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In this paper, we study simplicial complexes whose Stanley–Reisner rings are almost Gorenstein and have a-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen–Macaulay simplicial complexes. A d-dimensional simplicial complex Δ is said to be uniformly Cohen–Macaulay if it is Cohen–Macaulay and, for any facet F of Δ, the simplicial complex Δ ∖ { F } is Cohen–Macaulay of dimension d. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen–Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension ≤2.

The character degree graph Γ ( G ) of a finite group G has long been studied as a means of understanding the structural properties of G. For example, a result of Manz and Palfy states that the character degree graph of a finite solvable group has at most two connected components. In this paper, we introduce the character degree simplicial complex G ( G ) of a finite group G. We provide examples justifying the study of this simplicial complex as opposed to Γ ( G ) , and prove an analogue of Manz's Theorem on the number of connected components that is dependent upon the dimension of G ( G ) .

Let Δ be a simplicial complex on [ n ] , S = K [ x 1 , … , x n ] the polynomial ring in n-variables over a field K and K [ Δ ] = S / I Δ the Stanley–Reisner ring of Δ with respect to K. It is proved that the Stanley Conjecture holds for K [ Δ ] , i.e., sdepth S ( K [ Δ ] ) ≥ depth S ( K [ Δ ] ) , when I Δ has four associated prime ideals. It is also obtained that sdepth S ( K [ Δ ] ) ≥ size S ( I Δ ) .

Let S = k [ x 1 , x 2 , … , x n ] be a polynomial ring. Let I be a Stanley–Reisner ideal in S of a pure simplicial complex of dimension one. In this paper, we study the Buchsbaum property of S / I r for any integer r > 0 . Our first purpose is giving a characterization of Ext-modules Ext S p ( S / m t , S / J ) for any monomial ideal J, where m t = ( x 1 t , x 2 t , … , x n t ) , in terms of certain simplicial complexes. Then we consider the Buchsbaum property of S / I r . The main tool to check the Buchsbaumness is the surjectivity criterion. We see the behavior of the canonical map from Ext S p ( S / m t , S / I r ) to H m p ( S / I r ) from the view point of reduced cohomology groups of simplicial complexes.

A linear ball is a simplicial complex whose geometric realization is homeomorphic to a ball and whose Stanley--Reisner ring has a linear resolution. It turns out that the Stanley--Reisner ring of the sphere which is the boundary complex of a linear ball satisfies the multiplicity conjecture. A class of shellable spheres arising naturally from commutative algebra whose Stanley--Reisner rings satisfy the multiplicity conjecture will be presented., 19 Pages

We use algorithms developed recently for the study of linear groups to investigate a sequence of matrix groups defined over GF ( 2 ) ; these are images of representations of certain finitely presented groups considered by Soicher in a study of simplicial complexes related to the Suzuki sequence graphs.

The zero divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero divisors of the semigroup, with two distinct vertices joined by an edge in case their product in the semigroup is zero. We continue the study of this construction and its extension to a simplicial complex.

We study the local cohomology modules HiIΣ(k[Δ]) of the Stanley–Reisner ring k[Δ] of a simplicial complex Δ with support in the ideal IΣ⊂k[Δ] corresponding to a subcomplex Σ⊂Δ. We give a combinatorial topological formula for the multigraded Hilbert series, and in the case where the ambient complex is Gorenstein, compare this with a second combinatorial formula that generalizes results of Mustata and Terai. The agreement between these two formulae is seen to be a disguised form of Alexander duality. Other results include a comparison of the local cohomology with certain Ext modules, results about when it is concentrated in a single homological degree, and combinatorial topological interpretations of some vanishing theorems.

Let M be an ordered matroid and C••(M) be an exterior algebra over its underlying set E, graded by both corank and nullity. Then C•0(M) is the simplicial chain complex of IN(M), the simplicial complex whose simplices are indexed by the independent sets of the matroid. Dually, C0•(M) is the cochain complex of IN(M*). We give a combinatorial description of a basis of eigenvectors for the combinatorial Laplacian of a family of boundary maps on the double complex, extending work by W. Kook, V. Reiner, and D. Stanton [2000, J. Amer. Math. Soc.13, 129–148] on IN(M). The eigenvalues are enumerated by a weighted version of the Tutte polynomial, using an identity of G. Etienne and M. Las Vergnas [1998, Discrete Math.179, 111–119]. As an application, we prove a duality theorem for the cohomology of Orlik–Solomon algebras.

We demonstrate that the simplicial complex formed from the poset of nontrivial subpairs for a given p-block of a finite group G is disconnected if and only if G has the block analogue of a strongly p-embedded subgroup.

In this paper we prove: • In characteristic > 0 every simplicial toric affine or projective variety with full parametrization is a set-theoretic complete intersection. This extends previous results by R. Hartshorne (1979, , 380–383) and T. T. Moh (1985, , 217–220). • In any characteristic, every simplicial toric affine or projective variety with full parametrization is an almost set-theoretic complete intersection. This extends previous known results by M. Barile and M. Morales (1998, , 1907–1912) and A. Thoma (, to appear). • In any characteristic, every simplicial toric affine or projective variety of codimension two is an almost set-theoretic complete intersection. Moreover the proofs are constructive and the equations we find are binomial ones.

This paper describes explicitly all non-regular non-degenerate simplicial stochastic Bernstein algebras. Consequently, the Bernstein problem (S. N. Bernstein, Science Ukraine 1 (1992), 14–19) in the non-degenerate case is settled, since the regular and exceptional cases have already been examined by Y. Lyubich in the 1970s. Notice that from this result it is possible to explicitly describe every non-regular simplicial algebra ( A , Δ) since the simplicial subalgebra (〈supp( A 2 )〉, [supp( A 2 )]) is non-degenerate. Also we prove the relevant Lyubich's conjecture (1992, Yu I. Lyubich, Biomathematics 22 , 232) in an affirmative way: all normal simplicial stochastic Bernstein algebras are regular.

Resume Let G be a finite group. The main result of this note shows that the reduced chain complex of an acyclic finite dimensional simplicial G-complex is a split acyclic complex of Z G-modules. The proof requires the extension of well known results on finitely generated p-permutation modules to arbitrary p-permutation modules. The key is a theorem stating that a bounded complex of p-permutation modules is split acyclic if and only if all its Brauer quotients are acyclic.

We explore the connection between ideals of fat points (which correspond to subschemes of P nobtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on ad-dimensional simplicial complex Δ embedded inRd. Using the inverse system approach introduced by Macaulay [ 11 ], we give a complete characterization of the free resolutions possible for ideals ink[x, y] generated by powers of homogeneous linear forms (we allow the powers to differ). We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed Δ, the dimension of the vector space of splines on Δ of degree less than or equal tok. We use this relationship and the results above to derive a formula which gives the number of planar (mixed) splines in sufficiently high degree.

A simplicial poset, a poset with a minimal element and whose every interval is a Boolean algebra, is a generalization of a simplicial complex. Stanley defined a ring A P associated with a simplicial poset P that generalizes the face-ring of a simplicial complex. If V is the set of vertices of P , then A P is a k [ V ]-module; we find the Betti polynomials of a free resolution of A P , and the local cohomology modules of A P , generalizing Hochster's corresponding results for simplicial complexes. The proofs involve splitting certain chain or cochain complexes more finely than in the simplicial complex case. Corollaries are that the depth of A P is a topological invariant, and that the depth may be computed in terms of the Cohen-Macaulayness of skeleta of P , generalizing results of Munkres and Hibi.

Let G be a connected algebraic Q-group, S, the Steinberg representation of G= G(Q). Recall that SG may be realized on the reduced integral homology of the Tits building of parabolic Q-subgroups of G (see Section 1). In this article, we combine facts about SG with the Borel-Serre Duality Theorem (see (3.1)) and basic Lie theory to derive new results on the cohomology of arithmetic subgroups r of G. These are summarized below. Our sharpest results apply to H’(T, -) (v is the virtual cohomological dimension of r) where G is split over E. Some of our work generalizes that of Ash, Ash and Rudolph, and Lee and Szczarba on &5,(Z). See [Al, A-R, L-S, L-S1 1. In earlier papers on this topic, an important ingredient has been a simplicial complex Y of dimension equal to vcd S&(E) on which &5,(Z) acts cocompactly with finite cell stabilizers. The existence of such a Y for a general Chevalley group has not been verified. (There has been recent success with SP,(Z) in [M-M].) Roughly speaking, we avoid this issue by using SG as a substitute for the top chain group of Y. Assume for now the G is semisimple, split over H, and has no factor of We 4.

We study certain subcomplexes Δ′ of an arbitrary simplicial complex Δ such that Hmi(k[Δ])∼-Hmi(k[Δ′]) for any 0⩽i

In .this brief note we show that the canonical module of the Stanley-Reisner ring of a doubly Cohen-Macaulay ordered set is isomorphic to a certain ideal of the same ring. For a general finite partially ordered set, the corresponding ideal is isomorphic to a submodule of the canonical module. For an introduction to Cohen-Macaulay ordered sets from the ringtheoretical point of view see Garsia [4] and Baclawski-Garsia [3]. Doubly Cohen-Macaulay ordered sets were introduced in Baclawski [2]. Let A be a finite simplicial complex of rank r (dimension r 1) on vertex set V. We write A, for {(T E A 1 (u 1 = k}. We assume that A can be colored, i.e., there exists a map c: V-r [r] = ( 1,2 ,..., r} such that for every u E A, I c@)l = I Iu , an d we fix a choice of coloring henceforth. The most important example of a colored complex is the simplicial complex A(P) of chains of a partially ordered set (poset)P. For example, the map c: P -+ [r] given by

In this paper we study various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise noncommuting (resp. commuting) sets of nontrivial elements in G.We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply connected.Our main result is a simplicial decomposition formula for BNC(G) which follows from a result of A. Björner, M. Wachs and V. Welker, on inflated simplicial complexes (2000, A poset fiber theorem, preprint). As a corollary we obtain that if G has a nontrivial center or if G has odd order, then the homology group Hn−1(BNC(G)) is nontrivial for every n such that G has a maximal noncommuting set of order n.We discuss the duality between NC(G) and C(G) and between their p-local versions NCp(G) and Cp(G). We observe that Cp(G) is homotopy equivalent to the Quillen complexes Ap(G) and obtain some interesting results for NCp(G) using this duality.Finally, we study the family of groups where the commutative relation is transitive, and show that in this case BNC(G) is shellable. As a consequence we derive some group theoretical formulas for the orders of maximal noncommuting sets.

We show that all monomial ideals in the polynomial ring in at most 3 variables are pretty clean and that an arbitrary monomial ideal $I$ is pretty clean if and only if its polarization $I^p$ is clean. This yields a new characterization of pretty clean monomial ideals in terms of the arithmetic degree, and it also implies that a multicomplex is shellable if and only the simplicial complex corresponding to its polarization is (non-pure) shellable. We also discuss Stanley decompositions in relation to prime filtrations.

In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alternative) proofs of classical criteria due to Hochster, Reisner, and Stanley., Comment: Minor revision. 15 pages, Plain TeX with epsf.tex, 8 PostScript figures, PostScript file available also at http://www.math.columbia.edu/~psorin/eprints/monbetti.ps

In this paper, we investigate the fundamental group of the Quillen complex of the symmetric group Δ A p ( S n ) . We show that when p is an odd prime the simplicial complex Δ A p ( S n ) is simply connected if and only if 3 p + 2 ⩽ n p 2 or n ⩾ p 2 + p . Furthermore, we determine the fundamental group π 1 ( A p ( S n ) ) in all cases except those where p ⩾ 5 and n ∈ { 3 p , 3 p + 1 } and that where p = 3 and n = 10 .