The Use of Ultraproducts in Commutative Algebra
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 ultraprodu...
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| Author / Creator: | |
|---|---|
| Other Corporate Authors / Creators: | SpringerLink (Online service) |
| Format: | Electronic eBook |
| Language: | English |
| Edition: | 1st ed. 2010. |
| Imprint: | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2010. |
| Series: | Lecture Notes in Mathematics,
1999 |
| Subjects: | |
| Online Access: | Available in Springer Mathematics and Statistics eBooks 2010 English/International. |
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| 008 | 100716s2010 gw | o |||| 0|eng d | ||
| 020 | |z 9783642133671 | ||
| 020 | |a 9781280391781 (online) | ||
| 020 | |a 9783642133688 (online) | ||
| 035 | |a (EBZ)ebs669019e | ||
| 040 | |d EBZ | ||
| 042 | |a msc | ||
| 050 | 4 | |a QA251.3 | |
| 100 | 1 | |a Schoutens, Hans. |e author. |4 aut |4 http://id.loc.gov/vocabulary/relators/aut | |
| 245 | 1 | 4 | |a The Use of Ultraproducts in Commutative Algebra |h [electronic resource] |c by Hans Schoutens. |
| 250 | |a 1st ed. 2010. | ||
| 264 | 1 | |a Berlin, Heidelberg : |b Springer Berlin Heidelberg : |b Imprint: Springer, |c 2010. | |
| 490 | 1 | |a Lecture Notes in Mathematics, |x 1617-9692 ; |v 1999 | |
| 505 | 0 | |a Ultraproducts and ?o?’ Theorem -- Flatness -- Uniform Bounds -- Tight Closure in Positive Characteristic -- Tight Closure in Characteristic Zero. Affine Case -- Tight Closure in Characteristic Zero. Local Case -- Cataproducts -- Protoproducts -- Asymptotic Homological Conjectures in Mixed Characteristic. | |
| 520 | |a 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. | ||
| 650 | 0 | |a Commutative algebra. | |
| 650 | 0 | |a Commutative rings. | |
| 650 | 0 | |a Algebraic geometry. | |
| 650 | 1 | 4 | |a Commutative Rings and Algebras. |
| 650 | 2 | 4 | |a Algebraic Geometry. |
| 710 | 2 | |a SpringerLink (Online service) | |
| 773 | 0 | |t Springer Mathematics and Statistics eBooks 2010 English/International |d Springer Nature | |
| 776 | 0 | 8 | |i Printed edition: |z 9783642133671 |
| 776 | 0 | 8 | |i Printed edition: |z 9783642133695 |
| 776 | 1 | |t The Use of Ultraproducts in Commutative Algebra | |
| 830 | 0 | |a Lecture Notes in Mathematics, |x 1617-9692 ; |v 1999 | |
| 856 | 4 | 0 | |3 Full text available |z Available in Springer Mathematics and Statistics eBooks 2010 English/International. |u https://ezproxy.wellesley.edu/login?url=https://link.springer.com/10.1007/978-3-642-13368-8 |