Uniform bounds

Abstract

In this chapter, we will discuss our first application of ultraproducts: the existence of uniformbounds over polynomial rings. Themethod goes back to A. Robinson, but really gained momentum by the work of Schmidt and van den Dries in [86], where they brought in flatness as an essential tool. Most of our applications will be concerned with affine algebras over an ultra-field. For such an algebra, we construct its ultra-hull as a certain faithfully flat ultra-ring. As we will also use this construction in our alternative definition of tight closure in characteristic zero in Chapter 6, we study it in detail in §4.3. In particular, we study transfer between the affine algebra and its approximations. We conclude in §4.4 with some applications to uniform bounds, in the spirit of Schmidt and van den Dries.