相关论文: A Generalized Macaulay Theorem and Generalized Fac…

We present a short proof of Reisner's Theorem, characterizing which simplicial complexes have a Cohen-Macaulay face ring. In some cases, we can also express some homological invariants of the face ring in terms of the reduced homology of…

The main invariant to study the combinatorics of a simplicial complex $K$ is the associated face ring or Stanley-Reisner algebra. Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face…

We classify the complementary vectors of doubly Cohen-Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid.…

A colored complex of type $\mathbf{a} = (a_1, \dots, a_n)$ is a simplicial complex ${\Delta}$ on a vertex set $V$, together with an ordered partition $(V_1, \dots, V_n)$ of $V$, such that every face $F$ of ${\Delta}$ satisfies $|F \cap V_i|…

We provide two examples which show that for an affine semigroup ring the set of Cohen-Macaulay classes can be larger than the set of conic classes.

Let K be the face ring of the independence complex of a matroid. We show that if T is a generic linear system of parameters, then K/T satisfies a weak form of the Hard Lefschetz Theorem. As a result, the first half of the h-vector of the…

In a recent work [2] with Datta, we introduced the mu vector (with respect to a given field) of simplicial complexes and used it to study tightness and lower bounds. In this paper, we modify the definition of mu vectors. With the new…

Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property,…

We prove a generalisation of the correspondence, due to Resende and Lawson--Lenz, between \'etale groupoids---which are topological groupoids whose source map is a local homeomorphisms---and complete pseudogroups---which are inverse monoids…

We construct a generalize Ishida complex to compute the local cohomology with monomial support of modules over quotients of polynomial rings by cellular binomial ideals. As a consequence, we obtain a combinatorial criterion to determine…

We investigate categorical and amalgamation properties of the functor Idc assigning to every partially ordered abelian group G its semilattice of compact ideals Idc G. Our main result is the following. Theorem 1. Every diagram of finite…

Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases these sets can be described as the…

For each finite subgroup G of SL(n, C), we introduce the generalized Cartan matrix C_{G} in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices…

We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a Cohen-Macaulay ring with a dualizing module, and we…

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let \alpha \colon G \to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^{\alpha} be the fixed point…

We extend the construction of moment-angle complexes to simplicial posets by associating a certain T^m-space Z_S to an arbitrary simplicial poset S on m vertices. Face rings Z[S] of simplicial posets generalise those of simplicial…

We studies the nearly Gorenstein property for Veronese subalgebras of (semi-)standard graded algebras. We introduce a condition~$(\natural)$ for Cohen--Macaulay semi-standard graded rings, motivated by the study of Ehrhart rings. We show…

We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) simplicial posets with a group action. The action on the complex induces an action on the face ring, and we…

Macaulay posets are posets in which an analog of the Kruskal-Katona Theorem holds. Macaulay rings (also called Macaulay-Lex rings) are rings in which an analog of Macaulay's Theorem for lex ideals holds. The study of both of these objects…

We investigate the structure and properties of symmetric ideals generated by general forms in the polynomial ring under the natural action of the symmetric group. This work significantly broadens the framework established in our earlier…