Local Projective Model Structures on Simplicial Presheaves

We introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Gro-then-dieck site T and give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis,A1) and SH((Sch/k)cdh,A1) are equivalent. This result was first proven by Voevodsky and our proof uses many of his techniques, but it does not use his theory of -closed classes.     

A note on determinants,minors and a boundary map

We give an explicit formula for the first non-trivial boundary map of the Akin–Buchsbaum–Weyman resolution (Schur functors and Schur complexes, Adv. Math 1982;44:207-278). The role of the Laplace expansion formula is emphasized.     

Lyubeznik's Resolution and Rooted Complexes

We describe a new family of free resolutions for a monomial ideal I, generalizing Lyubeznik's construction. These resolutions are cellular resolutions supported on the rooted complexes of the lcm-lattice of I. Our resolutions are minimal for the matroid ideal of a finite projective space.     

Gorenstein FP-内射复形

引给出了Gorenstein FP-内射复形的概念,进而研究了它的一些性质.     

复形的$D_C$-投射维数

设$R$是交换环, $C$是半对偶$R$-模. 本文引入并研究同调下有界复形的$D_C$-投射维数.     

Rigid Dualizing Complexes Over Commutative Rings

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211–3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes. This research was supported by the US–Israel Binational Science Foundation. The second author was partially supported by the US National Science Foundation.     

Pre-Triangulations and Liftable Complexes

We introduce the concept of pre-triangulations, a relaxation of triangulations that goes beyond the frequently used concept of pseudo-triangulations. Pre-triangulations turn out to be more natural than pseudo-triangulations in certain cases. We show that pre-triangulations arise in three different contexts: In the characterization of polygonal complexes that are liftable to three-space in a strong sense, in flip sequences for general polygonal complexes, and as graphs of maximal locally convex functions. Research supported by the FWF Joint Research Project ‘Industrial Geometry’ S9205-N12.     

Finite-Type Invariants of Cubic Complexes

The paper is for a general audience and may serve as a preliminary introduction to the theory of finite-type invariants.     

TheK-theory of finite fields, revisited

A new, short proof is given for the Quillen theorem that calculates theK-theory of finite fields. This proof uses the Gabber rigidity theorem and the homotopy theory of simplicial presheaves.Research supported by NSERC.     

Complexes of directed trees and independence complexes

First we prove that certain complexes on directed acyclic graphs are shellable. Then we study independence complexes. Two theorems used for breaking and gluing such complexes are proved and applied to generalize the results by Kozlov.An interesting special case is anti-Rips complexes: a subset P of a metric space is the vertex set of the complex, and we include as a simplex each subset of P with no pair of points within distance r. For any finite subset P of R the homotopy type of the anti-Rips complex is determined.     

Linear resolutions of quadratic monomial ideals

We study the minimal free resolution of a quadratic monomial ideal in the case where the resolution is linear. First, we focus on the squarefree case, namely that of an edge ideal. We provide an explicit minimal free resolution under the assumption that the graph associated with the edge ideal satisfies specific combinatorial conditions. In addition, we construct a regular cellular structure on the resolution. Finally, we extend our results to non-squarefree ideals by means of polarization.     

Algebraic Shifting and Basic Constructions on Simplicial Complexes

We try to understand the behavior of algebraic shifting with respect to some basic constructions on simplicial complexes, such as union, coning, and (more generally) join. In particular, for the disjoint union of simplicial complexes we prove Δ(K ˙∪ L) = Δ(Δ(K) ˙∪ Δ(L)) (conjectured by Kalai [6]), and for the join we give an example of simplicial complexes K and L for which Δ(K*L)≠Δ(Δ(K)*Δ(L)) (disproving a conjecture by Kalai [6]), where Δ denotes the (exterior) algebraic shifting operator. We develop a ‘homological’ point of view on algebraic shifting which is used throughout this work.     

Special Cube Complexes

We introduce and examine a special class of cube complexes. We show that special cube-complexes virtually admit local isometries to the standard 2-complexes of naturally associated right-angled Artin groups. Consequently, special cube-complexes have linear fundamental groups. In the word-hyperbolic case, we prove the separability of quasiconvex subgroups of fundamental groups of special cube-complexes. Finally, we give a linear variant of Rips’s short exact sequence. Received: October 2005, Accepted: September 2006     

Cores of Simplicial Complexes

We introduce a method to reduce the study of the topology of a simplicial complex to that of a simpler one. Applying this method to complexes arising from graphs, we give topological meaning to classical graph invariants. As a consequence, we answer some questions raised in (Ehrenborg and Hetyei in Eur. J. Comb. 27(6):906–923, 2006) on the independence complex and the dominance complex of a forest and obtain improved algorithms to compute their homotopy types.     

闭环供应链竞争的博弈分析与链内协调合同设计

假定生产商委托零售商回收废旧产品以进行再制造活动,研究两条分别由单生产商和单零售商组成的闭环供应链的竞争和链内协调。对应两条闭环供应链均为分散式供应链、均为集中式供应链、一条为分散式供应链另一条为集中式供应链,运用博弈论和均衡分析方法分别建立了三种情况下刻画闭环供应链竞争的带均衡约束的均衡模型、纳什均衡模型和带均衡约束的优化模型,进行了模型求解。基于合同理论,提出供应链竞争下协调链内零售商的零售价和回收价格决策的批发价加回收补贴合同。最后的算例验证了模型的合理性和协调合同的有效性。研究表明:产品零售价、需求量与产品替代效应正相关,废旧产品供给量、供应链利润与回收价格敏感性正相关;供应链协调是供应链竞争下的占优策略,有利于提高供应链的运作效率,但两条协调供应链之间的竞争将更激烈,存在囚徒困境现象。     

Extending the Concept of Chain Geometry

We introduce the chain geometry (K,R) over a ring R with a distinguished subfield K, thus extending the usual concept where R has to be an algebra over K. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of K is normal in the group of units of R. This condition is not equivalent to R being a K-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.     

Virtual resolutions of points in P1×P1

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of finite sets of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Specifically, we describe a virtual resolution for a sufficiently general set of points X in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ that only depends on $|X|$. We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$; more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in ${\mathbb{P}}^1\times {\mathbb{P}}^1$.     

Additive Functions and Chain Complexes of Projective Modules

We study additive functions given on a category of finitely generated projective modules. Using these functions, we define p-minimal epimorphisms and give a necessary and sufficient condition for their existence. We prove results concerning the structure of p-minimal chains of projective modules.