Čech cohomology of semiring schemes

A semiring scheme generalizes a scheme in such a way that the underlying algebra is that of semirings. We generalize Čech cohomology theory and invertible sheaves to semiring schemes. In particular, when X=PMn, a projective space over a totally ordered idempotent semifield M, we show that Hˇm(X,OX) is in agreement with the classical computation for all m. Finally, we classify all invertible sheaves on X=PMn by computing Pic(X) explicitly.