Line Bundles of Rational Degree Over Perfectoid Space

Abstract

In this thesis we lay the foundation for rational degree d as an element of Z[1/p] by using perfectoid analogue of projective space, and consider power series instead of polynomials. We start the groundwork by proving Weierstrass theorems for perfectoid spaces which are analogues of standard Weierstrass theorems in complex analysis. We then move onto defining sheaves for Projective perfectoid analogue and prove perfectoid analogues of Gorthendieck's classication theorem on projective line, Serre's theorem on Cohomology of line bundles. As intermediate results we also compute Picard groups and describe Cartier and Weil divisors for Perfectoid.