{"title":"度公式","authors":"C. Haesemeyer, C. Weibel","doi":"10.2307/j.ctv941tx2.13","DOIUrl":null,"url":null,"abstract":"This chapter uses algebraic cobordism to establish some degree formulas. It presents δ as a function from a class of smooth projective varieties over a field 𝑘 to some abelian group. Here, a degree formula for δ is a formula relating δ(𝑋), δ(𝑌), and deg(𝑓) for any generically finite map 𝑓 : 𝑌 → 𝑋 in this class. The formula is usually δ(𝑌)=deg(𝑓)δ(𝑋). These degree formulas are used to prove that any norm variety over 𝑘 is a ν\n n−1-variety. Using a standard result for the complex bordism ring 𝑀𝑈*, which uses a gluing argument of equivariant bordism theory, this chapter establishes Rost's DN (Degree and Norm Principle) Theorem for degrees, and defines the invariant η(𝑋/𝑆) of a pseudo-Galois cover.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"36 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Degree Formulas\",\"authors\":\"C. Haesemeyer, C. Weibel\",\"doi\":\"10.2307/j.ctv941tx2.13\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This chapter uses algebraic cobordism to establish some degree formulas. It presents δ as a function from a class of smooth projective varieties over a field 𝑘 to some abelian group. Here, a degree formula for δ is a formula relating δ(𝑋), δ(𝑌), and deg(𝑓) for any generically finite map 𝑓 : 𝑌 → 𝑋 in this class. The formula is usually δ(𝑌)=deg(𝑓)δ(𝑋). These degree formulas are used to prove that any norm variety over 𝑘 is a ν\\n n−1-variety. Using a standard result for the complex bordism ring 𝑀𝑈*, which uses a gluing argument of equivariant bordism theory, this chapter establishes Rost's DN (Degree and Norm Principle) Theorem for degrees, and defines the invariant η(𝑋/𝑆) of a pseudo-Galois cover.\",\"PeriodicalId\":145287,\"journal\":{\"name\":\"The Norm Residue Theorem in Motivic Cohomology\",\"volume\":\"36 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-06-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Norm Residue Theorem in Motivic Cohomology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2307/j.ctv941tx2.13\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Norm Residue Theorem in Motivic Cohomology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2307/j.ctv941tx2.13","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
This chapter uses algebraic cobordism to establish some degree formulas. It presents δ as a function from a class of smooth projective varieties over a field 𝑘 to some abelian group. Here, a degree formula for δ is a formula relating δ(𝑋), δ(𝑌), and deg(𝑓) for any generically finite map 𝑓 : 𝑌 → 𝑋 in this class. The formula is usually δ(𝑌)=deg(𝑓)δ(𝑋). These degree formulas are used to prove that any norm variety over 𝑘 is a ν
n−1-variety. Using a standard result for the complex bordism ring 𝑀𝑈*, which uses a gluing argument of equivariant bordism theory, this chapter establishes Rost's DN (Degree and Norm Principle) Theorem for degrees, and defines the invariant η(𝑋/𝑆) of a pseudo-Galois cover.
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