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

摘要

本章进一步讨论了Borel-Moore同调群𝐻𝑀−1，−1(𝑋)的性质。结果表明，对于有限平面映射，它在𝑋中是逆变的，对于固有映射，它有一个泛函推进。如果𝑋平滑且合适(特征为0)，那么𝐻𝑀−1，−1(𝑋)与𝐻2 +1，+1(𝑋，0)一致，并且具有很好的表现形式，本章将对此进行更深入的探讨。本章主要结果的命题:如果𝑋是规范各种ª和𝑘𝓁-special的形象𝐻𝐵𝑀−1−1(𝑋)→𝑘×的集团单位𝑏这样ª∪𝑏消失在𝐾𝑀𝑛+ 1(𝑘)/𝓁。同样，本章还探讨了其方程的历史轨迹。

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The Motivic Group H−1,−1BM

This chapter develops some more of the properties of the Borel–Moore homology groups 𝐻𝐵𝑀 −1,−1(𝑋). It shows that it is contravariant in 𝑋 for finite flat maps, and has a functorial pushforward for proper maps. If 𝑋 is smooth and proper (in characteristic 0), 𝐻𝐵𝑀 −1,−1(𝑋) agrees with 𝐻2𝒅+1,𝒅+1(𝑋, ℤ), and has a nice presentation, which this chapter explores in more depth. The main result in this chapter is the proposition that: if 𝑋 is a norm variety for ª and 𝑘 is 𝓁-special then the image of 𝐻𝐵𝑀 −1,−1(𝑋) → 𝑘× is the group of units 𝑏 such that ª ∪ 𝑏 vanishes in 𝐾𝑀 𝑛+1(𝑘)/𝓁. Again, this chapter also explores the historic trajectory of its equations.

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The Norm Residue Theorem in Motivic Cohomology

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