  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.
