On algebras associated to partially ordered sets

Abstract

We continue the work [2] 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 prove a rank-selection theorem which generalizes the well-known rank-selection theorem of Stanley-Reisner rings. Finally, we determine an explicit presentation of certain global rings of sections.

On Some Local Cohomology Spectral Sequences

“…When V = {A/I p } is the inverse system associated to a Stanley-Reisner ring A/I and T = m we recover Hochster's formula; more generally, when V = {A/I p } is the inverse system associated to A/I, where I is any monomial ideal we recover Takayama's formula as formulated by Brun and Römer in [10,Corollary 2.3]. In addition, this decomposition also recovers the Brun-Bruns-Römer decomposition [9,Theorem 1.3] for the so-called…”

On toric face rings

“…This was the main point of view in [4] where the local cohomology groups of rings of such type were studied systematically. A presentation of the toric face ring K [Σ ] was computed besides other things in [6], and initial ideals of the presentation ideals were considered in [5].…”

Ext and local cohomology modules of face rings of simplicial posets

“…It is worth noting that these face rings are also squarefree modules; these objects have been studied from a various viewpoints by Yanagawa (see, e.g., [Yan11b] and [Yan11a]). Furthermore, similar topics have been investigated from a sheaf-theoretic viewpoint by Brun and Römer in [BR08] and by Brun, Bruns, and Römer in [BBR07].…”