Perfect(oid) Algebraic Geometry
Résumé
Elementary Algebraic Geometry can be described as study of zeros of polynomials with integer degrees, this idea can be naturally carried over to 'polynomials' with degree Z[1/p], which leads to the perfectoid version of algebraic geometry. In this paper the similarities and differences between the two distinct cases are discussed. The first part of the paper constructs corresponding tangent space, projective space and proves the Nullstellensatz. The second part constructs Perfectoid Algebras algebraically and proves Weierstraß theorems and shows coherence under specific conditions. The line bundles of degree n ∈ Z[1/p] are constructed and theirČech cohomology computed. The last part of the paper constructs a modulo p analogue of Scholze's tilting functor.
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