Subdivisions of Toric Complexes

Abstract

Abstract. We introduce toric complexes as polyhedral complexes consisting of rational cones together with a set of integral generators for each cone, and we define their associated face rings. Abstract simplicial complexes and rational fans can be considered as toric complexes, and the face ring for toric complexes extends Stanley and Reisner's face ring for abstract simplicial complexes [20] and Stanley's face ring for rational fans [21]. Given a toric complex with defining ideal I for the face ring we give a geometrical interpretation of the initial ideals of I with respect to weight orders in terms of subdivisions of the toric complex generalizing a theorem of Sturmfels in [23]. We apply our results to study edgewise subdivisions of abstract simplicial complexes.

On algebras associated to partially ordered sets

“…In Theorem 5.6 we give a criterion, namely that T is sectioned and has a presentation , ensuring that the ring lim P T of global sections of an RP-algebra T is of the form K[F ]/I for a finite set F , where the ideal I is a sum I = I T + I of the Stanley-Reisner ideal of a simplicial complex related to the order complex of P and the sum I T of defining ideals for the stalks T x of T . A presentation of T roughly consists of a subset F x of F and a surjective homomorphism The paper is organized as follows: In Section 2 we present classes of KPalgebras associated to toric face rings (as introduced in [3]), monomial ideals and algebras with straightening laws (ASL's for short). In Section 3 we give a criterion on a KP-algebra with stalks given by ASL's to have an ASL as ring of global sections.…”

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].…”

Cohomology of partially ordered sets and local cohomology of section rings

“…For example, Bruns and Gubeladze studied such algebras in a series of papers [5,6]. Brun and Römer considered the relationship between initial ideals of the defining ideal of the face ring of a rational fan and subdivisions of that fan in [3]. This paper is organized as follows.…”