Planar lattices are lexicographically shellable

The special properties of planar posets have been studied, particularly in the 1970's by I. Rival and others. More recently, the connection between posets, their corresponding polynomial rings and corresponding simplicial complexes has been studied by Stanley and others. This paper, using work of Björner, provides a connection between the two bodies of work, by characterizing when planar posets are Cohen-Macaulay. Planar posets are lattices when they contain a greatest and a least element. We show that a finite planar lattice is lexicographically shellable and therefore Cohen-Macaulay iff it is rank-connected.     

A Krull-Schmidt theorem for one-dimensional rings of finite Cohen-Macaulay type

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal Cohen-Macaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen-Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.     

Homotopy type of posets and lattice complementation

This paper is concerned with homotopy properties of partially ordered sets, in particular contractibility. The main result is that a noncomplemented lattice with deleted bounds is contractible. The paper also presents (i) the homology of final sets and cutsets, (ii) a generalization to posets of Rota's crosscut theorem, (iii) contractibility proofs for some classes of posets of interest in fixed point theory, and (iv) a simple characterization of the Cohen-Macaulay property for dismantlable lattices.     

A Poset Fiber Theorem for Doubly Cohen-Macaulay Posets and Its Applications

This paper studies topological properties of the lattices of non-crossing partitions of types A and B and of the poset of injective words. Specifically, it is shown that after the removal of the bottom and top elements (if existent) these posets are doubly Cohen-Macaulay. This strengthens the well-known facts that these posets are Cohen-Macaulay. Our results rely on a new poset fiber theorem which turns out to be a useful tool to prove double (homotopy) Cohen- Macaulayness of a poset. Applications to complexes of injective words are also included.     

On sequentially Cohen-Macaulay complexes and posets

The classes of sequentially Cohen-Macaulay and sequentially homotopy Cohen-Macaulay complexes and posets are studied. First, some different versions of the definitions are discussed and the homotopy type is determined. Second, it is shown how various constructions, such as join, product and rank-selection preserve these properties. Third, a characterization of sequential Cohen-Macaulayness for posets is given. Finally, in an appendix we outline connections with ring-theory and survey some uses of sequential Cohen-Macaulayness in commutative algebra. Supported in part by National Science Foundation grants DMS 0302310 and DMS 0604562. Supported by Deutsche Forschungsgemeinschaft (DFG)     

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.     

Combinatorial methods in the theory of Cohen-Macaulay rings

Characterizations of Cohen-Macaulay posets are given in terms of the nonsingularity of certain incidence matrices. These results are applied to derive a purely combinatorial construction of basic systems for Stanley-Reisner rings of shellable posets. A procedure for transferring identities from one polynomial ring to another is then used to obtain basic systems for partition rings.     

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.     

Spherical complexes attached to symplectic lattices

To the integral symplectic group Sp(2g,\mathbbZ){{\rm Sp}(2g,\mathbb{Z})} we associate two posets of which we prove that they have the Cohen-Macaulay property. As an application we show that the locus of marked decomposable principally polarized abelian varieties in the Siegel space of genus g has the homotopy type of a bouquet of (g − 2)-spheres. This, in turn, implies that the rational homology of moduli space of (unmarked) principal polarized abelian varieties of genus g modulo the decomposable ones vanishes in degree ≤ g − 2. Another application is an improved stability range for the homology of the symplectic groups over Euclidean rings. But the original motivation comes from envisaged applications to the homology of groups of Torelli type. The proof of our main result rests on a refined nerve theorem for posets that may have an interest in its own right.     

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.     

Central Delannoy Numbers and Balanced Cohen-Macaulay Complexes

We introduce a new join operation on colored simplicial complexes that preserves the Cohen-Macaulay property. An example of this operation puts the connection between the central Delannoy numbers and Legendre polynomials in a wider context. On leave from the Rényi Mathematical Institute of the Hungarian Academy of Sciences. Received April 18, 2005     

Cohen-Macaulay and Gorenstein property of Rees algebras of non-singular equimultiple prime ideals

We give criteria for the Cohen-Macaulay and Gorenstein property of Rees algebras of height 2 non-singular equimultiple prime ideals in terms of explicite representations of the associated graded rings. As consequences, we show that in general, the Cohen-Macaulay resp. Gorenstein property of such Rees algebras does not imply the Cohen-Macaulay resp. Gorenstein property of the base ring and that these properties depend upon the characteristic. Dedicated to the memrory of Professor Lê Van Thiêm Professor Lê Van Thiêm was the first directorof Hanoi Institute of Mathematics     

The lexicographic sum of Cohen-Macaulay and shellable ordered sets

We study the problem of determining when the lexicographic sum ∑ _q∈Q P _q of a family of posets {P _q/qεQ} over a posetQ is Cohen-Macaulay or shellable. Our main result, a characterization of when the lexicographic sum is Cohen-Macaulay, is proven using combinatorial methods introduced by Garsia. A similar characterization for when the lexicographic sum is CL (chainwise-lexicographically)-shellable, is derived using the recursive atom ordering method due to Björner and Wachs.     

On Cohen-Macaulay Posets, Koszul Algebras and Certain Modules Associated to Schubert Varieties

In the first part of this paper, some cohomological propertiesof the incidence algebra A_k(X) of a finite poset X over a fieldk are studied. In particular, it is shown that A_k(X) is Koszulif and only if every open interval of X is Cohen-Macaulay overk. In the second part, certain categories of representationsof a Borel subgroup B of a semisimple connected algebraic groupG are studied. It is shown that these categories have the samecohomological properties as certain Cohen-Macaulay posets relatedto the Weyl group of G, and the higher Ext-groups between certainB-modules are then computed, using results from the first part.     

Wild hypersurfaces

Complete hypersurfaces of dimension at least 2 and multiplicity at least 4 have wild Cohen-Macaulay type.     

Exact categories,big Cohen-Macaulay modules and finite representation type

One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterisation of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce “big objects” and prove an Auslander-type “splitting-big-objects” theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type.     

Graphes et ordonnés démontables, propriété de la clique fixe

This work extends to dismantable graphs many properties of dismantable posets dealing with products, exponentiation, existence of paths in a graph of all homomorphisms, fixed clique property, etc. We show that a poset is dismantlable in the sense of Rival if and only if its comparability graph is dismantlable, thus proving that, for posets, dismantlability is a comparability invariant. We establish the analogues for graphs of results of Duffus, Poguntke and Rival about fixed point sets and cores in posets.     

Frankl’s Conjecture and the Dual Covering Property

We have proved that the Frankl’s Conjecture is true for the class of finite posets satisfying the dual covering property. This research was supported by the Board of College and University Development, University of Pune, via the projects BCUD/494 and SC-66.     

Rudin性质与拟Z-连续Domain

对一般子集系统 Z,引入了 Rudin性质,给出了它的映射式刻划,作为拟连续偏序集和Z-连续偏序集的公共推广,引入了拟Z-连续Domain的概念,讨论了拟Z-连续Domain的基本性质,特别地,给出了 Rudin性质及其映射式刻划在拟 Z-连续Domain方面的若干应用,将关于拟连续偏序集的主要结果推广至了拟 Z-连续 Domain情形。     

On forbidden configurations for strong posets

The concept of strong elements in posets is introduced. Several properties of strong elements in different types of posets are studied. Strong posets are characterized in terms of forbidden structures. It is shown that many of the classical results of lattice theory can be extended to posets. In particular, we give several characterizations of strongness for upper semimodular (USM) posets of finite length. We characterize modular pairs in USM posets of finite length and we investigate the interrelationships between consistence, strongness, and the property of being balanced in USM posets of finite length. In contrast to the situation in upper semimodular lattices, we show that these three concepts do not coincide in USM posets.