Combinatorial topology and the global dimension of algebras arising in combinatorics

Abstract

We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra.Résumé. Nous présentons une magnifique interaction entre la topologie combinatoire et l'algèbre homologique d'une classe de monoïdes qui figurent naturellement dans la combinatoire algébrique. Nous explorons plusieurs applications de cette interaction. Par exemple, nous introduisons une nouvelle interprétation du nombre de Leray d'un complexe de clique en termes de la dimension globale d'une certaine algèbre non commutative.

Moment categories and operads

“…The most prominent example of a left regular band is the face monoid L A of a hyperplane arrangement A in Euclidean space, cf. [28]. In this case, the quotientlattice L A / ≃ coincides with the so-called intersection lattice I A of the hyperplane arrangement.…”

“…From a purely abstract point of view, moment categories can be considered as categorification of left regular bands, well known in semigroup literature, cf. [28]. It is remarkable that the construction of the universal commutative quotient of a left regular band extends to moment categories.…”

“…Such monoids are known in the semigroup literature as left regular bands, cf. [28]. More precisely, a band is a semigroup consisting of idempotent elements, and a band is said to be left regular if the Schützenberger relation holds.…”

On left regular bands and real conic–line arrangements

“…Preliminaries: The left regular band and the face semigroup of a hyperplane arrangement. In this section, we recall the notion of a left regular band (LRB) and its connections to the combinatorics of hyperplane arrangements (see also surveys in [11,12,24]). Define the partially ordered set of faces as:…”

Order complex of ideals in a commutative ring with identity

“…Since posets occur in many areas, so do order complexes. For some recent and interesting occurrences of order complexes in algebraic contexts, the reader may consult [1], [3], [5], [6], [8]- [10].…”