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Uniform Bounds

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1999)

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.

Keywords

  • Prime Ideal
  • Local Ring
  • Maximal Ideal
  • Polynomial Ring
  • Universal Property

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Hans Schoutens .

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Schoutens, H. (2010). Uniform Bounds. In: The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics(), vol 1999. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13368-8_4

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