The Use of Ultraproducts in Commutative Algebra

כריכה קדמית
Springer, 16 ביולי 2010 - 210 עמודים
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In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.
 

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תוכן

1 Introduction
1
2 Ultraproducts and Łos Theorem
7
3 Flatness
29
4 Uniform Bounds
51
5 Tight Closure in Positive Characteristic
64
6 Tight Closure in Characteristic Zero Affine Case
81
7 Tight Closure in Characteristic Zero Local Case
97
8 Cataproducts
113
9 Protoproducts
126
10 Asymptotic Homological Conjectures
149
A Henselizations
171
B Boolean Rings
179
References
193
Index
199
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