Related papers: Stanley-Reisner rings for quasi-arithmetic matroid…
In this paper, we provide a simple proof for the fact that two simplicial complexes are isomorphic if and only if their associated Stanley-Reisner rings, or their associated facet rings are isomorphic as $K$-algebras. As a consequence, we…
Simplicial complexes X provide commutative rings A(X) via the Stanley-Reisner construction. We calculated the cotangent cohomology, i.e., T1 and T2 of A(X) in terms of X. These modules provide information about the deformation theory of the…
One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this…
A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible $h$-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve…
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.…
Brown has shown that the Stanley-Reisner ring of the broken circuit complex of a graph has a linear system of parameters which is defined in terms of the circuits and cocircuits of the graph. Later on Brown and Sagan conjectured a special…
Unprojection theory aims to analyze and construct complicated commutative rings in terms of simpler ones. Our main result is that, on the algebraic level of Stanley-Reisner rings, stellar subdivisions of non-acyclic Gorenstein simplicial…
We prove that the Stanley--Reisner ideal of the Alexander dual of the subword complexes in Coxeter groups has linear quotients with respect to the lexicographical order of the minimal monomial generators. As a consequence, we obtain a…
We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the…
In this short note we give an elementary description of the linear part of the minimal free resolution of a Stanley-Reisner ring of a simplicial complex $\Delta$. Indeed, the differentials in the linear part are simply a compilation of…
Using linear algebra methods we study certain algebraic properties of monomial rings and matroids. Let I be a monomial ideal in a polynomial ring over an arbitrary field. If the Rees cone of I is quasi-ideal, we express the normalization of…
We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations for the Chow ring, recovering…
Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, {\em quasi-graphic matroids}, that unifies these two existing classes. Unlike frame matroids and…
We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen-Macaulay semigroup ring by the "shape" of the canonical module. We also characterize the seminormality of a toric…
We introduce the notion of power lattices that unifies and extends the equicardinal geometric lattices, Cartesian products of subspace lattices, and multiset subset lattices, among several others. The notions of shellability for simplicial…
This paper studies properties of simplicial complexes for which the m-th symbolic power of the Stanley-Reisner ideal equals to the m-th ordinary power for a given m > 1. The main results are combinatorial characterizations of such complexes…
To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By…
We prove that all the symbolic powers of a Stanley-Reisner ideal are Cohen-Macaulay if and only if the associated simplicial complex is a matroid.
We continue the study on sheaves of rings on finite posets. We present examples where the ring of global sections coincide with toric faces rings, quotients of a polynomial ring by a monomial ideal and algebras with straightening laws. We…
Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals…