Showing papers on "Global dimension published in 1992"
TL;DR: In this article, the grade of a finitely generated (left or right) module is defined by the grade matrix of a noetherian ring, which is defined as a function of the dimension of the module's injective dimension.
Abstract: Let A be a noetherian ring. When A is commutative (of finite Krull dimension), A is said to be Gorenstein if its injective dimension is finite. If A has finite global dimension, one says that A is regular. If A is arbitrary, these hypotheses are not sufficient to obtain similar results to those of the commutative case. To remedy this problem, M. Auslander has introduced a supplementary condition. Before stating it, we recall that the grade of a finitely generated (left or right) module is defined by
253 citations
Journal Article•
TL;DR: In this paper, it was shown that these Sklyanin algebras are Noetherian domains and that they are regular graded rings of global dimension four, and that much of the machinery developed in (ATV) to deal with 3-dimensional algesbras has an analogue in dimension four.
Abstract: The notation of a (non-commutative) regular, graded algebra is introduced in (AS). The results of that paper, combined with those in (ATV1, 2), give a complete description of the regular graded rings of global dimension three. This paper considers certain algebras that were defined by Sklyanin (Sk 1, 2) in connection with his work on the Quantum Inverse Scattering Method. We prove that these Sklyanin algebras are regular graded algebras of global dimension four, and are Noetherian domains. Moreover, we show that much of the machinery developed in (ATV) to deal with 3-dimensional algebras has an analogue in dimension four.
163 citations
136 citations
TL;DR: In this paper, it was shown that the Ratliff-Rush ideal is the largest ideal for which, for sufficiently large positive integers n, (n) = I and hence that ̃̃ I = Ĩ.
Abstract: Ratliff and Rush show in particular that Ĩ is the largest ideal for which, for sufficiently large positive integers n, (Ĩ) = I and hence that ̃̃ I = Ĩ. We call regular ideals I for which I = Ĩ Ratliff–Rush ideals, and we call Ĩ the Ratliff–Rush ideal associated with I. It is easy to see that an element a of I : I is integral over I, in the sense that there is an equation of the form a + b1a k−1 + . . . + bk = 0, where bi ∈ I for i = 1, . . . , k. Therefore, the ideal Ĩ is always between I and the integral closure I ′ of I, and hence integrally closed ideals are Ratliff–Rush ideals. Ratliff and Rush observe [RR, (2.3.4)] that the powers of an invertible ideal are Ratliff–Rush ideals, so any principal ideal generated by a nonzerodivisor is a Ratliff–Rush ideal. They also prove the interesting fact that for any regular ideal I of R, there is a positive integer n such that for all k ≥ n, Ĩk = I [RR, (2.3.2)], i.e., all sufficiently high powers of a regular ideal are Ratliff–Rush. A regular ideal I is always a reduction of its associated Ratliff–Rush ideal Ĩ, in the sense that I(Ĩ) = (Ĩ) for some positive integer n. For the basic facts on reductions and reduction numbers of ideals, we refer the reader to [NR], [H1], and [H2]. In particular, if there is an element a of an ideal I for which aI = I then aR is called a principal reduction of I and the smallest n for which this equation holds is called the reduction number of I. We will call a regular ideal I stable iff it has a principal reduction with reduction number at most one, i.e., iff there is an element a of I for which
82 citations
TL;DR: A finite dimensional monomial relation algebra has finite global dimension if and only if the relative cyclic homology of its radical is equal to zero as mentioned in this paper, i.e., the relation algebra is monomial in length.
23 citations
TL;DR: In this paper, the injective dimension of the ring R as a left R-module, injdim,(R), and the tinitistic global dimensions, lfPD(R) and lFPD (R), are defined.
21 citations
01 Jan 1992
TL;DR: In this article, the authors give a survey of what is known about the structure of the indecomposable injective modules over a Noetherian ring R and suggest some directions for future research in these areas.
Abstract: The purpose of this article is to give a survey of part of what is known about the structure of the indecomposable injective modules over a Noetherian ring R, to indicate how this structure depends on the nature of certain bimodules within the ring which afford “links” between the prime ideals of R, and to suggest some directions for future research in these areas.
11 citations
TL;DR: In this article, the authors compare the two characterizations using Happel's description of the derived category of modules over a finite-dimensional algebra of finite global dimension, which can be interpreted in terms of tilting theory.
Abstract: Bernstein-Gelfand-Gelfand showed that the derived category of coherent sheaves on the projective n-space Db(cohPn) is equivalent to the stable ategory of the category of Z-graded finite-dimensional modules over the exterior algebra. At the same time Beilinson gave a description of Db(cohPn) which can be interpreted in terms of tilting theory. We compare these two characterizations using Happel's description of the derived category of modules over a finite-dimensional algebra of fnite global dimension.
9 citations
8 citations
TL;DR: In this article, the authors studied finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals, and established the following results for finite extension without trace conditions.
Abstract: Finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals are studied. This setting includes finite free extensions and extensions arising from actions of finite groups when the order of the group is invertible. In this setting we establish the following results which were previously obtained (for finite extension without trace conditions) only under strong restrictions on the rings involved. Let R⊂S be an extension of Noetherian rings such that S is finitely generated as a left R-module and such that the left trace ideal of S in R is equal to R. If S is right fully bounded, or is a Jacobson ring, then R has the same property; furthermore, R and S have the same classical Krull dimension
6 citations
TL;DR: The left global dimension of a semiprime ring, with left Krull dimension α≥1 is the supremum of the projective dimensions of the p-critical cyclic modules where β≤α as mentioned in this paper.
Abstract: The left global dimension of a semiprime ring, with left Krull dimension α≥1 is found to be the supremum of the projective dimensions of the p-critical cyclic modules where β≤α. A similar result is true for upper triangular matrix rings whose entries come from a domain with Krull dimension. In addition if R is a a ring of the form where S is a semiprime ring with left Krull dimension α≥1, T is any ring with l.K dim T≤α, and A is an S-T bimodule such that sA has Krull dimension then the left global dimension of R is the supremum of the projective dimensions of the -critical cyclic left R-modules where β<αa. These results are used to compute homological dimensions of rings with Krull dimension. Some analogues are given for weak dimension and for rings with Gabriel dimension.
01 Jan 1992
TL;DR: In this paper, examples are presented of finite-dimensional algebras that admit no positive grading (that is, a nontrivial grading indexed by the natural numbers) and some of these examples have finite global dimension (they are even quasihereditary), and yield a negative answer to a question of Anick and Green.
Abstract: Examples are presented of finite-dimensional algebras that admit no positive grading (that is, a nontrivial grading indexed by the natural numbers). Some of these examples have finite global dimension (they are even quasihereditary), and yield a negative answer to a question of Anick and Green. A finite-dimensional algebra A over a field K has a positive semisimple grading in case there is a K-decomposition
TL;DR: In this article, it was shown that a right invertible maximal ideal in a semiprime prime polynomial identity ring is also left-invertible (Theorem 3.5).
Abstract: The aim of this paper is to generalise the results of [7] from the prime to the semiprime case. It was shown, for instance, that if M is the annihilator of a simple right module S of projective dimension 1 over a Noetherian prime polynomial identity (PI) ring R then M is either an invertible ideal or an idempotent ideal [7, Proposition 4.2]. One of the main applications of this result was that a prime Noetherian affine PI ring of global dimension less than or equal to 2 is a finite module over its centre. It turns out that this theorem is valid more generally when the ring is semiprime [1, Theorem A]. Clearly this requires [7, Proposition 4.2] also to be strengthened to the semiprime case. We do this by showing that a right invertible maximal ideal in a semiprime Noetherian PI ring is also left invertible (Theorem 3.5).
01 Jan 1992
TL;DR: In this article, the authors explore connections between Lie theory, relative homological algebra, invariant theory, and representations of finite dimensional algebras and explore some intriguing connections between lie theory and relative homology.
TL;DR: The notion of a Koszul algebra was introduced in this article, which is the universal algebra of invariant differential operators acting on differential forms on a riemannian (resp. kahlerian) manifold.
TL;DR: In this article, the right Krull dimension of the algebra Q op ⊗ F Q for certain semisimple Artinian F -algebras Q is computed explicitly and directly.
Abstract: . For F a field we compute, explicitly and directly, the right Krull dimension of the algebra Q op ⊗ F Q for certain semisimple Artinian F -algebras Q . (Here Q op denotes the opposite ring of Q .) We use our calculation to give alternative proofs of some theorems of J. T. Stafford and A. I. Lichtman. Our methods involve a detailed study of skew polynomial rings.
TL;DR: In this paper, the global dimension of Noetherian rings and rings with Gabriel and Krull dimension was computed by taking a subclass of cyclic modules determined by the Gabriel filtration in the lattice of hereditary torsion theories.
Abstract: In this paper we compute the global dimension of Noetherian rings and rings with Gabriel and Krull dimension by taking a subclass of cyclic modules determined by the Gabriel filtration in the lattice of hereditary torsion theories.