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The Norm Residue Theorem in Motivic Cohomology最新文献

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Motives over S 动机大于S
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.23943/princeton/9780691191041.003.0006
C. Haesemeyer, C. Weibel
{"title":"Motives over S","authors":"C. Haesemeyer, C. Weibel","doi":"10.23943/princeton/9780691191041.003.0006","DOIUrl":"https://doi.org/10.23943/princeton/9780691191041.003.0006","url":null,"abstract":"This chapter shows that the operation φ‎\u0000 𝑉 of the definition introduced in the previous chapter extends to a cohomology operation over 𝑘, and that it satisfies the recognition criterion of a theorem, so that φ‎\u0000 𝑉 must be β‎𝑃𝑏. This construction of the cohomology operation utilizes the machinery of motives over a simplicial noetherian scheme. The chapter first presents this scheme in three parts, initially summarizing the basic theory of motives over a scheme 𝑆 before discussing motives over a simplicial scheme and over a smooth simplicial scheme. It then presents the slice filtration and generalizes from simplicial scheme 𝔛 to embedded schemes. Finally, this chapter defines the operations φ‎\u0000 𝑖 and φ‎\u0000 𝑉.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126918891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Overview of the Proof 证明概述
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.23943/princeton/9780691191041.003.0001
C. Haesemeyer, C. Weibel
{"title":"An Overview of the Proof","authors":"C. Haesemeyer, C. Weibel","doi":"10.23943/princeton/9780691191041.003.0001","DOIUrl":"https://doi.org/10.23943/princeton/9780691191041.003.0001","url":null,"abstract":"This chapter provides the main steps in the proof of Theorems A and B regarding the norm residue homomorphism. It also proves several equivalent (but more technical) assertions in order to prove the theorems in question. This chapter also supplements its approach by defining the Beilinson–Lichtenbaum condition. It thus begins with the first reductions, the first of which is a special case of the transfer argument. From there, the chapter presents the proof that the norm residue is an isomorphism. The definition of norm varieties and Rost varieties are also given some attention. The chapter also constructs a simplicial scheme and introduces some features of its cohomology. To conclude, the chapter discusses another fundamental tool—motivic cohomology operations—as well as some historical notes.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131340609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of Norm Varieties 范数变体的存在性
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.15
C. Haesemeyer, C. Weibel
{"title":"Existence of Norm Varieties","authors":"C. Haesemeyer, C. Weibel","doi":"10.2307/j.ctv941tx2.15","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.15","url":null,"abstract":"This chapter constructs norm varieties for symbols ª = {𝑎1, ...,𝑎𝑛} over a field 𝑘 of characteristic 0, and starts the proof that norm varieties are Rost varieties. It first recalls the definition of a norm variety for a symbol ª in 𝐾𝑀\u0000 𝑛(𝑘)/𝓁; if 𝑛 ≥ 2 and 𝑘 is 𝓁-special, norm varieties are geometrically irreducible. Next, the chapter uses the Chain Lemma to produce a specific ν‎\u0000 n−1-variety ℙ(𝒜), and a pencil Q of splitting varieties over 𝔸1—{0} whose fibers 𝑄𝑊 are fixed point equivalent to ℙ (𝒜). Using a bordism result, this chapter shows that any equivariant resolution 𝑄(ª) of 𝑄𝑊 is a ν‎\u0000 n−1-variety. Next, one of Rost's degree formulas is used to show that any norm variety for ª is ν‎\u0000 n−1 because 𝑄(ª) is. Finally, a norm variety for ª is constructed by induction on 𝑛, making use of the global inductive assumption that BL(n − 1) holds.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133727843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Acknowledgments 致谢
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.4
C. Haesemeyer, Charles A. Weibel
{"title":"Acknowledgments","authors":"C. Haesemeyer, Charles A. Weibel","doi":"10.2307/j.ctv941tx2.4","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.4","url":null,"abstract":"","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"114 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123651460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Model Structures for the 𝔸¹-homotopy Category 函数的模型结构
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.17
{"title":"Model Structures for the 𝔸¹-homotopy Category","authors":"","doi":"10.2307/j.ctv941tx2.17","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.17","url":null,"abstract":"","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127039968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rost Motives and H90 Rost的动机和H90
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.9
C. Haesemeyer, C. Weibel
{"title":"Rost Motives and H90","authors":"C. Haesemeyer, C. Weibel","doi":"10.2307/j.ctv941tx2.9","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.9","url":null,"abstract":"This chapter introduces the notion of a Rost motive, which is a summand of the motive of a Rost variety 𝑋. It highlights the theorem that, assuming that Rost motives exist and H90(n − 1) holds, then 𝐻𝑛+1\u0000 ét(𝑘, ℤ(𝑛)) injects into 𝐻𝑛+1\u0000 ét(𝑘(𝑋), ℤ(𝑛)). While there may be many Rost varieties associated to a given symbol, there is essentially only one Rost motive. The Rost motive captures the part of the cohomology of a Rost variety 𝑋. Since a Rost motive is a special kind of symmetric Chow motive, the chapter begins by recalling what this means. It then introduces the notion of 𝔛-duality. This duality plays an important role in the axioms defining Rost motives, as well as a role in the construction of the Rost motive in the next chapter. Finally, this chapter assumes that Rost motives exist and proves a key theorem.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129952508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Degree Formulas 度公式
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.13
C. Haesemeyer, C. Weibel
{"title":"Degree Formulas","authors":"C. Haesemeyer, C. Weibel","doi":"10.2307/j.ctv941tx2.13","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.13","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\u0000 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":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122886674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Index 指数
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.23
{"title":"Index","authors":"","doi":"10.2307/j.ctv941tx2.23","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.23","url":null,"abstract":"","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124589763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Overview of the Proof 证明概述
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.6
M. Hill, Mike Hopkins, D. Ravenel
{"title":"An Overview of the Proof","authors":"M. Hill, Mike Hopkins, D. Ravenel","doi":"10.2307/j.ctv941tx2.6","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.6","url":null,"abstract":"","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133898285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rost’s Chain Lemma 罗斯特链式引理
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.14
Christian Haesemeyer, Charles A. Weibel
{"title":"Rost’s Chain Lemma","authors":"Christian Haesemeyer, Charles A. Weibel","doi":"10.2307/j.ctv941tx2.14","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.14","url":null,"abstract":"This chapter states and proves Rost's Chain Lemma. The proof (due to Markus Rost) does not use the inductive assumption that BL(n − 1) holds. Throughout this chapter, 𝓁 is a fixed prime, and 𝑘 is a field containing 1/𝓁 and all 𝓁th roots of unity. It fixes an integer 𝑛 ≥ 2 and an 𝑛-tuple (𝑎1, ..., 𝑎𝑛) of units in 𝑘, such that the symbol ª = {𝑎1, ..., 𝑎𝑛} is nontrivial in the Milnor 𝐾-group 𝐾𝑀\u0000 𝑛(𝑘)/𝓁. The chapter produces the statement of the Chain Lemma by first proving the special case 𝑛 = 2. The notion of an 𝓁-form on a locally free sheaf over 𝑆 is then introduced, before the chapter shows how 𝓁-forms may be used to define elements of 𝐾𝑀\u0000 𝑛(𝑘(𝑆))/𝓁.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126994119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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