On algebras associated to partially ordered sets

A finite poset $P$ is called "simplicial", if it has the smallest element $\hat{0}$, and every interval $[\hat{0}, x]$ is a boolean algebra. The face poset of a simplicial complex is a typical example. Generalizing the Stanley-Reisner ring of a simplicial complex, Stanley assigned the graded ring $A_P$ to $P$. This ring has been studied from both combinatorial and topological perspective. In this paper, we will give a concise description of a dualizing complex of $A_P$, which...

The main goal of this paper is to obtain upper bounds for the regularity of graded deficiency modules in the spirit of the one obtained by Kumini--Murai in the monomial case building upon the spectral sequence formalism developed by \`Alvarez Montaner, Boix and Zarzuela. This spectral sequence formalism allows us not only to recover Kumini--Murai's upper bound for monomial ideals, but also to extend it for other types of rings, which include toric face rings and some binomial...

In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture made by M. Katzman about the openness of the finitely generated locus of such Frobenius algebras. This conjecture can be formulated in a precise manner: Let us define \begin{equation*} U=\{P\in \text{Spec}(R)~:~\mathcal{F}(E_{R_{p}})~\te...

Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that the principal order filter at every vertex is isomorphic to the whole poset. Well-known examples include k-array trees, the grid graphs, and the Stern poset. Very little is known about upho posets in general. In this paper, we construct upho posets with Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions have rational poles and zeros. W...

Let P be a poset, O(P) the order polytope of P and C(P) the chain polytope of P. In this paper, we study the canonical ideal of the Ehrhart ring K[C(P)] of C(P) over a field K and characterize the level (resp. anticanonical level) property of K[C(P)] by a combinatorial structure of P. In particular, we show that if K[C(P)] is level (resp. anticanonical level), then so is K[O(P)]. We exhibit examples which show the converse does not hold. Moreover, we show that the symbolic ...

For a standard graded algebra $R$, we consider embeddings of the the poset of Hilbert functions of quotients of $R$ into the poset of ideals of $R$, as a way of classification of Hilbert functions. There are examples of rings for which such embeddings do not exist. We describe how the embedding can be lifted to certain ring extensions, which is then used in the case of polarization and distraction. A version of a theorem of Clements--Lindstr\"om is proved. We exhibit a condit...

We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that ...

In this paper we consider the classical problem of computing linear extensions of a given poset which is well known to be a difficult problem. However, in our setting the elements of the poset are multivariate polynomials, and only a small "admissible" subset of these linear extensions, determined implicitly by the evaluation map, are of interest. This seemingly novel problem arises in the study of global dynamics of gene regulatory networks in which case the poset is a Boole...

The homogeneous coordinate ring of a Schubert variety (a Schubert cycle for short) is an algebra with straightening law generated by a distributive lattice. This paper gives a simple method to study the set of all the join-irreducible elements of this distributive lattice, and gives a simple proof of the criterion of the Gorenstein property of Schubert cycles.

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or rig...