Kato homology of arithmetic schemes and higher class field theory over local fields

Abstract

For arithmetical schemes $X$, K. Kato [J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)] introduced certain complexes $C^{r,s}(X)$ of Gersten-Bloch-Ogus type whose components involve Galois cohomology groups of all the residue fields of $X$. For specific $(r,s)$, he stated some conjectures on their homology generalizing the fundamental isomorphisms and exact sequences for Brauer groups of local and global fields. We prove some of these conjectures in small degrees and give applications to the class field theory of smooth projective varieties over local fields, and finiteness questions for some motivic cohomology groups over local and global fields.