On cohomology and vector bundles over monoid schemes

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).