Some remarks on algebras with straightening laws
type: Item
instance of: scholarly article
title: Some remarks on algebras with straightening laws
zbMATH Open document ID: 0816.13018
DOI: 10.1016/0022-4049(94)90120-1
author: Naoki Terai
published in: Journal of Pure and Applied Algebra
publication date: 1994-09-22T00:00:00Z
review text: The author considers the following problem. Given a finite poset (partially ordered set) \(H\), which ASL's (algebra with straightening law) exist on \(H\)? A moduli space for the ASL's on \(H\) is constructed as an algebraic scheme over an algebraically closed field \(k\) whose closed points correspond to ASL's on \(H\) over \(k\). If \(\text{rank} H = 2\) (the rank is the maximal number of elements in a chain in \(H)\), the moduli space is given quite explicitly with equations. Let \(k[H]\) be the discrete ASL on \(H\), i.e. the Stanley-Reisner ring where all ``right hand sides'' of the straightening relations are 0. Then it is shown that for any ASL on \(H\) the inequality \(\text{depth} k [H] \leq \text{depth} A \leq \text{depth} k[H] + 1\). \textit{Hibi} has conjectured: If there is an ASL on \(H\) which is an integral domain, then \(k[H]\) is Cohen-Macaulay. A counterexample is given, but the conjecture is proved in some special cases.
Mathematics Subject Classification ID: 13F50
Mathematics Subject Classification ID: 06A11
zbMATH DE Number: 643794
zbMATH Keywords: algebra with straightening law
zbMATH Keywords: depth
zbMATH Keywords: ASL
zbMATH Keywords: Stanley-Reisner ring
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90120-1
OpenAlex ID: W2020953861
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description: scientific article; zbMATH DE number 643794