Abstract

The aim of the script is to act as a course on Scheme theory from the internal perspective
of the topos Sh(X), therefore showing that the internal logic of sheaf topoi is a strong
enough foundation to build the whole theory on without necessarily referring back to the
usual methods.
In this thesis we define what it means to work from the internal perspective:
We define elementary topoi and how to build and interpret formulas in the internal
logic.
After, we move to the specific case that is the category of sheaves on either a topological
space or a locale, and explicit the semantics of that language.
Weshow that the logic is intuitionistically solid and prove some results about geometric
formulas that apply to later constructions.
When that is done, we procede to rebuild some theory of schemes from this perspective:
First we define abelian groups, rings, local rings and modules over sheaves, and some
special cases.
Then we build the basics of scheme theory by defining affine schemes, general schemes,
coherent modules, and some special classes of morphism of schemes,
In the end we attempt to talk about relative schemes from this perspective and what
is needed to build the theory, then procede to show that it is well suited for a synthetic
approach to schemes through some exercises from Hartshorne’s Algebraic Geometry
chapter II