Local Picard Group of Pointed Monoids and Their Algebras

TL;DR

This paper provides an explicit formula for the cohomology of the sheaf of units on the punctured spectrum of Stanley-Reisner rings, linking algebraic and combinatorial properties to compute the local Picard group.

Contribution

It introduces a combinatorial approach to compute the local Picard group of Stanley-Reisner rings via explicit cohomology formulas.

Findings

Derived an explicit cohomology formula for sheaf units

Computed the local Picard group of Stanley-Reisner rings

Decomposed cohomology into combinatorial and field-dependent parts

Abstract

The main goal of this paper is to give an explicit formula for the cohomology of the sheaf of units of the punctured spectrum of a Stanley-Reisner ring. In particular we compute the local Picard group of $K [\triangle]$. To achieve this we study the corresponding purely combinatorial problem on the punctured spectrum of the pointed monoid defined by $\triangle$. The cohomology of the sheaf of units on ${\rm Spec}^ \bullet K[\triangle ]$ is then the direct sum of this combinatorial cohomology, which has a decomposition along the vertices, and another part depending on the field $K$.