Some Background Material from Algebraic Geometry

In this chapter we give a review of some basic tools which are needed in later chapters for the construction of Néron models. Assuming that the reader is familiar with Grothendieck’s definition of schemes and morphisms, we treat the concept of smooth and étale morphisms, of henselian rings, and of S-rational maps; moreover, we have included some facts on differential calculus and on flatness. Concerning the smoothness, we give a self-contained exposition of this notion, relating it closely to the Jacobi criterion. For the other topics we simply state results, sometimes without giving proofs. Most of the material presented in this chapter is contained in Grothendieck’s treatments [EGA IV₄] and [SGA 1].