Gl Dim Research Articles

Tilting up algebras of small homological dimensions

We consider algebras Λ which satisfy the property that for each indecomposable module Λ X, either its projective dimension pd ΛX is at most one or its injective dimension id ΛX is at most one and that the global dimension gl dim Λ is three. We will show that this class is in bijective correspondence with a class of algebras of global dimension two admitting a special tilting torsion pair.

On the Global Dimension of a Ring Modulo its Nilpotent Radical

For any preassigned integer $n \geq 3$, a Noetherian affine PI ring $R$ is constructed with $\operatorname {gl dim }R \leq 3$, but $\operatorname {gl dim}\left ( {R/N\left ( R \right )} \right ) = n$. A second similar ring is constructed with $\operatorname {gl dim }R \leq 5$ and $\operatorname {gl dim}\left ( {R/N\left ( R \right )} \right ) = \infty$.

On the global dimension of a ring modulo its nilpotent radical

For any preassigned integer n ≥ 3 n \geq 3 , a Noetherian affine PI ring R R is constructed with gl dim ⁡ R ≤ 3 \operatorname {gl dim }R \leq 3 , but gl dim ⁡ ( R / N ( R ) ) = n \operatorname {gl dim}\left ( {R/N\left ( R \right )} \right ) = n . A second similar ring is constructed with gl dim ⁡ R ≤ 5 \operatorname {gl dim }R \leq 5 and gl dim ⁡ ( R / N ( R ) ) = ∞ \operatorname {gl dim}\left ( {R/N\left ( R \right )} \right ) = \infty .

The graded Cartan matrix and global dimension of 0-relations algebras

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 < ∞⇒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 < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞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 < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

Some constructions of rings of finite global dimension

In the study of idealizer rings in [14], the global dimensions of certain subrings were calculated. This work was extended by Goodearl [9]. Here it is shown, in Section 1, that the techniques involved can be used rather more generally than was apparent there. The results concern a ring S with a right ideal A such that SA = S, and a subring R of S containing A. They describe gl dim R in terms of gl dim S and the projective dimensions of simple subfactors of S/R or R/A, provided that these R-modules are sufficiently well conditioned.

On the Cartan matrix of an artin algebra of global dimension two

The purpose of this paper is to show that if n is an artin algebra of g!obal dimension (81 dim) 2, then the determinant of its Cartan matrix equals + I. This generalizes previous results of Donovan and Freislich [2], Igusa and Todorov ]4 ] and Wilson [5]. We recall that if n is an artin algebra (for instance, a finite-dimensional algebra over a field), there are finitely many nonisomorphic indecomposable projective rl modules P,, P ,,..., P,,. We denote by /End,(P.) Hom,,(P,, P,,) the length of Hom,(P,. PI) as an End,(P,) module. Then the h x n matrix C(n j, having knd.,cPi, Hom.,(P,, P,i) as the (i,j)th entry is called the Cartan matrix of rl. Let ,d be now an artin algebra of finite representation typethat is, having finitely many nonisomorphic indecomposable ‘4 modules. Let H= u End,(,tr) Hom.,(M, N)) be the matrix with integral entries, where M, N range through the set of the nonisomorphic indecomposable II modules. Donovan and Freislich [2], have shown that if .4 = kG is a group algebra of finite representation type, then det H = + 1. Then, using associated graded algebras, Wilson showed that det H = + 1 if A is a finite-dimensional algebra of finite-representation type over an algebraically closed field [ 5 IO Independently the same result was proved by Igusa and Todorov [4] without making any assumption on the ground field. Let P be the Auslander algebra of n (see [ 1 ] for further details). that is. r = End,,(M, 0 MT . . . @ M,JoD, where M, ,..., M, are the nonisomorphic indecomposable n modules. Then, it has been shown that H = C(T), where. C(T) denotes the Cartan matrix of the Auslander algebra r. It is known that gl dim r 2. [We recall that

Cohomological dimension and global dimension of algebras

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.

Maximal orders over regular local rings of dimension two

Introduction. Auslander and Goldman [4] have studied maximal orders over discrete valuation rings. In this paper, relying heavily on their results, we investigate maximal orders over regular local rings of dimension two. The two theories are rather different. We begin ?5 by listing three theorems from [4]. Then we prove a partial generalization of two of them in dimension two (Theorem 5.4) and exhibit two examples which show that the remaining statements do not generalize. In [4] a structure theorem is proved for maximal orders over discrete valuation rings which is the analogue of the Wedderburn structure theorem for simple artin rings. In ?6 we extend this theorem, in a weakened form, to maximal orders over an arbitrary integrally closed noetherian domain R. We sharpen this somewhat when R is regular local of dimension two and the maximal order is well behaved. Various other structure theorems are given in this section. The first four sections are devoted to building up homological machinery, most of which is applied to orders in the last two sections. The setting is fairly general: (R, mn) is a commutative noetherian local ring with maximal ideal m, and A is an R-algebra which is finitely generated as an R-module. The best results (Theorems 1.10 and 2.16) are obtained when A is quasi-local (i.e., A/Rad A is a simple artin ,ring, where Rad A is the Jacobson radical of A). Such rings behave very much like commutative local rings. The author wishes to express his gratitude to Professor Maurice Auslander for his many helpful suggestions and his patient supervision of the research for this paper, which is the major portion of the author's doctoral dissertation. Many thanks also to Professor David Buchsbaum and Dr. Silvio Greco for useful and stimulating conversations. Notations and conventions. Throughout this paper all rings have units and all modules are unitary. We use the abbreviations pd, inj dim, and gl dim for projective, injective, and global dimension, respectively. Only when there is a possible leftright ambiguity will we write l.pd or r.inj dim, etc. R is a commutative noetherian ring and A is an R-algebra which is finitely generated as an R-module. For the first four sections R is local with maximal ideal m. By R& we will mean the completion of R in the m-adic topology, and if M is an

Pseudocompact algebras, profinite groups and class formations

A pseudocompact ring A is a complete Hausdorff topological ring which admits a system of open neighborhoods of 0 consisting of two sided ideals I for which A/I is an Artin ring. A complete HausdorR topological A-module M is said to be pseudocompact if it has a system of open neighborhoods of 0 consisting of submodules N for which M/N has finite length. The category G?,, of pseudocompact (I-modules is an Abelian category with exact inverse limits and enough projectives. Such generalities, which are more or less well known, are gathered in the first section for the convenience of the reader. To get more interesting results, we must introduce some commutativity by assuming that, in addition, (1 is a pseudocompact algebra over a commutative pseudocompact ring Q (see definition in Section 2). We may then define a tensor product on %A and introduce its derived functor 554. The category %+,, of discrete cl-modules, which is dual to VA0 by Proposition 2.3, also plays an important role through the bifunctor Horn : V, x 5BA + go and its derived functor &zt. We then have the proper setting for doing homological algebra. This is done in Section 3 which generalizes the elementary results on homological dimension in complete Noetherian semilocal rings. As an immediate application we find that if s2({xi}} is the algebra of noncommutative formal power series in {xi} over Q, then gl dim J2{{xi}} = gl dim Sa + 1 (Theorem 3.9). We recall that a profinite group is a compact totally disconnected topological group, i.e., an inverse limit of finite groups. We define the complete group

Pseudocompact algebras, profinite groups and class formations

A pseudocompact ring A is a complete Hausdorff topological ring which admits a system of open neighborhoods of 0 consisting of two sided ideals I for which A/I is an Artin ring. A complete HausdorR topological A-module M is said to be pseudocompact if it has a system of open neighborhoods of 0 consisting of submodules N for which M/N has finite length. The category G?,, of pseudocompact (I-modules is an Abelian category with exact inverse limits and enough projectives. Such generalities, which are more or less well known, are gathered in the first section for the convenience of the reader. To get more interesting results, we must introduce some commutativity by assuming that, in addition, (1 is a pseudocompact algebra over a commutative pseudocompact ring Q (see definition in Section 2). We may then define a tensor product on %A and introduce its derived functor 554. The category %+,, of discrete cl-modules, which is dual to VA0 by Proposition 2.3, also plays an important role through the bifunctor Horn : V, x 5BA + go and its derived functor &zt. We then have the proper setting for doing homological algebra. This is done in Section 3 which generalizes the elementary results on homological dimension in complete Noetherian semilocal rings. As an immediate application we find that if s2({xi}} is the algebra of noncommutative formal power series in {xi} over Q, then gl dim J2{{xi}} = gl dim Sa + 1 (Theorem 3.9). We recall that a profinite group is a compact totally disconnected topological group, i.e., an inverse limit of finite groups. We define the complete group