Local Picard Group of Binoids and Their Algebras

Abstract

We begin by introducing schemes of binoids, invertible $\mathcal{O}_M$-sets and cohomology of sheaves of abelian groups defined on schemes of binoids. We define the so-called punctured combinatorial \v{C}ech-Picard complex, whose first cohomology computes $\mathrm{Pic}^{\mathrm{loc}}(M)$, the local Picard group of a binoid. We look then at simplicial binoids, whose spectrum presents very nice combinatorial properties. We prove that the cohomology of a constant sheaf on this punctutured spectra can be computed entirely in terms of simplicial cohomology. Through some other results, we prove that we can compute cohomology of the sheaf of units (and thus the local Picard group) by meaning of reduced simplicial cohomology. We move from combinatorics to algebra and we try to use these tools to understand binoid algebras and cohomology of their sheaf of units. We introduce a new topology, that we call combinatorial topology on $\mathrm{Spec}\mathbb{K}[M]$. In this topology, if $M$ is a reduced, torsion-free and cancellative binoid, we can decompose the sheaf of units of its algebra in a direct sum of a constant part $\mathbb{K}^*$ and of the pushforward of the combinatorial units. We specialize to the case of the Stanley-Reisner rings, for which we are able to prove a vanishing result for the affine case. By meaning of the combinatorial topology we then prove that we can compute the Zariski cohomology of the sheaf of units entirely in terms of simplicial cohomology, both usual and reduced. This ultimately leads to a generalization to any monomial ideal, thus yielding us some purely combinatorial non-vanishing results for $\mathrm{Pic}^{\mathrm{loc}}(\mathbb{K}[M])$ in the last Corollary.