On Cohomology and vector bundles over monoid schmes

Abstract

The aim of this paper is to study the cohomology theory of monoid schemes in general and apply it to vector and line bundles. We will prove that over separated monoid schemes, any vector bundle is a coproduct of line bundles and then go on to study the line bundles in more detail. Amongst other things, we prove that over separated monoid schemes, ${\sf Pic}$ respects finite products. Next we will introduce the notion of $s$-cancellative monoids. They are monoids for which $ax=ay$ implies that $(xy)^nx=(xy)^ny, n\in\mathbb{N}$. This class is important since it is the biggest class of monoids for which $M^*_{\mathfrak{p}}$ maps injectively into its group of fractions for every prime ideal ${\mathfrak{p}}$. As we will see in section 6, this will enable us to embed $\mathcal{O}^*_X$ injectively in a constant sheaf provided $X$ is locally $s$-cancellative. We develop the theory of $s$-divisors and we prove that for an $s$-cancellative monoid scheme $X$, the group ${\sf Pic}(X)$ can be described in terms of $s$-divisors. For cancellative monoid schemes, $s$-divisors agree with the Cartier divisors. We then introduce the notion of $s$-smooth monoid schemes, which generalise smooth monoids schemes, and prove that for them $H^i(X,\mathcal{O}^*_X)=0$ for all $i\geq 2$. Furthermore we show that it is a local property and respects finite products. Finally we investigate the relationship between line bundles over a monoid scheme $X$ and over its geometric realisation $X_k$, where $k$ is a commutative ring. We prove that if $k$ is an integral domain (resp. principal ideal domain) and $X$ is a cancellative and torsion free (resp. seminormal and torsion-free) monoid scheme, then the induced map ${\sf Pic}(X)\to {\sf Pic}(X_k)$ is a monomorphism (resp. isomorphism).