Gurney, Lance Rory

### Description

We give a streamlined and self-contained construction of the category of schemes from the functor of points perspective. In the first part the necessary preliminary material is reviewed. We begin with some brief remarks on set theory in order to avoid size issues. Then the necessary facts from category theory are recalled, in particular the definition and standard properties of functor categories with values in the category of sets. Then the necessary facts from commutative algebra are...[Show more] recalled. We then introduce absolutely flat rings and some of their properties and applications are given. Finally, we recall the base properties of integral homomorphisms of rings and in particular norm and characteristic polynomial maps associated to a finite locally free homomorphism. In the second part the category of spaces is defined as a subcategory of the category functors from the category of rings to the category of sets. The basic properties of the category of spaces are then given and affine spaces and the affinisation functor are defined. We then introduce the two fundamental finiteness conditions: coherence and local finite presentation. These two concepts are what underlies two most important tools in the functor of points setting: descent and passage to the limit. We then consider subspaces in detail and describe several methods of construction for subspaces and of detecting their properties, making heavy use of absolutely flat spaces. We then extend some basic notions from commutative algebra (integral homomorphisms, norms and characteristic polynomials) to the setting of spaces. In the third and final part the category of schemes is defined as a subcategory of the category of spaces. The method of localisation is introduced as are the notions of flatness and quasi-finiteness. We then use the functorial machinery that has been set up to prove several non-trivial facts in the theory of schemes: the existence of certain quotients in the category of schemes, conditions for the image of a morphism of schemes to exist, and that the category of algebraic groups over a field is abelian. -- provided by Candidate.

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