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The Norm Residue Theorem in Motivic Cohomology最新文献

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Hilbert 90 for $K_{n}^{M}$ $K_{n}^{M}$的希尔伯特90
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.8
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引用次数: 0
Existence of Rost Motives 主要动机的存在
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.10
C. Haesemeyer, C. Weibel
{"title":"Existence of Rost Motives","authors":"C. Haesemeyer, C. Weibel","doi":"10.2307/j.ctv941tx2.10","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.10","url":null,"abstract":"This chapter fixes a Rost variety 𝑋 for a sequence. It constructs a Rost motive 𝑀 = (𝑋, 𝑒) with coefficients ℤ(𝓁) under the inductive assumption that BL(n − 1) holds and discusses three important axioms. It introduces a candidate for the Rost motive and demonstrates how a motive satisfies two axioms. To further aid in the proof, the chapter argues that End(𝑀) is a local ring and then verifies an axiom proving that 𝑀 is a Rost motive whenever 𝑋 is a Rost variety. Finally, the chapter considers the historical background behind these equations. It reveals the eponymous Rost motive and considers Voevodsky's own construction of the Rost motive.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"4 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":"134641940","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
The Motivic Group H−1,−1BM 动力组H−1，−1BM
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.23943/princeton/9780691191041.003.0007
C. Haesemeyer, C. Weibel
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引用次数: 0
The Motivic Group $H_{-1,-1}^{BM}(X)$ 动机组$H_{-1，-1}^{BM}(X)$
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.12
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引用次数: 0
Motivic Classifying Spaces 动机分类空间
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.20
C. Haesemeyer, C. Weibel
{"title":"Motivic Classifying Spaces","authors":"C. Haesemeyer, C. Weibel","doi":"10.2307/j.ctv941tx2.20","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.20","url":null,"abstract":"This chapter focuses on motivic classifying spaces. It first connects the motives 𝑆∞\u0000 tr(𝕃𝑛) to cohomology operations on 𝐻2𝑛, 𝑛, at least when char(𝑘)=0. This parallels the Dold–Thom theorem in topology, which identifies the reduced homology ̃𝐻*(𝑋, ℤ) of a connected space 𝑋 with the homotopy groups of the infinite symmetric product 𝑆∞𝑋. A similar analysis shows that 𝔾𝑚 represents 𝐻1,1(−, ℤ), which allows us to describe operations on 𝐻1,1. The chapter then introduces the notion of scalar weight operations on 𝐻2𝑛, 𝑛. Afterward, it develops formulas for 𝑆𝓁tr(𝕃𝑛). These formulas imply that 𝑆∞\u0000 tr(𝕃𝑛) is a proper Tate motive, so there is a Künneth formula for them. The chapter culminates in a theorem demonstrating that β‎𝑃𝑏 is the unique cohomology operation of scalar weight 0 in its bidegree.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"16 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":"132257896","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
Hilbert 90 for KnM 希尔伯特90是KnM
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.23943/princeton/9780691191041.003.0003
C. Haesemeyer, C. Weibel
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引用次数: 0
Relation to Beilinson-Lichtenbaum
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.23943/princeton/9780691191041.003.0002
C. Haesemeyer, Charles A. Weibel
{"title":"Relation to Beilinson-Lichtenbaum","authors":"C. Haesemeyer, Charles A. Weibel","doi":"10.23943/princeton/9780691191041.003.0002","DOIUrl":"https://doi.org/10.23943/princeton/9780691191041.003.0002","url":null,"abstract":"This chapter show how the Beilinson–Lichtenbaum condition is equivalent to the assertion that the norm residue is an isomorphism. A diagram featuring this proof is also included. The chapter first proves the reductions introduced in the previous chapter, BL(n) and H90(n), arguing that if BL(n) holds then BL(n − 1) holds and that H90(n) implies H90(n − 1). Next, the chapter turns to the cohomology of singular varieties, cohomology with supports, and rationally contractible presheaves. From there, it argues that Bloch–Kato implies Beilinson–Lichtenbaum and concludes the theorems by proving that H90(n) implies BL(n). The chapter concludes with some historical notes.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"334 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":"116529066","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 𝔸1-homotopy类别的模型结构
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.23943/princeton/9780691191041.003.0012
C. Haesemeyer, C. Weibel
{"title":"Model Structures for the 𝔸1-homotopy Category","authors":"C. Haesemeyer, C. Weibel","doi":"10.23943/princeton/9780691191041.003.0012","DOIUrl":"https://doi.org/10.23943/princeton/9780691191041.003.0012","url":null,"abstract":"This chapter provides the 𝔸1-local projective model structure on the categories of simplicial presheaves and simplicial presheaves with transfers. These model categories, written as Δ‎opPshv(Sm)𝔸1 and Δ‎op\u0000 PST(Sm)𝔸1, are first defined. Their respective homotopy categories are Ho(Sm) and the full subcategory DM\u0000 eff\u0000 nis\u0000 ≤0 of DM\u0000 eff\u0000 nis. Afterward, this chapter introduces the notions of radditive presheaves and ̅Δ‎-closed classes, and develops their basic properties. The theory of ̅Δ‎-closed classes is needed because the extension of symmetric power functors to simplicial radditive presheaves is not a left adjoint. This chapter uses many of the basic ideas of Quillen model categories, which is a category equipped with three classes of morphisms satisfying five axioms. In addition, much of the material in this chapter is based upon the technique of Bousfield localization.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"41 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":"115490298","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
Relation to Beilinson–Lichtenbaum 与贝林森-利希滕鲍姆的关系
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.7
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引用次数: 0
Symmetric Powers of Motives 动机的对称力量
The Norm Residue Theorem in Motivic Cohomology Pub Date : 2019-06-11 DOI: 10.2307/j.ctv941tx2.19
C. Haesemeyer, C. Weibel
{"title":"Symmetric Powers of Motives","authors":"C. Haesemeyer, C. Weibel","doi":"10.2307/j.ctv941tx2.19","DOIUrl":"https://doi.org/10.2307/j.ctv941tx2.19","url":null,"abstract":"This chapter develops the basic theory of symmetric powers of smooth varieties. The constructions in this chapter are based on an analogy with the corresponding symmetric power constructions in topology. If 𝐾 is a set (or even a topological space) then the symmetric power 𝑆𝑚𝐾 is defined to be the orbit space 𝐾𝑚/Σ‎𝑚, where Σ‎𝑚 is the symmetric group. If 𝐾 is pointed, there is an inclusion 𝑆𝑚𝐾 ⊂ 𝑆𝑚+1𝐾 and 𝑆∞𝐾 = ∪𝑆𝑚𝐾 is the free abelian monoid on 𝐾 − {*}. When 𝐾 is a connected topological space, the Dold–Thom theorem says that ̃𝐻*(𝐾, ℤ) agrees with the homotopy groups π‎\u0000 *(𝑆∞𝐾). In particular, the spaces 𝑆∞(𝑆 𝑛) have only one homotopy group (𝑛 ≥ 1) and hence are the Eilenberg–Mac Lane spaces 𝐾(ℤ, 𝑛) which classify integral homology.","PeriodicalId":145287,"journal":{"name":"The Norm Residue Theorem in Motivic Cohomology","volume":"11 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":"124209352","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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