The Use of Ultraproducts in Commutative AlgebraSpringer, 16 ביולי 2010 - 210 עמודים 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. |
תוכן
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 |
199 | |
מהדורות אחרות - הצג הכל
The Use of Ultraproducts in Commutative Algebra, מהדורה 1999 <span dir=ltr>Hans Schoutens</span> תצוגה מקדימה מוגבלת - 2010 |
מונחים וביטויים נפוצים
A-module affine proto-grade Artin Approximation assume balanced big Cohen-Macaulay big Cohen-Macaulay algebra Boolean called catapower cataproduct characteristic zero coefficients Commutative Algebra complete Conjecture contradiction Corollary criterion cyclically pure define definition denote discrete valuation ring domain elements embedding dimension equal characteristic equations exact sequence F-regular faithfully flat finitely generated ideal follows Frobenius geometric dimension Hence Hensel system homology homomorphism indeterminates integral closure intersection isomorphic K-affine algebra Lefschetz hull Lemma linear Ło´s Łoś Theorem Math maximal ideal minimal prime mixed characteristic module Monomial Moreover morphism n-Boolean ring Noetherian local ring Noetherian ring non-zero particular positive characteristic power series prime ideal prime number Proof Proposition proto-grade proto-graded ring protopower protoproduct prove R-algebra regular local ring residue field respectively result scalar extension solution subring subset system of parameters Theorem tight closure tuple ultra-hull ultra-ring ultrafilter ultrapower ultraproduct uniform bound whence