On Canonical Modules of Toric Face Rings

Abstract

Abstract. Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one can associate to a rational pointed fan Σ inis Cohen-Macaulay, the main result of this paper is to characterize the situation when its canonical module is isomorphic to a Z d -graded ideal of K [Σ]. From this result several algebraic and combinatorial consequences are deduced. As an application, we give a relation between the cleanness of K[Σ] and the shellability of Σ. §1. Introduction Let Σ be a rational pointed fan in R d , i.e. Σ is a finite collection of rational pointed cones in R d such that for C ′ ⊆ C with C ∈ Σ we have that C ′ is a face of C if and only if C ′ ∈ Σ, and if C, C ′ ∈ Σ, then C ∩ C ′ is a common face of C and C ′ . Stanley constructed in [20] the toric face ring K[Σ] over a field K associated to Σ as follows.As a K-vector spacex a · x b = x a+b if a and b are elements of a common face C ∈ Σ, 0 otherwise.