Showing papers on "Global dimension published in 2001"
TL;DR: In this paper, a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra was given, and it was shown that such a ring is a homomorphic image of a finite-dimensional Gorenstein ring.
Abstract: The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.
92 citations
TL;DR: In this paper, a generalization of Golod's theorem on the behaviour of G{sub K}-dimension with respect to a suitable module K under factorization by ideals of a special kind is obtained and a new form of the Avramov-Foxby conjecture on the transitivity of G-dimension is suggested.
Abstract: For finite modules over a local ring the general problem is considered of finding an extension of the class of modules of finite projective dimension preserving various properties. In the first section the concept of a suitable complex is introduced, which is a generalization of both a dualizing complex and a suitable module. Several properties of the dimension of modules with respect to such complexes are established. In particular, a generalization of Golod's theorem on the behaviour of G{sub K}-dimension with respect to a suitable module K under factorization by ideals of a special kind is obtained and a new form of the Avramov-Foxby conjecture on the transitivity of G-dimension is suggested. In the second section a class of modules containing modules of finite CI-dimension is considered, which has some additional properties. A dimension constructed in the third section characterizes the Cohen-Macaulay rings in precisely the same way as the class of modules of finite projective dimension characterizes regular rings and the class of modules of finite CI-dimension characterizes complete intersections.
77 citations
TL;DR: In this article, Kapranov studied the case of coherent sheaves over a smooth projective curve defined over a finite field and observed analogies with quantum affine algebras.
Abstract: To an abelian category A of homological dimension one satisfying certain finiteness conditions, one can associate an algebra, called the Hall algebra. Kapranov studied this algebra when A is the category of coherent sheaves over a smooth projective curve defined over a finite field, and observed analogies with quantum affine algebras. We recover here in an elementary way his results in the case when the curve is the projective line.
65 citations
TL;DR: It is known that the powers m n of the maximal ideal of a local Noetherian ring share certain homological properties for all sufficiently large integers n as discussed by the authors, and the smallest integers n for which such properties begin to hold.
55 citations
TL;DR: In this paper, the notion of good filtration dimension and Weyl dimension in a quasi-hereditary algebra were defined explicitly for all irreducible modules in SL 2 and SL 3 and it was shown that the global dimension of a Schur algebra for GL 2 and GL 3 is twice the good dimension.
41 citations
TL;DR: In this paper, it was shown that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2.
Abstract: We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b. 0. Introduction. Let Γ be a finite graph and n a natural number. The marked n-point configuration space of Γ is a subspace CnΓ in the nth cartesian power of Γ defined by CnΓ := {(x1, . . . , xn) ∈ Γ n : xi 6= xj for i 6= j}. Consider the natural free action of the symmetric group Sn on the space CnΓ defined by σ(x1, . . . , xn) = (xσ(1), . . . , xσ(n)) and put CnΓ := CnΓ/Sn. The space CnΓ is called the (unmarked) n-point configuration space of Γ . This paper reports on partial progress towards understanding the homology of configuration spaces of graphs, or even more generally of compact polyhedra. For another recent result in that direction, see [G]. We call a vertex v of Γ branched if it is adjacent to at least three edges. We denote by b = b(Γ ) the number of branched vertices in Γ . The main result of this paper is the following. 0.1. Theorem. Let Γ be a finite graph and n a natural number. (1) There exists a cube complex KnΓ of dimension min(b(Γ ), n) which embeds as a deformation retract into the configuration space CnΓ . (2) The fundamental group π1(CnΓ ) contains a subgroup isomorphic to the free abelian group Z with k = min(b(Γ ), [n/2]), where [x] denotes the integer part of x. 2000 Mathematics Subject Classification: Primary 55M10; Secondary 20J05, 51F99. The author was supported by the Polish State Committee for Scientific Research (KBN) grant 2 P03A 023 14.
40 citations
TL;DR: In this paper, the lower bound of Hilbert-Kunz multiplicity for non-regular unmixed local rings of Krull dimension $d$ containing a field of characteristic p>0 was investigated, and it was shown that such rings are isomorphic to the non-degenerate quadric hypersurface.
Abstract: In this paper, we investigate the lower bound $s_{HK}(p, d)$ of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension $d$ containing a field of characteristic $p>0$. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that $s_{HK}(p, 3) = 4/3$ and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity $4/3$ is isomorphic to the non-degenerate quadric hypersurface $k[[X, Y, Z, W]]/(X^{2}+Y^{2}+Z^{2}+W^{2})$ under mild conditions. Furthermore, we pose a generalization of the main theorem to the case of $\dim A \ge 4$ as a conjecture, and show that it is also true in case $\dim A = 4$ using the similar method as in the proof of the main theorem.
35 citations
TL;DR: In this article, the authors describe finitely generated modules over all commutative noetherian rings that do not have wild representation type (with a possible exception involving characteristic 2).
Abstract: This is the first of a series of four papers describing the finitely generated modules over all commutative noetherian rings that do not have wild representation type (with a possible exception involving characteristic 2). This first paper identifies the wild rings, in the complete local case. The second paper describes the finitely generated modules over the remaining complete local rings. The last two papers extend these results by dropping the "complete local" hypothesis.
25 citations
TL;DR: In this article, the Auslander-Reiten quivers of finite-dimensional K-algebras have been shown to admit families of stable tubes (in the sense of Ringel [17]).
Abstract: We introduce a new wide class of finite-dimensional algebras which admit families of standard stable tubes (in the sense of Ringel [17]). In particular, we prove that there are many algebras of arbitrary nonzero (finite or infinite) global dimension whose Auslander–Reiten quivers admit faithful standard stable tubes. Introduction. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finite-dimensional K-algebra (associative, with an identity), which we moreover assume to be basic. An algebra A can be written as a bound quiver algebra A ∼= KQ/I, where Q = QA is the Gabriel quiver of A and I is an admissible ideal in the path algebra KQ of Q. Equivalently, we will consider A as a K-category whose class of objects is the set of vertices of QA. For an algebra A, we denote by modA the category of finite-dimensional (over K) right A-modules, by rad(modA) the Jacobson radical of modA and by rad(modA) the infinite radical of modA. Recall that rad(modA) is generated by nonisomorphisms between indecomposable objects in modA, and rad(modA) is the intersection of all finite powers rad(modA), i ≥ 1, of rad(modA). By anA-module we mean an object of modA. For each vertex i of QA, we denote by SA(i) the simple A-module at i, and by PA(i) (respectively, IA(i)) the projective cover (respectively, injective envelope) of SA(i) in modA. Moreover, we denote by D the standard duality HomK(−,K) on modA. We shall denote by ΓA the Auslander–Reiten quiver of A and by τA and τ A the Auslander–Reiten translations DTr and TrD in ΓA, respectively. We do not distinguish between an indecomposable A-module and the vertex of ΓA corresponding to it. By a component of ΓA we mean a connected component of ΓA. For a family C of components in ΓA, we denote by suppA C the support of C and by annA C the annihilator of C. Recall that suppA C is the full subcategory of A given by all objects i such that SA(i) is a 2000 Mathematics Subject Classification: 16G10, 16G70, 18G05. Supported by the Foundation for Polish Science.
25 citations
TL;DR: In this article, the tensor Krull minimal (TKM) is defined for affine commutative F-algebras, where F is an uncountable algebraically closed field and σ is an affine automorphism.
24 citations
TL;DR: In this article, the concept of ample filter of invertible sheaves on a commutative noetherian ring was introduced and generalized to twisted homogeneous coordinate rings, which were previously known only when the scheme was projective over an algebraically closed field.
Abstract: Let $X$ be a scheme, proper over a commutative noetherian ring $A$. We introduce the concept of an ample filter of invertible sheaves on $X$ and generalize the most important equivalent criteria for ampleness of an invertible sheaf. We also prove the Theorem of the Base for $X$ and generalize Serre's Vanishing Theorem. We then generalize results for twisted homogeneous coordinate rings which were previously known only when $X$ was projective over an algebraically closed field. Specifically, we show that the concepts of left and right $\sigma$-ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be noetherian.
TL;DR: In this paper, a characterization of Artinian modules dual to Faith's Theorem is provided, and the authors also extend these results to the more general settings of dual Krull dimensions and Krull dimension respectively.
Abstract: This paper is motivated by a recent work of C. Faith [9] who has proved that a quotient finite dimensional module which satisfies the ascending chain condition on subdirectly irreducible submodules is Noetherian. A natural question to ask (also raised by Faith [11]) is whether its dual holds true. We answer this in the affirmative, and provide a characterization of Artinian modules dual to Faith's Theorem. We also extend these results to the more general settings of dual Krull dimension and Krull dimension respectively.
TL;DR: In this article, the Krull dimension of the generalized Weyl algebra T = R(σ, a), where R is a left noetherian ring, was obtained for the first time.
TL;DR: In this paper, the authors extend these results to rings R having relative Krull dimension with respect to a hereditary torsion theory τ on Mod-R such that any τ-torsion-free right R -module M has nonempty assassinator.
TL;DR: In this article, the authors discuss necessary and sufficient conditions for the cross product algebra S [midast ] G to be semi-hereditary, maximal or Azumaya over V.
Abstract: Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F , and let K be a finite Galois extension of F with group G , and S the integral closure of V in K . If, in the crossed product algebra K [midast ] G , the 2-cocycle takes values in the group of units of S , then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G . In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K / F , for the V -order S [midast ] G to be semihereditary, maximal or Azumaya over V .
TL;DR: In this article, the authors established the dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when R/p is regular.
Abstract: The recent work of Kurano and Roberts on Serre's positivity conjecture suggests the following dimension inequality: for prime ideals p and q in a local, Cohen-Macaulay ring (A, n) such that e(Ap) = e(A) we have dim(A/p) + dim(A/q) < dim(A). We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when R/p is regular.
TL;DR: In this article, it was shown that every module over a commutative ring R has a divisible envelope when every S-torsion free R-module has a corresponding envelope, where S is the set of nonzero divisors of R.
Abstract: In this article we characterize Gorenstein rings of Krull dimension at most one by using divisible envelopes. Some of these results were known in [6], where the given ring is an integral domain. We prove these results hold over commutative rings which may not be domains. The motivation of the present discussion is partially from [1], where modules over Gorenstein rings of Krull dimension at most one were studied (see also [[14], Theorem 5.19] for a completely different homological characterization of these rings). We also show that every module over a commutative ring R has a divisible envelope when every S-torsion free R-module has a divisible envelope which is S-torsion free, where S is the set of nonzero-divisors of R.
TL;DR: In this paper, a ring of differential operators with coefficients from a regular commutative affine domain of Krull dimension 2 is defined and simple holonomic Λ-modules are described.
Abstract: Let K be an algebraically closed field of characteristic zero. Let Λ be the ring of (K-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension 2 which is the tensor product of two regular commutative affine domains of Krull dimension 1. Simple holonomic Λ-modules are described. Let a K-algebra D be a regular affine commutative domain of Krull dimension 1 and D(D) be the ring of differential operators with coefficients from D. We classify (up to irreducible elements of a certain Euclidean domain) simple D(D)-modules (the field K is not necessarily algebraically closed).
TL;DR: In this paper, it was shown that H ∞ (D ) is a Hermite regular coherent ring of weak global dimension 2, and that it is stable by elementary algebraic operations.
TL;DR: In this paper, the global dimension of a ring R is denoted by GD(R), and the projective dimension of an R is defined as the dimension of the projection of R to the projectivity of R.
Abstract: Throughout this paper, rings considered are commutative with identity, all modules are assumed unital. The global dimension of a ring R will be denoted by GD(R), and the projective dimension of an ...
TL;DR: In this article, a criterion is presented which is easy to check in specific examples and provides information as to when a given noetherian ring is not embeddable in an artinian one.
TL;DR: In this paper, it was shown that an infinite product of epimorphisms in the category of commutative rings need not be an epimomorphism, and that for rings with finitely many primes and integral extensions of noetherian rings of dimension 1, one can find a ring of regularity degree n + 1.
Abstract: Elements of the universal (von Neumann) regular ring T(R) of a commutative semiprime ring R can be expressed as a sum of products of elements of R and quasi-inverses of elements of R. The maximum number of terms required is called the regularity degree, an invariant for R measuring how R sits in T(R). It is bounded below by 1 plus the Krull dimension of R. For rings with finitely many primes and integral extensions of noetherian rings of dimension 1, this number is precisely the regularity degree. For each n ≥ 1, one can find a ring of regularity degree n + 1. This shows that an infinite product of epimorphisms in the category of commutative rings need not be an epimorphism. Finite upper bounds for the regularity degree are found for noetherian rings R of finite dimension using the Wiegand dimension theory for Patch R. These bounds apply to integral extensions of such rings as well.
01 Jan 2001
TL;DR: In this paper, it was shown that all polynomials in two variables over a Noetherian Q-domain of Krull dimension one are projectively tame, and that all automorphisms over a reduced ring of dimension zero are tame.
Abstract: In this paper all coordinates in two variables over a Noetherian Q-domain of Krull dimension one are proved to be projectively tame. In order to do this, some results concerning projectively-tameness of polynomials in general are shown. Furthermore, we deduce that all automorphisms in two variables over a Noetherian reduced ring of dimension zero are tame.
TL;DR: In this article, the power stably free dimension of a ring is defined and a classification of rings with power-stably-free dimension 0, 1 is given. But it is not known whether a ring can be represented by a ring with a finite power-stable free dimension, and it is unknown whether all projective modules over R[x 1, …, xn ] are power free.
Abstract: In this paper, we define the power stably free dimension for rings. Using its relations with other dimensions, we get a classification of rings. Moreover, we give the equivalent characterizations of a ring with power stably free dimension 0, 1 respectively. For a commutative ring R in which each f. g. module has a finite power stably free dimension, we show that R[x 1, …, xn ] is connected and all f. g. projective modules over R[x 1, …, xn ] are power free.
TL;DR: In this paper, it was shown that the commutative Noetherian rings that have a torsion free cover with a flat kernel are the Gorenstein rings with Krull dimension at most one.
Abstract: Over Pruufer domains every module has a torsion free cover with a flat kernel. In this paper we show that the commutative Noetherian rings that have this property are the Gorenstein rings with Krull dimension at most one. Here we use the torsion theory defined by the nonzero-divisors of the commutative ring R.
01 Jan 2001
Journal Article•
TL;DR: In this paper, it was shown that the global dimensions of R and the crossed product R#σH are the same, when R is finite-dimensional semisimple and cosemisimple Hopf algebra.
Abstract: In this paper, the author obtains that the global dimensions of R and the crossed product R#σH are the same; meantime, their weak dimensions are also the same, when H is finite-dimensional semisimple and cosemisimple Hopf algebra.
TL;DR: In this article, a basis of invariants for the fundamental representations of the exceptional Lie groups E(6) and E(7), up to degree 18, was generated by computer.
Abstract: We generate by computer a basis of invariants for the fundamental representations of the exceptional Lie groups E(6) and E(7), up to degree 18. We discuss the relevance of this calculation for the study of supersymmetric gauge theories, and revisit the self-dual exceptional models. We study the chiral ring of G(2) to degree 13, as well as a few classical groups. The homological dimension of a ring is a natural estimator of its complexity and provides a guideline for identifying theories that have a good chance to be amenable to a solution.