Rings with lexicographic straightening law

Abstract

The concept of a “straightening law” as a means of analyzing the structure of particular commutative algebras has appeared independently in the work of a number of authors. The purpose of this paper is to introduce a systematic theory of algebras endowed with a structure of a lexicographic straightening law based on a partially ordered set (or “lexicographic ring” for short). Much of the work in the literature dealing with such rings fits quite naturally in the context of our theory, and we sketch some of this work. Moreover, some of our later results, in particular the Betti number bound (in Section 6), may actually be new results even for the special cases of the known examples of lexicographic rings. The concept of a lexicographic ring as an interesting area of study was suggested to this author by DeConcini, who conjectured that a lexicographic ring is Cohen-Macaulay if the underlying partially ordered set is so. We prove this result in two different ways. It has recently come to our attention that this conjecture has also been proved by DeConcini et al. [ 131, using deformation theory methods. The results of this paper are arranged in two parts. The first part, consisting of Sections 2 through 4, is more elementary and uses the technique of “combinatorial decompositions” as developed by Garsia and this author. The second part, Sections 6 and 7, requires some knowledge of homological algebra methods in ring theory. The intermediate Section 5 discusses some of the known examples of lexicographic rings. In this section we also sketch some of the ways that our theory may be extended to more general contexts. The main result of the first part is the Transfer Theorem 4.3, which enables one to transfer combinatorial decompositions from the underlying partially ordered set (or more precisely from the corresponding Stanley-Reisner ring) to the lexicographic ring. As a result those concepts that are expressible in terms of combinatorial decompositions are also