Max Zeuner: Geometric and functorial schemes from the constructive predicative viewpoint

Abstract

Schemes are usually defined using locally ringed spaces but for algebraic geometers it can be more convenient to work with the functor of points associated to a scheme. In fact, Grothendieck favored the approach of defining schemes directly as certain well-behaved functors from rings to sets. These "functorial schemes" are easily shown equivalent to the "geometric schemes" defined through locally ringed spaces. I this talk we look at how much of this picture can be recovered when working constructively and predicatively.