03467nam 22006255 450 99646651250331620200702111416.01-280-39178-297866135697073-642-13368-110.1007/978-3-642-13368-8(CKB)2550000000015812(SSID)ssj0000450490(PQKBManifestationID)11298204(PQKBTitleCode)TC0000450490(PQKBWorkID)10434894(PQKB)11501715(DE-He213)978-3-642-13368-8(MiAaPQ)EBC3065515(PPN)149078560(EXLCZ)99255000000001581220100716d2010 u| 0engurnn|008mamaatxtccrThe Use of Ultraproducts in Commutative Algebra[electronic resource] /by Hans Schoutens1st ed. 2010.Berlin, Heidelberg :Springer Berlin Heidelberg :Imprint: Springer,2010.1 online resource (X, 210 p.) Lecture Notes in Mathematics,0075-8434 ;1999Bibliographic Level Mode of Issuance: Monograph3-642-13367-3 Includes bibliographical references (p. 193-197) and index.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.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.Lecture Notes in Mathematics,0075-8434 ;1999Commutative algebraCommutative ringsAlgebraic geometryCommutative Rings and Algebrashttps://scigraph.springernature.com/ontologies/product-market-codes/M11043Algebraic Geometryhttps://scigraph.springernature.com/ontologies/product-market-codes/M11019Commutative algebra.Commutative rings.Algebraic geometry.Commutative Rings and Algebras.Algebraic Geometry.51260G5160E0760J8045K0565N3028A7860H0560G5760J7526A33mscSchoutens Hansauthttp://id.loc.gov/vocabulary/relators/aut478944BOOK996466512503316Use of ultraproducts in commutative algebra261768UNISA