Related papers: The Multiplicity Conjecture for Barycentric Subdiv…
A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to…
The Stanley's Conjecture on Cohen-Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley's Conjecture…
For a simplicial complex $\Delta$, the graded Betti number $\beta_{i,j}(k[\Delta])$ of the Stanley-Reisner ring $k[\Delta]$ over a field $k$ has a combinatorial interpretation due to Hochster. Terai and Hibi showed that if $\Delta$ is the…
The Multiplicity Conjecture (MC) of Huneke and Srinivasan provides upper and lower bounds for the multiplicity of a Cohen-Macaulay algebra $A$ in terms of the shifts appearing in the modules of the minimal free resolution (MFR) of $A$. All…
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 introduce the concepts of an amazing hypercube decomposition and a double shortcut for it, and use these new ideas to formulate a conjecture implying the Combinatorial Invariance Conjecture of the Kazhdan--Lusztig polynomials for the…
We introduce the concept of Stanley decompositions in the localized polynomial ring $S_f$ where $f$ is a product of variables, and we show that the Stanley depth does not decrease upon localization. Furthermore it is shown that for monomial…
In this paper we show that a simplicial complex can be determined uniquely up to isomorphism by its barycentric subdivision or comparability graph. At the end, it is summarized several algebraic, combinatorial and topological invariants of…
We define a transcendence degree for division algebras, by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the…
Let $\Delta$ be a stable simplicial complex on $n$ vertexes. Over an arbitrary base field $K$, the symmetric algebraic shifted complex $\Delta^s$ of $\Delta$ is defined. It is proved that the Betti numbers of the Stanley-Reisner ideals in…
We apply Miller's theory on multigraded modules over a polynomial ring to the study of the Stanley depth of these modules. Several tools for Stanley's conjecture are developed, and a few partial answers are given. For example, we show that…
We show that the maximal shifts in the minimal free resolution of the quotients of a polynomial ring by a monomial ideal are subadditive as a function of the homological degree. This answers a question that has received some attention in…
Zaremba's Conjecture concerns the formation of continued fractions with partial quotients restricted to a given alphabet. In order to answer the numerous questions that arrive from this conjecture, it is best to consider a semi-group, often…
The purpose of this paper is to prove that certain limits of polynomial rings are themselves polynomial rings, and show how this observation can be used to deduce some interesting results in commutative algebra. In particular, we give two…
This paper provides a self-contained exploration of subdivisions of simplicial complexes, with emphasis on barycentric subdivision. We present formal definitions of subdivisions, show how the realization of a complex is preserved under…
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 investigate generalized notions of the nerve complex for the facets of a simplicial complex. We show that the homologies of these higher nerve complexes determine the depth of the Stanley-Reisner ring $k[\Delta]$ as well as the…
Given an infinite field $\mathbb{k}$ and a simplicial complex $\Delta$, a common theme in studying the $f$- and $h$-vectors of $\Delta$ has been the consideration of the Hilbert series of the Stanley--Reisner ring $\mathbb{k}[\Delta]$…
In this note we compare the a-invariant of a homogeneous algebra B to the a-invariant of a subalgebra A. In particular we show that if $A \subset B$ is a finite homogeneous inclusion of standard graded domains over an algebraically closed…
We present a conjecture on multiplicity of irreducible representations of a subgroup $H$ contained in the irreducible representations of a group $G$, with $G$ and $H$ having the same derived groups. We point out some consequences of the…