Degree as a monoid

Abstract

“Polynomials” with degree as an ordered monoid say $\mathrm{\Delta }$ are constructed along with corresponding schemes and line bundles $\mathcal{𝒪}\left(d\right)$, $d\in \mathrm{\Delta }$. The cohomology of these line bundles is then computed using Čech complex and new proof of zero cohomology of affine schemes is given. The last section of the paper applies the theory developed for the construction and computation of affine cohomology of perfectoid Tate algebras.