Quasi-isomorphisms of Koszul complexes

Let be a surjective homomorphism of noetherian local commutative rings that induces an isomorphism between the first Koszul homology modules and an epimorphism between the second Koszul homology modules. Then induces isomorphisms between Koszul homology modules in all dimensions.

     

Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal in a polynomial ring and for every we consider the Koszul homology with respect to a sequence of of generic linear forms. The Koszul-Betti number is, by definition, the dimension of the degree part of . In characteristic , we show that the Koszul-Betti numbers of any ideal are bounded above by those of the gin-revlex of and also by those of the Lex-segment of . We show that iff is componentwise linear and that and iff is Gotzmann. We also investigate the set of all the gin of and show that the Koszul-Betti numbers of any ideal in are bounded below by those of the gin-revlex of . On the other hand, we present examples showing that in general there is no is such that the Koszul-Betti numbers of any ideal in are bounded above by those of .

     

Selfinjective Koszul algebras of finite complexity

In this paper, we study selfinjective Koszul algebras of finite complexity. We prove that the complexity is a nonnegative integer when it is finite; and that the category Yt of modules with complexity less or equal to t, is resolving and coresolving. We show that for each 0 ≤ 1 ≤ m there exist a family of modules of complexity 1 parameterized by G（l, m）, the Grassmannian of l-dimensional subspaces of an m-dimensional vector space V, for the exterior algebra of V. Using complexity, we also give a new approach to the representation theory of a tame symmetric algebra with vanishing radical cube over an algebraically closed field of characteristic 0, via skew group algebra of a finite subgroup of SL（2, C） over the exterior algebra of a 2-dimensional vector space.     

The Stanley Depth in the Upper Half of the Koszul Complex

Let R = K[X₁, ?c, X_n] be a polynomial ring over some field K. In this article, we prove that the kth syzygy module of the residue class field K of R has Stanley depth n ? 1 for ?n/2? ≤k < n, as it had been conjectured by Bruns et al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than 1. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra.     

Stable Projective Homotopy Theory of Modules,Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.     

Generalizing the notion of Koszul algebra

We introduce a generalization of the notion of a Koszul algebra, which includes graded algebras with relations in different degrees, and we establish some of the basic properties of these algebras. This class is closed under twists, twisted tensor products, regular central extensions and Ore extensions. We explore the monomial algebras in this class and we include some well-known examples of algebras that fall into this class.     

On the Koszul Modules of Exterior Algebras

We prove that the Koszul modules over an exterior algebra can be filtered by the cyclic Koszul modules. We also introduce the cyclic dimension vector as invariants for studying the Koszul modules over an exterior algebra.     

On the Category of Koszul Modules of Complexity One of an Exterior Algebra

In this paper, we prove that there is a natural equivalence between the category F1（x） of Koszul modules of complexity 1 with filtration of given cyclic modules as the factor modules of an exterior algebra A = ∧V of an m-dimensional vector space, and the category of the finite-dimensional locally nilpotent modules of the polynomial algebra of m - 1 variables.     

An Extension of the Roots Separation Theorem

Let T _n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues ₁<₂<< _n and W _k is an (n–1)×(n–1) submatrix by deleting the kth row and the kth column from T _n , k=1,2,...,n. Let _{12 n–1} be the eigenvalues of W _k . It is proved that if W _k has no multiple eigenvalue, then ₁<₁<₂<₂<< _n–1< _n–1< _n ; otherwise if _i = _i+1 is a multiple eigenvalue of W _k , then the above relationship still holds except that the inequality _i < _i+1< _i+1 is replaced by _i = _i+1= _i+1.     

Computing Eigenvalues of Lommel-Type Equations by the Sampling Method

Using the semiclassical approximation for where and as , we shall show that theeigenvalues are the zeros of a certain function in a Paley–Wienerspace, which allows us to use the Whittaker–Shannon–Kotelnikovsampling theorem to approximate the eigenvalues. We show how thedistribution of the eigenvalues depends on the asymptotics of thecoefficients and as . After abrief discussion on the truncation error, numerical examples are provided.     

测度链上p-Laplacian边值问题的三个正对称解

该文研究了p-Laplacian动力边值问题（g（u^△（t）））△＋a（t）f（t，u（t））=0，t∈[0，T]T,u（0）=u（T）=w，u△（0）=-u^△（T）正解的存在性．其中W是非负实数，g（v）=｜v｜p-2v1 P＞1．根据对称技巧和五泛函不动点定理，证明了边值问题至少有三个正的对称解，同时，给出了一个例子验证了我们的结果。     

Computing the tame kernel of quadratic imaginary fields

J. Tate has determined the group (called the tame kernel) for six quadratic imaginary number fields where Modifying the method of Tate, H. Qin has done the same for and and M. Skaba for and

In the present paper we discuss the methods of Qin and Skaba, and we apply our results to the field

In the Appendix at the end of the paper K. Belabas and H. Gangl present the results of their computation of for some other values of The results agree with the conjectural structure of given in the paper by Browkin and Gangl.

     

Exact sequences for the homology of the matching complex

Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex Mn, which is the simplicial complex of matchings in the complete graph Kn. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of Mn. First, we demonstrate that there is nonvanishing 3-torsion in whenever , where . By results due to Bouc and to Shareshian and Wachs, is a nontrivial elementary 3-group for almost all n and the bottom nonvanishing homology group of Mn for all n≠2. Second, we prove that is a nontrivial 3-group whenever . Third, for each k?0, we show that there is a polynomial fk(r) of degree 3k such that the dimension of , viewed as a vector space over Z₃, is at most fk(r) for all r?k+2.     

Computing the Wedderburn decomposition of group algebras by the Brauer-Witt theorem

We present an alternative constructive proof of the Brauer-Witt theorem using the so-called strongly monomial characters that gives rise to an algorithm for computing the Wedderburn decomposition of semisimple group algebras of finite groups.

     

Five-torsion in the homology of the matching complex on 14 vertices

J.L. Andersen proved that there is 5-torsion in the bottom nonvanishing homology group of the simplicial complex of graphs of degree at most two on seven vertices. We use this result to demonstrate that there is 5-torsion also in the bottom nonvanishing homology group of the matching complex on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we conclude that the case n=14 is exceptional; for all other n, the torsion subgroup of the bottom nonvanishing homology group has exponent three or is zero. The possibility remains that there is other torsion than 3-torsion in higher-degree homology groups of when n≥13 and n≠14. Research of J. Jonsson was supported by European Graduate Program “Combinatorics, Geometry, and Computation”, DFG-GRK 588/2.     

Two Splittings of a Square Matrix

In this note it is shown that any square matrix AC ^n×n can be represented as the sum A= , where is complex symmetric and rank . The corresponding persymmetric result can be used in finding the terms of a small rank perturbed Toeplitz matrix via an O(n ²) computation. This allows one to perform fast matrix–vector products in case n is large.     

Pseudo-differential operators with meromorphic symbols and systems of complex differential equations

The paper is devoted to general boundary value problems for systems of complex differential and pseudo-differential equations with meromorphic symbols. As the solution spaces, special classes of analytic and exponential functions and functionals are introduced and the existence and uniqueness theorems in these spaces are proved.