On cohomology and vector bundles over monoid schemes

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 that the Pic functor respects finite products.Next we will introduce the notion of s-cancellative monoids and show that if X is locally of that type, OX⁎ can be embed injectively in a constant sheaf. We develop the theory of s-divisors, which generalise the Cartier divisors, and we prove that for s-cancellative monoid schemes, s-divisors describe the group Pic(X).We then generalise smooth monoid schemes with s-smooth monoid schemes, and prove that for them Hi(X,OX⁎)=0 for all i≥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 Xk, where k is a commutative ring. We prove that if k is an integral domain (resp. PID) and X is a cancellative and torsion-free (resp. seminormal and torsion-free) monoid scheme, then the induced map Pic(X)→Pic(Xk) is a monomorphism (resp. isomorphism).