Showing papers in "Communications in Algebra in 2000"
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TL;DR: In this paper, it was shown that for a ring R such that every direct sum of a lifting module and a simple module is lifting, every simple R-module is small M-projective for any lifting module.
Abstract: Let R be a ring with identity and let be a finite direct sum of relatively protective R-modules Mi Then it is proved that M is lifting if and only if M is amply supplemented and Mi is lifting for all 1 ≤ i ≤ n.Let be a finite direct sum of R-modules Mi . We prove that M is (quasi-) discrete if and only if are relatively projective (quasi-) discrete modules. We also prove that, for an amply supplemented R-module M=M 1⊕M 2 such that M 1 and M 2 have the finite exchange propertyM is lifting if and only if M 1 and M 2 are lifting and relatively small projective R-modules and every co-closed submodule N of M with M= N+M 1 = N+M 2 is a direct summand of M.Finally, we prove that, for a ring Rsuch that every direct sum of a lifting R-module and a simple R-module is lifting, every simple R-module is small M-projective for any lifting R-module M.
185 citations
TL;DR: The homological theory of contravariantly finite subcategories:auslander-buchweitz contexts, gorenstein categories and (co-)stabilization as mentioned in this paper.
Abstract: (2000). The homological theory of contravariantly finite subcategories:auslander-buchweitz contexts, gorenstein categories and (co-)stabilization. Communications in Algebra: Vol. 28, No. 10, pp. 4547-4596.
167 citations
TL;DR: In this paper, Fanggui and McCasland showed that if D satisfies ACC on integral ★ w -ideals, L ∗w (D) is a Noether lattice and hence primary decomposition, the Krull intersection theorem, and the principal ideal theorem hold for ∗ w-ideals of D.
Abstract: Let D be an integral domain with quotient field K, let (F(D) (f(D)) be the set of nonzero (finitely generated) fractional ideals of D, and let ★ be a star-operation on F(D).For A ∊ F(D) and there exists J∊f(D) such that J★=D, and xJ ⊆ A}.Then A ★w = {x ∊ K | exists J ∊ f(D) such that J ★ = D, and xJ ⊆ A}. Then and ★w are star-operations on F(D) that satisfy . Moreover, is the greatest (finite character) star-operation Δ ≤ ★ with (A ∩B)Δ=A Δ∩ B Δ.We also show that ★ w -Max(D)= ★ s -Max(D) and A ★w =∩{AP | P ∊★ s -Max(D)}.Let L ★w (D) = {A | A is an integral ★ w -ideal}∪{0}. Then L ★w (D) forms an r-lattice. If D satisfies ACC on integral ★ w -ideals,L ∗w (D) is a Noether lattice and hence primary decomposition, the Krull intersection theorem, and the principal ideal theorem hold for ∗ w -ideals of D. For the case of ★=υ,★ w is the w-operation introduced by Wang Fanggui and R.L. McCasland.
116 citations
TL;DR: The multiplier ideal is a universal test ideal as mentioned in this paper, and the multiplier ideal can be used as a test ideal for any algebraic metric function, including the multiplicative ideal, as well.
Abstract: (2000). The multiplier ideal is a universal test ideal. Communications in Algebra: Vol. 28, No. 12, pp. 5915-5929.
112 citations
TL;DR: The theorv of Doi-Hopf modules was generalized to Weak Hopf Algebras as mentioned in this paper, where weak hopf modules were generalized to DoiHopf module theory.
Abstract: The theorv of Doi-Hopf modules [8,11] is generalized to Weak Hopf Algebras [1, 14, 2].
108 citations
TL;DR: In this paper, a simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertextransitive, i.e., it is not vertex-to-vertical.
Abstract: A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. This paper gives a classification of semisymmetric graphs of order 2pq where p and q are distinct primes. It is shown that there are 143 examples of such graphs, 131 of which are biprimitive.
88 citations
TL;DR: In this paper, the authors partially extend results on the finiteness properties of local cohomology modules from the case of a regular local ring containing a field to the unramified case of regular local rings of mixed characteristic.
Abstract: In this note we partially extend results on the finite-ness properties of local cohomology modules from the case of a regular local ring containing a field to the unramified case of a regular local ring of mixed characteristic.
83 citations
TL;DR: In this paper, the authors define act,ions and coactions for quasi-Hopf algebras and introduce the notion of coactions of quasi-hopf algebra.
Abstract: The construct,ion of quasi-Hopf algebras obtains its importance from its signficance in Physics, cf.[l], 151, 161. Since for usual quantum groups and Hopf algebras actions and coactions of Hopf algebras play an important r81e, it is natmal to try to define act,ions and coactions for quasi-Hopf algebras. Now, coactions of quasi-Hopf algebras have been introduced by I?. Hansser and F. Nil1 in [8], [9]. If H is a quasi-Hopf algebra and B is an (associative) algebra, then, roughly speaking, B is a right H-comodule algebra if there exists an algebra map p : B -+ B @ H which turns B into an \"almost\" H-comodule
75 citations
TL;DR: The Auslander dual and k-torsionless modules were introduced in this paper, and the Universal pushforward was used to define the Gorenstein rings in exact sequences.
Abstract: 0. Introduction 1. The Auslander dual and k-torsionless modules k-torsionless modules Universal pushforward 2. Gorenstein dimension of modules Characterization of Gorenstein rings in t e r m of G-dimension Gorenstein dimension in exact sequences Gorenstein dimension and depth Regular elements and Gorenstein dimension Examples Weak Gorenstein dimension 3. kth syzygies of finite Gorenstein dimension Evans-Griffith presentations of syzygies References
73 citations
TL;DR: For the Hermitian curve H defined over the finite field, a complete classification of Galois coverings of H of prime degree was given in this paper, where the corresponding quotient curves turn out to be special cases of wider families of curves -covered by H arising from subgroups of the special linear group SL(2,F q ) or from sub groups in the normaliser of the Singer group of the projective unitary group.
Abstract: For the Hermitian curve H defined over the finite field , we give a complete classification of Galois coverings of H of prime degree. The corresponding quotient curves turn out to be special cases of wider families of curves -covered by H arising from subgroups of the special linear group SL(2,F q ) or from subgroups in the normaliser of the Singer group of the projective unitary group . Since curves -covered by H are maximal over , our results provide some classification and existence theorems for maximal curves having large genus, as well as several values for the spectrum of the genera of maximal curves. For every q 2, both the upper limit and the second largest genus in the spectrum are known, but the determination of the third largest value is still in progress. A discussion on the “third largest genus problem“ including some new results and a detailed account of current work is given.
70 citations
TL;DR: In this article, it was shown that in a large class of rings, containing Noetherian reduced rings, zero-dimensional rings, polynomials over reduced rings and C(X), every ideal consisting of zero-divisors is contained in a prime z-ideal.
Abstract: An ideal Iin a commutative ring Ris called a z°-ideal if Iconsists of zero-divisors and for each a∊ Ithe intersection of all minimal prime ideals containing ais contained in I.We prove that in a large class of rings, containing Noetherian reduced rings, Zero-dimensional rings, polynomials over reduced rings and C(X), every ideal consisting of zero-divisors is contained in a prime z°-ideal. It is also shown that the classical ring of quotients of a reduced ring is regular if and only if every prime z°-ideal is a minimal prime ideal and the annihilator of a f.g. ideal consisting of zero-divisors is nonzero. We observe that z°-ideals behave nicely under contractions and extensions.
TL;DR: In this article, functional identities of the form for all s 1,s 2,⋯,sm ϵ S (where, etc) were studied for prime rings with maximal right ring of quotients Q and with extended centroid C.
Abstract: LetA be a prime ring with maximal right ring of quotients Q and with extended centroid C. Further, let S be a set, let αS→ A be a map, let m be a positive integer and let Ei Fi S m-1, be maps. We study functional identities of the form for all s 1,s 2,⋯,sm ϵ S (where , etc). If Sα is an (m+ l)-free subset of Q (for example, if S α = A and A is not algebraic of bounded degree m+1 over C, or S α is a ncentral Lie ideal of A and A is not algebraic of bounded degree m + 2 over C), definitive results are obtained.
TL;DR: In this article, it was shown that a map in several variables on a prime ring satisfying an identity of polynomial type must be a quasi-polynomial (i.e., a Polynomial in noncommutative variables whose coefficients are Martindale centroid valued functions) provided that the ring does not satisfy a standard identity of low degree.
Abstract: We show that a map in several variables on a prime ring satisfying an identity of polynomial type must be a quasi-polynomial (i.e., a polynomial in noncommutative variables whose coefficients are Martindale centroid valued functions)provided that the ring does not satisfy a standard identity of low degree. Obtained results have applications to the study of Lie maps of prime rings (Lie ideals of prime rings and skew elements of prime rings with involution)and to the study of Lie-admissible algebras and Lie homomorphisms of Lie algebras of Poisson algebras.
TL;DR: In this paper, a complete classification of the 32-dimensional pointed Hopf algebras over an algebraically closed field k with chark k ≠ 2 is given.
Abstract: We give a complete classification of the 32-dimensional pointed Hopf algebras over an algebraically closed field k with chark k ≠ 2. It turns out that there are infinite families of isomorphism classes of pointed Hopf algebras of dimension 32. In [AS1], [BDG] and [Ge] are given families of counterexamples for the tenth Kaplansky conjecture. Up to now, 32 is the lowest dimension where Kaplansky conjecture fails.
TL;DR: Buchweitz and Schreyer as discussed by the authors considered vector bundles without intermediate cohomology on some Fano 3-folds with second Betti number b2 = 1 and showed that there are only three vector bundles of rank greater than two.
Abstract: A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964;
Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate
cohomology if and only if it decomposes as a direct sum of line bundles. It is also known
that only on projective spaces and quadrics there is, up to a twist by a line bundle,
a finite number of indecomposable vector bundles with no intermediate cohomology
[see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182
(1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl
0617.14033)].
In the paper under review the authors deal with vector bundles without intermediate
cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds
they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5
and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of
rank two vector bundles without intermediate cohomology on such 3-folds is given. In
fact the authors prove that, up to a twist, there are only three indecomposable vector
bundles without intermediate cohomology. Vector bundles of rank greater than two are
also considered. Under an additional technical condition, the authors characterize the
possible Chern classes of such vector bundles without intermediate cohomology.
TL;DR: In this article, it was shown that up to a translation each automorphism of the derived category D b X of coherent sheaves on a weighted projective line X, equiv-alently of D b A of finite dimensional modules over a derived canonical algebra A, is composed of tubular mutations and automorphisms of X.
Abstract: We show that up to a translation each automorphism of the derived category D b X of coherent sheaves on a weighted projective line X, equiv-alently of the derived category D b A of finite dimensional modules over a derived canonical algebra A, is composed of tubular mutations and automorphisms of X. In the case of genus one this implies that the automorphism group is a semi-direct product of the braid group on three strands by a finite group. Moreover we prove that most automorphisms lift from the Grothendieck group to the derived category.
TL;DR: In this paper, the authors studied the Hochschild cohomology ring H *(⋀) of algebras of the form ⋀ = kZe /JN, where Ze is an oriented cycle with e vertices and J is the ideal generated by the arrows.
Abstract: The purpose of this paper is to study the Hochschild cohomology ring H *(⋀)of algebras of the form ⋀ = kZe /JN , where Ze is an oriented cycle with e vertices and J is the ideal generated by the arrows, N≥2. We provide a new description of the Yoneda product in H *(⋀). and prove that this is a finitely generated infinite dimensional ring. In addition we show that algebras of the form ⋀ = kZe /JN are not derived equivalent unless they are isomorphic.
TL;DR: In this paper, it was shown that the invariant ring A,K[V]G is generated by homogeneous invariants of degrees at most dim (V) of a cyclic group G = Z p ×H with |H| coprime to p.
Abstract: Let G = Z p be a cyclic group of prime order p with a representation G → GL(V) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert series of the invariant ring K[V]G. Following Almkvist and Fossum in broad outline, we start by giving a shorter, self-contained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our results generalize to groups of the type Z p ×H with |H| coprime to p. Moreover, we prove that for any finite group G whose order is divisible by p but not by p 2 the invariant ring A,K[V]G is generated by homogeneous invariants of degrees at most dim (V).(|G| – 1).
TL;DR: For a large class of endomorphisms of finitely generated free groups, the authors proved that their mapping tori groups are word-hyperbolic if and only if they contain Baumslag-Solitar subgroups.
Abstract: For al large class of endomorphisms of finitely generated free groups we prove that their mapping tori groups are word-hyperbolic if and only if they don’t contain Baumslag-Solitar subgroups.
TL;DR: In this article, a criterion is given as to when the category of indecomposable weight and generalized weight modules with supports from a fixed orbit is tame for a class of generalized Weyl algebras.
Abstract: For a class of generalized Weyl algebras which includes the Weyl algebras An a criterion is given as to when the category of indecomposable weight and generalized weight modules with supports from a fixed orbit is tame. In tame cases indecomposable modules are described.
TL;DR: In this paper, structural properties of the monoids of all injective order preserving partial transformations on a chain with n elements are studied, and the main aim is to give a presentation for these monoids.
Abstract: In this paper we study several structural properties of the monoids \poin of all injective order preserving partial transformations on a chain with n elements. Our main aim is to give a presentation for these monoids.
TL;DR: In this article, the authors analyse the structure of the multiplier ring M(R) of a (nonuni-tal)Von Neumann regular ring R and show that every principal right ideal is generated by two idempotents.
Abstract: We analyse the structure of the multiplier ring M(R) of a(nonuni-tal)Von Neumann regular ring R. We show that M(R) is not regular in general, but every principal right ideal is generated by two idempotents. This, together with Riesz Decomposition on idempotents of M(R), furnishes a description of the monoid V(M(R)) of Murray-Von Neumann equivalence classes of idempotents which is used to effectively examine the lattice of ideals of M(R). The techniques developed here allow other applications to the category of projective modules over regular rings.
TL;DR: In this article, it was shown that if S is a commutative R-algebra and ψ: M→an R-module homomorphism, then Sψ(M) is a multiplication S-module.
Abstract: Let R bea commutative ring with identity. An R-module (ideal of R) A is called a multiplication module (ideal) if for each submodule N of A there exists an ideal I of R with N = I A. We give several characterizations of multiplication modules. Using the method of idealization we show how to reduce questions concerning multiplication modules to multiplication ideals. For example, we show that if S is a commutative R-algebra and ψ: M→an R-module homomorphism where M is a multiplication R-module and N is an S-module, then Sψ(M) is a multiplication S-module.
TL;DR: In this paper, for which a,b,c, and k there exists complexes with α injective and β surjective complexes, when α degenerates in P k in expected codimension, which is b-a-c+ 1
Abstract: We classify for which a,b,c, and k there exists complexes with α injective and β surjective. Furthermore we show that when it exists, we may assume that α degenerates in P k in expected codimension, which is b-a-c+ 1 We give applications to instanton bundles, space curves, and surfaces in .P 4.
TL;DR: In this paper, the authors proved that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.
Abstract: We prove that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.
TL;DR: In this article, the authors studied locally Cohen-Macaulay curves in P 3 which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two.
Abstract: We study in detail locally Cohen-Macaulay curves in P 3 which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes Hd,g(2H) of lo- cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that Hd,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva- lence classes.
TL;DR: In this paper, the order of the unitary subgroup of a finite p-group over a finite field K of characteristic p is computed when G is either an extraspecial a-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.
Abstract: Let KG be a group algebra of a finite p-group G over a finite field K of characteristic p. We compute the order of the unitary subgroup of the group of units when G is either an extraspecial a-group or the central product of such a group with a cyclic group of order 4 or G has an abelian subgroup A of index 2 and an element b such that b inverts each element of A.
Abstract: We show how to lift any monomial ideal J in n variables to a saturated ideal J of the same codimension in n -+ t variables. We show that I has the same graded Betti numbers as J and we show how to obtain the matrices for the resolution of I. The cohornology of l is described. Making general choices for our lifting, we show that l is the ideal of a reduced union of linear varieties with singularities that are "as small as possible" given the cohomological constraints. The case where J is Artinian is the nicest. In the case of curves we obtain stick figures for l, and in the case of points we obtain certain k-configurations which we can describe in a very precise way.
TL;DR: In this article, the concept of dominions in several varieties of nilpotent groups was investigated and the existence of nontrivial dominions was established in the category of all groups.
Abstract: We investigate the concept of dominion (in the sense of Isbell) in several varieties of nilpotent groups. We obtain a complete description of dominions in the variety of nilpotent groups of class at most two. Then we look at the behavior of dominions of subgroups of groups in N 2 when taken in the context of Nc for c>2. Finally, we establish the existence of nontrivial dominions in the category of all nilpotent groups.