Subalgebras and finitistic dimensions of Artin algebras

Let A be an Artin algebra.We investigate subalgebras of A with certain conditions and obtain some classes of algebras whose finitistic dimensions are finite.     

几乎优越扩张与同调维数

设环S是环R的几乎优越扩张．本文证明了R和S具有相同的f．f．P．维数以及finitistic维数．若M_S是右S－模,则FP－id（M_S）＝FP－id（M_R）．若G是有限群,R是G分次环且|G|^－1∈R,则Smash积R＃G^*和R具有相同的f．f．P．维数,finitistic维数,以及FP－整体维数．     

关于有限维数猜想的一些新进展

在Artin代数的表示理论中,有一个著名的有限维数猜想：任意给定一个Artin代数,它的有限维数都是有限的．这个猜想已有45年的历史,至今悬而未决．本文主要综述它的一些历史发展情况,并介绍关于有限维数猜想的一些最新进展．     

Squared cycles in monomial relations algebras

Let be an algebraically closed field. Consider a finite dimensional monomial relations algebra of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra . There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern in the syzygies of a specific factor module of the corresponding indecomposable projective module is found, allowing us to conclude that the square of any cycle must lie in the ideal I.     

Homological dimensions of gentle algebras via geometric models

Let A=kQ/I be a finite-dimensional basic algebra over an algebraically closed field k,which is a gentle algebra with the marked ribbon surface(S_A,M_A,Γ_A). It is known that S_A can be divided into some elementary polygons {Δ_i|1} ≤ d} by ΓA, which has exactly one side in the boundary of S_A. Let■(Δ_i)be the number of sides of Δ_i belonging to Γ_A if the unmarked boundary component of S_A is not a...     

Homological integral of Hopf algebras

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

     

Global dimensions of dual extension algebras

Dedicated to the memory of Maurice Auslander     

Homological behavior of idempotent subalgebras and Ext algebras

Let A be a(left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ :=(1-e)A(1-e) and the semi-simple right A-module Se := e A/e rad A. In this paper, we investigate the relationship between the global dimensions of A and Γ, by using the homological properties of Se. More precisely, we consider the Yoneda ring Y(e) := Ext_A~*(Se, Se) of e. We prove that if Y(e) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both A and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that Y(e) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if Y(e) has finite global dimension, then the Cartan determinants of A and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.     

Homological dimension and representation type of algebras under base field extension

We prove that for finite-dimensional associative algebras the finitistic global dimension in the sense of H. Bass and also the self-injective dimension are preserved under arbitrary extensions of the base field. Further, the global dimension, and as a consequence also finite representation type, are preserved under base field extensions which are separable in the sense of S. MacLane. The results are derived with the aid of ultraproduct-techniques and are applied to the model theory of finite dimensional algebras.     

First,Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.     

Graph representation of projective resolutions

We generalize the concept — dimension tree and the related results for monomial algebras to a more general case — relations algebras Λ by bringing Gröbner basis into play. More precisely, we will describe the minimal projective resolution of a left Λ-module M as a rooted ‘weighted’ diagraph to be called the minimal resolution graph for M. Algorithms for computing such diagraphs and applications as well will be presented.     

Homological properties of sklyanin algebras

In their recent paper [13], Tate and Van den Bergh studied certain quadratic algebras, called the “Sklyanin algebras”. They proved that these algebras have the Hilbert series of a polynomial algebra, are Noetherian and Koszul, and satisfy the Auslander-Gorenstein and Cohen-Macaulay conditions. This paper gives an alternative proof of these results, as suggested in [13], and thereby answering a question in their paper.     

Homological characterization of lean algebras

Certain classes of lean quasi-hereditary algebras play a central role in the representation theory of semisimple complex Lie algebras and algebraic groups. The concept of a lean semiprimary ring, introduced recently in [1] is given here a homological characterization in terms of the surjectivity of certain induced maps between Ext¹-groups. A stronger condition requiring the surjectivity of the induced maps between Ext ^k -groups for allk≥1, which appears in the recent work of Cline, Parshall and Scott on Kazhdan-Lusztig theory, is shown to hold for a large class of lean quasi-hereditary algebras. Research partially supported by NSERC of Canada and by Hungarian National Foundation for Scientific Research grant no. 1903 Research partially supported by NSERC of Canada     

On strict homological dimensions of algebras of continuous functions

We prove that, for each nonnegative integer n and n = ∞, there exists a compact topological space Ω such that the strict global dimension and the strict bidimension of the Banach algebra C(Ω) of all continuous functions on Ω are equal to n. We also obtain several “additivity formulas” for the strict homological dimensions of strict Banach algebras.     

A central extension theorem for essential dimensions

Let be an arbitrary field and a finite group. We will denote by the essential dimension of over . A generalization of the central extension theorem of Buhler and Reichstein (Compositio Math. 106 (1997) 159-179, Theorem 5.3) is obtained.

     

PBW deformations of quadratic monomial algebras

A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a Poincaré-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions equivalent to the Braverman-Gaitsgory Theorem to efficiently determine PBW deformations of quadratic monomial algebras. In particular, a graphical interpretation is presented for this result, and we discuss circumstances under which some of the conditions of this theorem need not be checked. Several examples are also provided. Finally, with these tools, we show that each quadratic monomial algebra admits a nontrivial PBW deformation.     

Candidates for nonzero Betti numbers of monomial ideals

Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d?1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fröberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.