Minimal Resolutions and the Homology of Matching and Chessboard Complexes

We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 × 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre and quadratic Veronese subalgebras. Our motivation is two-fold: We immediately deduce from these results the irreducible decomposition for the symmetric group action on the rational homology of all chessboard complexes and complete graph matching complexes as studied by Björner, Lovasz, Vreica and ivaljevi. This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes. The class of modules over the Segre rings and quadratic Veronese rings which we consider is closed under the operation of taking canonical modules, and hence exposes a pleasant symmetry inherent in these Betti numbers.     

On Duality in the Homology Algebra of a Koszul Complex

The homology algebra of the Koszul complex K(x ₁, ..., x _n ; R) of a Gorenstein local ring R has Poincare duality if the ideal I = (x ₁, ..., x _n ) of R is strongly Cohen-Macaulay (i.e., all homology modules of the Koszul complex are Cohen-Macaulay) and under the assumption that dim R - grade I ⩽ 4 the converse is also true.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 77–81, 2003.     

On the Betti Numbers of Chessboard Complexes

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vreica, and ivaljevi in [2]. Our formula also shows that all eigenvalues of the Laplacians of the simplicial complexes are integers, and it gives a formula (involving partitions) for the multiplicities of the eigenvalues.     

Certain Homology Cycles of the Independence Complex of Grids

Let G be an infinite graph such that the automorphism group of G contains a subgroup K ?? ^d with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles,” the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G,K) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of degree exactly r?|G/I|?1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q(G,K) turns out to be an interval of the form [a,b]∩?={r∈?:a≤r≤b}. For example, for the square grid, we obtain the interval $[\frac{1}{5},\frac{1}{4}]\cap \mathbb{Q}Let G be an infinite graph such that the automorphism group of G contains a subgroup K ≅ℤ ^d with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles,” the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G,K) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of degree exactly r⋅|G/I|−1; |G/I| denotes the size of the vertex set of G/I. In each of the three cases, Q(G,K) turns out to be an interval of the form [a,b]∩ℚ={r∈ℚ:a≤r≤b}. For example, for the square grid, we obtain the interval [\frac15,\frac14]?\mathbbQ[\frac{1}{5},\frac{1}{4}]\cap \mathbb{Q}.     

Bestvina's Normal Form Complex and the Homology of Garside Groups

A Garside group is a group admitting a finite lattice generating set . Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice , and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvina's weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually Abelian.     

On the Homology of the Virasoro Algebra

This paper gives the structure of the homology of the Witt algebra and the Virasoro algebra with coeffieionts in a Verma module. Let s_k=(3k^2+k)/2, t_k=(3k^2—k)/2,k\in Z_+, and P={—s_k,-t_k|k\in Z_+}. Then the author obtains the homology of the Witt algebra with coefficients in an irreducible module L(\Lambda) with highest weight \Lambda \notin P, and the homology of the Virasoro algebra with coefficients in some irreducible modules.     

On the Homology of Completion and Torsion

Let A be a commutative ring, and ${\mathfrak{a}}$ a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM equivalence, which is an equivalence between the category of cohomologically ${\mathfrak{a}}$ -adically complete complexes and the category of cohomologically ${\mathfrak{a}}$ -torsion complexes. These are triangulated subcategories of the derived category of A-modules. Our work extends earlier work by Alonso–Jeremias–Lipman, Schenzel and Dwyer–Greenlees.     

Homology of a Perturbation of a Complex Projective Hypersurface

A nonsingular hypersurface X in with n3 is studied. The main result of the paper says that the homology coming from the affine part of a hypersurface of smaller degree forms a direct summand in the homology of X, which is independent over integers with the class of a multiple hyperplane section. The proof is outlined. Bibliography: 3 titles.     

棋盘图的几种染色

染色问题是图论的重要研究内容之一,采用一种全新的方法给出了一类特殊图——棋盘图的邻点可区别边染色和邻点可区别全染色,并给出了相应的色数.     

5-Torsion Points on Curves of Genus 2

Let C be a smooth proper curve of genus 2 over an algebraicallyclosed field k. Fix a Weierstrass point in C(k) and identifyC with its image in its Jacobian J under the Albanese embeddingthat uses as base point. For any integer N1, we write J_N forthe group of points in J(k) of order dividing N and for the subset of J_N of points oforder N. It follows from the Riemann–Roch theorem thatC(k)J₂ consists of the Weierstrass points of C and that C(k) and C(k) are empty (see [3]). The purpose of this paper is to study curvesC with C(k) non-empty.     

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.     

On f-Vectors and Relative Homology

We find strong necessary conditions on the f-vectors, Betti sequences, and relative Betti sequence of a pair of simplicial complexes. We also present an example showing that these conditions are not sufficient. If only the difference between two Betti sequences is specified, and not the individual Betti sequences, then the characterization is complete, and the characterization of all pairs of simplicial complexes matches the characterization of pairs of near-cones. Our necessary conditions rely upon a combinatorial decomposition of pairs of simplicial complexes that reflects the homology and relative homology of the complexes.     

谈谈与图有关的几种复形的同调群

我们从组合拓扑方法在图论的应用中,着重介绍与图有关的几种复形的近期研究动态,论述其中一些基础性的问题,并提出一些可供研究的新问题。     

On Balance for Relative Homology

The Hodge spectrum is an important analytic invariant of singularities encoding the Hodge filtration and the monodromy of the Milnor fiber. However, explicit formulas exist in only a few cases. In this article, the main result is a combinatorial formula for the Hodge spectrum of any homogeneous polynomials in three variables whose zero locus is a projective curve arrangement having only ordinary multiple points.     

On Generalized Homology of Artin Groups

The Morava K-theory and the generalized BrownPeterson homology of groups related to the classical braid groups are calculated. Examples of such groups are the Artin groups and braid groups in handlebodies. Bibliography: 13 titles.     

On the Local Homology Theory for Artinian Modules

Let A be a non-zero Artinian R-module. For an arbitrary ideal I of R, we show that the local homology module Hpx(A) is independent of the choice of x whenever 0 : AI = 0 : A(x1,...,xr). Thus, identifying these modules, we write HpI(A). In this paper we prove that there is a certain duality between HiI(A) and the local cohomology modules and provide some information about the vanishing local homology module HiI(A) which gives an improved form of the main results of [22].     

Corrigendum: On the Floer Homology of Cotangent Bundles

We fix an orientation issue that appears in our previous paper about the isomorphism between Floer homology of cotangent bundles and loop space homology. When the second Stiefel‐Whitney class of the underlying manifold does not vanish on 2‐tori, this isomorphism requires the use of a twisted version of the Floer complex. © 2014 Wiley Periodicals, Inc.