Regular ℤ-graded local rings and graded isolated singularities

A commutative Noetherian local ring of finite global dimension is a regular local ring and the structure of such a ring is well described 1n many texts (see §1.9). It is perhaps surprising that very little seems to be known about the corresponding non-commutat1ve rings and it is our aim 1n this thesis to examine some of the ways in which the commutative theory can, and cannot be extended to a right Noetherian local ring of finite right global dimension. Chapter 1 contains basic definitions and, in Chapter 2, results are obtained on the projective and Injective dimensions of modules over right Noetherian local rings. Commutative regular local rings are domains and we begin Chapter 3 by considering the question of when a right Noetherian local ring R of finite right global dimension is a prime ring. An example of Stafford's shows that R is not necessarily prime; however, by examining the Nilpotent radical of R, we are able to extend results of Ramras [57] and Walker [75] and show that R is indeed prime when certain prime ideals are localisable. In Chapter 4 we consider the lattice of prime ideals of R when R is an AR-r1ng and provide theorems which generalise some of the basic results from the theory of commutative regular local rings. Section 4.3 contains examples which not only illustrate points arising in Chapters 3 and 4 but also show that some of the techniques which are mainstays of the commutative theory, fall dramatically in a non-commutat1ve setting. In Chapter 5, we generalise the concepts of regular sequences and Cohen Macaulay rings enabling us to prove, in §5.2 and chapter 6, that a right Noetherian local ring of finite right global dimension, which 1s integral over its centre, is a prime ring and exhibits many properties similar to those enjoyed by commutative regular local rings. Examples are provided which show that some alternative approaches are not applicable to the rings considered in these chapters. Regular local rings are Gorensteln rings and, as such, are the subject of an elegant structure theorem due to Bass [5]. In Chapter 7, we generalise his theorem to the situation of rings Integral over their centres. Each chapter begins with a summary of the results contained in that chapter.

Let $R$ be a Cohen-Macaulay ring and $M$ a maximal Cohen-Macaulay $R$-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and van den Bergh we study the question of when the endomorphism ring of $M$ has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced $1$-dimensional hypersurfaces.

Generalized Koszul algebra and Koszul duality

Let R be a regular local ring and A an algebra over R which is an R-progenerator. Defining the cohomological dimension of A as R-dim A=l hdAe(A), one obtains the Hochschild cohomological dimension of A as an R-algebra. We show the following under the additional hypothesis that R-dim A is finite: (1) R-dim A =n iff A/N is R-separable and l hdA(A/N)=n+gl R; (2) A= R-dim A +gl R; (3) A is R-separable iff Ae=gl R. The purpose of this note is to extend a result of Eilenberg on the Hochschild cohomological dimension of associative algebras. The results obtained will relate the global dimension of an algebra A which is a progenerator as an R-module, the global dimension of the ground ring R, and the cohomological dimension of the algebra. Throughout we will assume that all rings have one. We shall say that an R-algebra A is an R-progenerator in case A is finitely generated, projective, and faithful as a module over the commutative ring R. N will denote the Jacobson radical of the algebra A. We will always mean by hd and gl dim the left homological dimension and the left global dimension. Recall that if a ring is commutative or noetherian, the left and right global dimensions coincide. We define R-dimA=hdA (A), where Ae=A ORA* and A* is the algebra antiisomorphic to A. Since A*=A, its left and right global dimensions also coincide. Hence for the main theorems, global dimension is well defined. We require two well-known results, the first of Eilenberg, Rosenberg, and Zelinsky [4, Proposition 2] and the second due to Kaplansky [5, p. 172]: RESULT 1. If A and B are R-algebras and A is R-flat, then (a) B?A <R-dim A +gl B. If further, A is also R-projective and contains R as an R-direct summand, then (b) B<gl BOA. Received by the editors December 21, 1970. AMS 1970 subject classifications. Primary 18H15, 16A46, 13H05; Secondary 13D05, 16A62.

A directed graph is associated to any basic tiled order, and it turns out that the graph is connected for all known examples of tiled orders of finite global dimension. It is proved that the minimal connected tiled orders of finite global dimension in a fixed algebra are of global dimension two, and that up to isomorphism, these minimal orders are characterized by their unoriented graph which is a tree. Their irreducible representations are in one-to-one correspondence with the possible orientations of this tree.

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A &lt; ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A &lt; ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A &lt; ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A &lt; ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

Coherent algebras and noncommutative projective lines

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber–Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

Let T be a compact complex torus, dim T>2. We show that the category of coherent sheaves on T is independent of the choice of the complex structure, if this complex structure is generic. The proof is independent of math.AG/0205210, where the same result was proven for K3 surfaces and even-dimensional tori.

A Commutative Local Ring with Finite Global Dimension and Zero Divisors

We prove that ifR is a commutative ring such that each localization at a prime ideal has finite weak global dimension then everyR-module has a flat envelope if and only ifR is coherent and has weak global dimension less than or equal to two.

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

In this paper, we consider several homological dimensions of crossed products AασG, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of AσαG are classified: global dimension of AσαG is either infinity or equal to that of A, and finitistic dimension of AσαG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ≤ G, we show that A and AασG share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).

Let A be a quasi-local ring of global dimension n < <*>.Assume that its center R is a noetherian domain, that A is finitely generated torsionfree as an .R-module, and that R is an R-direct summand of A. Then R is integrally closed in its quotient field K and Macauley of dimension n.Furthermore, when n = 2, A is a maximal Rorder in the central simple K-algebra A ®^j K.This extends an earlier result of the author, in which R was assumed to have global dimension 2. Examples are given to show that in the above situation R can have infinite global dimension.

Multilinear forms and graded algebras