TY - BOOK ID - 8283891 TI - The use of ultraproducts in commutative algebra PY - 2010 SN - 3642133673 9786613569707 1280391782 3642133681 PB - New York : Springer, DB - UniCat KW - Commutative algebra KW - Ultraproducts KW - Mathematics KW - Physical Sciences & Mathematics KW - Algebra KW - Mathematical Theory KW - Commutative algebra. KW - Ultraproducts. KW - Prime products KW - Products, Prime KW - Products, Ultra KW - -Ultra-products KW - Mathematics. KW - Algebraic geometry. KW - Commutative rings. KW - Commutative Rings and Algebras. KW - Algebraic Geometry. KW - Rings (Algebra) KW - Algebraic geometry KW - Geometry KW - Math KW - Science KW - Model theory KW - Algebra. KW - Geometry, algebraic. KW - Mathematical analysis UR - https://www.unicat.be/uniCat?func=search&query=sysid:8283891 AB - 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. ER -