Local Picard group of pointed monoids and their algebras

Abstract

The main goal of this article is to give an explicit formula for H-Zar(j) (Spec center dot K[Delta], O-K[Lambda](+) in terms of simplicial and reduced simplicial cohomology, where K[Delta] is the Stanley- Reisner ring of the simplicial complex Delta. In particular, we compute the local Picard group of K[Delta]. To achieve this, we study the corresponding purely combinatorial problem on the punctured spectrum of the pointed monoid defined by Delta. The cohomology of the sheaf of units on Spec.K[Delta] is then the direct sum of this combinatorial cohomology, which has a decomposition along the vertices, and another part depending on the field K.