The use of ultraproducts in commutative algebra
Although the first important application of ultraproducts to algebra and number theory has been made already in the sixties by \textit{J. Ax} and \textit{S. Kochen} [Am. J. Math. 87, 605--630, 631--648 (1965; Zbl 0136.32805)] who used them to obtain essential progress in a problem of Artin, it took some time before this method has been applied to other algebraic problems. The aim of this book is to show several important applications of ultraproducts and their modifications (cataproducts and protoproducts) in the theory of commutative rings without using the language of model theory. The first few chapters introduce the principal notions and results, including a version of \textit{J. Łoś}'s theorem [in: Math. interpretation of formal systems, 98--113 (1955; Zbl 0068.24401)] and a kind of Lefschetz principle. In Chapter 3 flat modules and homomorphisms are considered, and a series of flatness criteria are given. The first applications of ultraproducts appear in the next chapter, where the existence of uniform bounds over polynomial rings is considered, following the approach of \textit{K. Schmidt} and \textit{L. van den Dries} [Invent. Math. 76, 77--91 (1984; Zbl 0539.13011)]. In Chapter 5 one finds the theory of tight closure in characteristic \(p>0\), exposed in an axiomatic way, and this is applied in the next two chapters to construct this theory in characteristic zero. In chapters 8 and 9 cataproducts and protoproducts are defined, studied and applied to uniform bounds (cataproducts are certain factors of ultraproducts of local Noetherian rings and have the nice property that if these rings have bounded embedding dimension, then their cataproduct is Noetherian, and protoproducts are certain subrings of ultraproducts). The last chapter deals with homological conjectures in the mixed characteristic case and brings solutions for some of them when the residual characteristic is sufficiently large. In his Introduction the author writes: ``[The ultraproducts] were conceived as non-algebraic, belonging to the alien universe of set-theory and non-standard arithmetic, a universe in which most mathematicians did not, and still do not feel too comfortable. The present book intends to debunk this common perception of ultraproducts \dots''. In the opinion of the reviewer he succeeded in his task admirably. Reviewer's remark: The MSC classification given in the book has nothing to do with its contents, and seems to be mistakenly copied from another book.
scientific article; zbMATH DE number 5768468