The Motivic Group H−1,−1BM

Abstract

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.