter regular algebras.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
E. Kirkman, R. Won, J. J. Zhang
Summary: We study semisimple Hopf algebra actions on Artin-Schelter regular algebras and prove upper bounds on the degrees of minimal generators of the invariant subring, as well as the degrees of syzygies of modules over the invariant subring. These results resemble those obtained by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds for group actions on commutative polynomial rings.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Ruipeng Zhu
Summary: This article provides a formula for computing the discriminant of skew Calabi-Yau algebra over a central Calabi-Yau algebra, and applies this method to study the Jacobian and discriminant for reflection Hopf algebras.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Hui-Xiang Chen, Ding-Guo Wang, James J. Zhang
Summary: This paper classifies all inner-faithful U-actions on noetherian Koszul Artin-Schelter regular algebras of global dimension up to three.
ALGEBRAS AND REPRESENTATION THEORY
(2023)
Article
Mathematics
E. Kirkman, R. Won, J. J. Zhang
Summary: This paper introduces and studies weighted sums of homological and internal degrees of cochain complexes of graded A-modules, providing weighted versions of Castelnuovo-Mumford regularity, Tor-regularity, Artin-Schelter regularity, and concavity. In some cases, an infinite invariant can be replaced with a weighted invariant that is finite, and several homological invariants of complexes can be expressed as weighted homological regularities. The paper proves several weighted homological identities that unify different classical homological identities and generate interesting new ones.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Runar Ile
Summary: In this paper, we established an axiomatic parametrised Cohen-Macaulay approximation method, which was mainly applied to pairs consisting of a finite type flat family of Cohen-Macaulay rings and modules. We studied the induced maps of deformation functors and deduced properties like smoothness and injectivity under general, mainly cohomological conditions on the module.
JOURNAL OF ALGEBRA
(2021)
Article
Mathematics, Applied
Jun Li
Summary: This paper investigates Artin-Schelter regular algebras of dimension 5 with three generators in degree 1, under the assumption that GKdim >= 4. It determines the degree types of the relations for the number of the generating relations less than five. The study proves that the only possible degree type for three generating relations is (2, 2, 3), and the only possible degree type for four generating relations is (2, 2, 3, 4).
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2022)
Article
Mathematics
Liran Shaul
Summary: In this study, a Cohen-Macaulay version of a result about Gorenstein rings was proven and extended to commutative DG-rings. A new technique for studying the dimension theory of a Noetherian ring A was developed, showing results about A by utilizing a Cohen-Macaulay DG-ring. Applications include proving the Cohen-Macaulay property of homotopy fibers in certain schemes and generalizing the miracle flatness theorem, with extensions to derived algebraic geometry.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Applied
Haijun Cao, Fang Xiao
Summary: The main aim of this study is to characterize affine weak k-algebra H with regular nilpotent structure. As preparation, we investigate some properties of weak Hopf algebra morphisms and prove the adjunction between the category C of weak Hopf algebras whose weak antipodes are anti-algebra morphisms. Then, we prove the main result of this study: the bijective correspondence between the category of affine algebraic k-regular monoids and the category of finitely generated commutative reduced weak k-Hopf algebras.
Article
Mathematics
Simon Crawford
Summary: In this paper, we study the action of a semisimple Hopf algebra on an m-Koszul Artin-Schelter regular algebra and propose a method to calculate the homological determinant of the action. Using this method, we prove that the smash product of the algebra is also a derivation-quotient algebra and generalize a result by constructing a quiver algebra lambda and applying the Auslander map. Several examples are computed, and our techniques are applied to derive results for quantum Kleinian singularities studied by Chan-Kirkman-Walton-Zhang.
ALGEBRAS AND REPRESENTATION THEORY
(2022)
Article
Mathematics
Xin Tang, Helbert J. Venegas Ramirez, James J. Zhang
Summary: This study focuses on a noncommutative version of the Zariski cancellation problem for certain classes of connected graded Artin-Schelter regular algebras with global dimension three.
PACIFIC JOURNAL OF MATHEMATICS
(2021)
Article
Mathematics
Harshit Yadav
Summary: In this article, the discussions mainly revolve around finite tensor categories and exact left C-module categories. The concept of unimodular module categories is introduced, and various characterizations, properties, and examples are provided. Two applications of unimodular module categories are presented: the construction of (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category, and the classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra.
ADVANCES IN MATHEMATICS
(2023)
Article
Mathematics, Applied
Yongyun Qin
Summary: A 4-recollement of derived categories of algebras leads to a 2-recollement of the corresponding f-Cohen-Macaulay Auslander-Yoneda algebras. This generalizes the main theorems of Pan on f-Cohen-Macaulay Auslander-Yoneda algebras and provides a useful method for constructing a new recollement of derived module categories from a given one.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Mathematics
Ali Mahin Fallah
Summary: In this article, Auslander-Reiten duality theorem is extended to certain R-algebras, and a criterion for projective modules over certain R-algebras is obtained in terms of vanishing of Ext modules through the argument presented.
COMMUNICATIONS IN ALGEBRA
(2021)
Article
Mathematics, Applied
Alex Chirvasitu, Ryo Kanda, S. Paul Smith
Summary: The paper introduces the algebras Q(n,k)(E, t) as generalizations of the 4-dimensional Sklyanin algebras introduced by Feigin and Odesskii. These algebras are quadratic algebras parametrized by coprime integers n > k= 1, a complex elliptic curve E, and a point t ? E. The main result of the paper is that Q(n,k)(E, t) has the same Hilbert series as the polynomial ring on n variables when t is not a torsion point. It is also shown that Q(n,k)(E, t) is a Koszul algebra of global dimension n when t is not a torsion point, and, for all but countably many t, Q(n,k)(E, t) is Artin-Schelter regular. The proofs utilize the fact that the space of quadratic relations defining Q(n,k)(E, t) is the image of an operator R-t (t) that belongs to a family of operators R-t (z): C-n ? C-n? C-n ? C-n, z ? C, which satisfy the quantum Yang-Baxter equation with spectral parameter.
SELECTA MATHEMATICA-NEW SERIES
(2023)
