Chains of prime ideals in tensor products of algebras

This paper investigates the length of particular chains of prime ideals in tensor products of algebras over a field k. As an application, we compute dim(A⊗_kA) for a new family of domains A that are k-algebras.     

Trace polynomials and invariant theory

LetA be a finite-dimensional simple (non-associative) algebra over an algebraically closed fieldF of characteristic 0. LetG be the group of its automorphisms which acts onkA, the direct sum ofk copies ofA. SupposeA is generated byk elements. In this paper, generators of the field of rational invariantF(kA) ^G are described in terms of operations of the algebraA.     

On the Homology and Fiber Cone of Ideals

In this article, we give a unified approach for several results concerning the fiber cone. Our novel idea is to use the complex C(x _k , ? _{I 1; I 2} , (1, n)). We improve earlier results obtained by several researchers and get some new results. We give a more general definition of ideals of minimal multiplicity and of ideals of almost minimal multiplicity. We also compute the Hilbert series of the fiber cone for these ideals.     

A numerical characterization of the S2-ification of a Rees algebra

Let A be a local ring with maximal ideal . For an arbitrary ideal I of A, we define the generalized Hilbert coefficients . When the ideal I is -primary, j_k(I)=(0,…,0,(−1)^ke_k(I)), where e_k(I) is the classical kth Hilbert coefficient of I. Using these coefficients we give a numerical characterization of the homogeneous components of the S₂-ification of S=A[It,t⁻¹], extending previous results obtained by the author to not necessarily -primary ideals.     

The Picard group of algebras over an algebraically nonclosed field

Let k be a field that is not algebraically closed, and let A be a k-algebra, whose each maximal ideal has residue field k. We prove that each element of the Picard group of A is of finite order and give an optimal upper bound for its order.     

On the equations of the edge cone of a graph¶and some applications

Let G be a graph such that none of its components is bipartite. We describe the facets of the cone generated by the columns of the incidence matrix of G. Let k[G] be the subring generated by the monomials of degree two defining the edges of G, where k is a field. Some estimates for the a-invariant of k[G] are shown when G is the cone of a normal connected non bipartite graph or G is the join of two normal connected non bipartite graphs. Received: 24 July 1997 / Revised version: 3 March 1998     

Lyubeznik numbers of projective schemes

Let X be a projective scheme over a field k and let A be the local ring at the vertex of the affine cone of X under some embedding . We prove that, when char(k)>0, the Lyubeznik numbers λi,j(A) are intrinsic numerical invariants of X, i.e., λi,j(A) depend only on X, but not on the embedding.     

The central simple modules of Artinian Gorenstein algebras

Let A be a standard graded Artinian K-algebra, with char K=0. We prove the following.

1.

A has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form z of A.

2.

If A is Gorenstein, then A has the Strong Lefschetz Property if and only if there exists a linear form z of A such that all central simple modules of (A,z) have the Strong Lefschetz Property.

As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property.     

Derived Picard Groups of Finite-Dimensional Hereditary Algebras

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic _k (A) is the group of triangle auto-equivalences of D> ^b( mod A) induced by two-sided tilting complexes. We study the group DPic _k (A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic _k , as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic _k (A) on a certain infinite quiver ^irr. This representation is faithful when the quiver of A is a tree, and then DPic _k (A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic _k (A). When A is hereditary, DPic _k (A) coincides with the full group of k-linear triangle auto-equivalences of D^b( mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^b( mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.     

Positivity and Convexity in Rings of Fractions

Given a commutative ring A equipped with a preordering A⁺ (in the most general sense, see below), we look for a fractional ring extension (= “ring of quotients” in the sense of Lambek et al. [L]) as big as possible such that A⁺ extends to a preordering R⁺ of R (i.e. with A ∩ R⁺ = A⁺) in a natural way. We then ask for subextensions A ⊂ B of A ⊂ R such that A is convex in B with respect to B⁺ : = B ∩ R⁺. Supported by DFG. A short form of this article has been delivered at the conference Carthapos 2006 at Carthago (Tunisia).     

On Krull and Valuative Dimension of Tensor Products of Algebras

This article is concerned with the dimension theory of tensor products of algebras over a field k. In fact, we provide formulas for the Krull and valuative dimension of A? _k B when A and B are k-algebras such that the polynomial ring A[n] is an AF-domain for some positive integer n. Also, we compute dim _v (A? _k B) in the case where A ? B.     

Attracting current and equilibrium measure for attractors on ℙ<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

Let f be a holomorphic endomorphism of ℙ ^k having an attracting setA. We construct an attracting current and an equilibrium measure associated toA. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy onA.     

Some results on the cartan matrix of a frobenius algebra

Let F be a field and A a Frobenius algebra over F. The Jacobson radical of A is denoted by J = J(A) and the kth socle of A by S _k (A). Let [Abar]=A/J ^k or A/S _k (A). This article gives new interesting relations between the Cartan matrix of A and that of [Abar]. From these results we prove that the Cartan matrix of A is diagonal if A/Soc(A) is a symmetric algebra. Let G be a finite group. If A is a block of F|G] with the above condition, then the Cartan matrix of A is (n), where n is the order of the defect group of A and the least integer such that Jⁿ (A)=0.     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

Degree sequences of <Emphasis Type="Italic">k</Emphasis>-multi-hypertournaments

Let n and k（n ≥ k 〉 1） be two non-negative integers.A k-multi-hypertournament on n vertices is a pair（V,A）,where V is a set of vertices with ｜V｜=n,and A is a set of k-tuples of vertices,called arcs,such that for any k-subset S of V,A contains at least one（at most k!） of the k! k-tuples whose entries belong to S.The necessary and suffcient conditions for a non-decreasing sequence of non-negative integers to be the out-degree sequence（in-degree sequence） of some k-multi-hypertournament are given.     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of R_n , It is shown that if A,BεR_n , det(A) det(B:) ≠ 0, then d_k (AB) ≡ 0 mod d_k (A) d_k (B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k (AB) = d_k (A) d_k (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

Real Prüfer extensions of commutative rings

Let A⊂R be an extension of commutative rings with 1. We show that A is totally real (i.e. all maximal ideals of A are real) and A⊂R is a Prüfer extension if and only if R is totally real and the holomorphy ring H(R/A) of R over A is A. Received: 2 January 2001 / Revised version: 23 April 2001     

Classification of integral trees

A finite poset Q is called integral over a field k if there exists an ASL (algebra with straightening laws) domain on Q{–} over k. We classify all trees (rank one connected posets without cycles) which are integral over an infinite field.     

Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA ₁, ...,A _k are distinct subsets of a set of cardinalityn, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, andk>n, then some two ofA ₁, ...,A _k have empty intersection. We prove a strengthening, that at leastk /n ofA ₁, ...,A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440