On injective Jordan semi-triple maps of matrix algebras

We show that every injective Jordan semi-triple map on the algebra M_n(F) of all n × n matrices with entries in a field F (i.e. a map Φ:M_n(F)→M_n(F) satisfying

Φ(ABA)=Φ(A)Φ(B)Φ(A)     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras

In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Gröbner-Shirshov bases of free Rota-Baxter algebra, free λ-differential algebra and free λ-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to the recent results obtained by K. Ebrahimi-Fard-L. Guo, and L. Guo-W. Keigher by using other methods.     

Decomposition of some pointed Hopf algebras given by the canonical Nakayama automorphism

Every finite dimensional Hopf algebra is a Frobenius algebra, with Frobenius homomorphism given by an integral. The Nakayama automorphism determined by it yields a decomposition with degrees in a cyclic group. For a family of pointed Hopf algebras, we determine necessary and sufficient conditions for this decomposition to be strongly graded.     

The rationality of the Hilbert-Kunz multiplicity in graded dimension two

We show that the Hilbert-Kunz multiplicity is a rational number for an R₊−primary homogeneous ideal I=(f₁, . . . , f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. More specific, we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f₁, . . . , f_n) on the projective curve Y=ProjR.     

On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ(B)=n. We investigate conditions guaranteeing that κ(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V,c) associated to a braided vector space (V,c), providing meaningful examples such that κ(T(V,c))≤1.     

Noncommutative symmetric systems over associative algebras

This paper is the first of a sequence of papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] on the (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134]; the systems over the Grossman-Larson Hopf algebras [R. Grossman, R.G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1) (1989) 184-210. [MR1023294]; L. Foissy, Les algèbres de Hopf des arbres enracinés décorés I, II, Bull. Sci. Math. 126 (3) (2002) 193-239; (4) 249-288. See also math.QA/0105212. [MR1909461]] of labeled rooted trees [W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136]; as well as their connections and applications to the inversion problem [H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982) 287-330. [MR 83k:14028]; A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, in: Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. [MR1790619]] and specializations of NCSFs [W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint]. In this paper, inspired by the seminal work [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348. See also hep-th/9407124. [MR1327096]] on NCSFs (noncommutative symmetric functions), we first formulate the notion of systems over associative Q-algebras. We then prove some results for systems in general; the systems over bialgebras or Hopf algebras; and the universal system formed by the generating functions of certain NCSFs in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348. See also hep-th/9407124. [MR1327096]]. Finally, we review some of the main results that will be proved in the following papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] as some supporting examples for the general discussions given in this paper.     

Cohen-Macaulayness and computation of Newton graded toric rings

Let H⊆Z^d be a positive semigroup generated by A⊆H, and let K[H] be the associated semigroup ring over a field K. We investigate heredity of the Cohen-Macaulay property from K[H] to both its A-Newton graded ring and to its face rings. We show by example that neither one inherits in general the Cohen-Macaulay property. On the positive side, we show that for every H there exist generating sets A for which the Newton graduation preserves Cohen-Macaulayness. This gives an elementary proof for an important vanishing result on A-hypergeometric Euler-Koszul homology. As a tool for our investigations we develop an algorithm to compute algorithmically the Newton filtration on a toric ring.     

On certain finite-dimensional algebras generated by two idempotents

This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)^m=(QP)^m and (PQ)^m-1≠(QP)^m-1. The main result is the classification of all these algebras, implying that for each m?2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given.     

Generic residual intersections and intersection numbers of movable components

Given a sequence x of elements of a commutative equidimensional noetherian ring R  , cycles _zi(x,R)$z_i(\mathbf{x},R)$ (i∈N$i\in \mathbb{N}$) in the cycle group of polynomial rings over R are defined by generic residual intersections. The study of these cycles gives new insight into the theory for excess intersections in projective space developed by Stückrad and Vogel, in particular concerning the contribution to the intersection cycle of embedded components not defined over the base field.     

The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t^-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.     

Buchsbaum homogeneous algebras with minimal multiplicity

In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous k-algebras A in terms of the dimension d, the codimension c, the initial degree q, and the length of the local cohomology modules of A. Next, we introduce the notion of Buchsbaum k-algebras with minimal multiplicity of degree q, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.     

Epsilon multiplicity for graded algebras

The notion of ε-multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative ε-multiplicity of reduced standard graded algebras over an excellent local ring exists as a limit. We also obtain some important special cases of Cutkosky's results concerning ε-multiplicity, as corollaries of our main theorem.     

Improving the bounds of the multiplicity conjecture: The codimension 3 level case

The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra A in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of A. All the examples studied so far have lead to conjecture (see [J. Herzog, X. Zheng, Notes on the multiplicity conjecture. Collect. Math. 57 (2006) 211-226] and [J. Migliore, U. Nagel, T. Römer, Extensions of the multiplicity conjecture, Trans. Amer. Math. Soc. (preprint: math.AC/0505229) (in press)]) that, moreover, the bounds of the MC are sharp if and only if A has a pure MFR. Therefore, it seems a reasonable-and useful-idea to seek better, if possibly ad hoc, bounds for particular classes of Cohen-Macaulay algebras.In this work we will only consider the codimension 3 case. In the first part we will stick to the bounds of the MC, and show that they hold for those algebras whose h-vector is that of a compressed algebra.In the second part, we will (mainly) focus on the level case: we will construct new conjectural upper and lower bounds for the multiplicity of a codimension 3 level algebra A, which can be expressed exclusively in terms of the h-vector of A, and which are better than (or equal to) those provided by the MC. Also, our bounds can be sharp even when the MFR of A is not pure.Even though proving our bounds still appears too difficult a task in general, we are already able to show them for some interesting classes of codimension 3 level algebras A: namely, when A is compressed, or when its h-vector h(A) ends with (…,3,2). Also, we will prove our lower bound when h(A) begins with (1,3,h₂,…), where h₂≤4, and our upper bound when h(A) ends with (…,h_c−1,h_c), where h_c−1≤h_c+1.     

Generic initial ideals and graded Artinian-level algebras not having the Weak-Lefschetz Property

We find a sufficient condition that is not level based on a reduction number. In particular, we prove that a graded Artinian algebra of codimension 3 with Hilbert function cannot be level if h_d≤2d+3, and that there exists a level O-sequence of codimension 3 of type for h_d≥2d+k for k≥4. Furthermore, we show that is not level if , and also prove that any codimension 3 Artinian graded algebra A=R/I cannot be level if . In this case, the Hilbert function of A does not have to satisfy the condition h_d−1>h_d=h_d+1.Moreover, we show that every codimension n graded Artinian level algebra having the Weak-Lefschetz Property has a strictly unimodal Hilbert function having a growth condition on (h_d−1−h_d)≤(n−1)(h_d−h_d+1) for every d>θ where

h₀<h₁<<h_α==h_θ>>h_s−1>h_s.