DOCUMENTA MATHEMATICA, Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 479-538

Uwe Jannsen and Shuji Saito

Kato Homology of Arithmetic Schemes and Higher Class Field Theory over Local Fields

For arithmetical schemes $X$, K. Kato 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 projecive varieties over local fields, and finiteness questions for some motivic cohomology groups over local and global fields.

2000 Mathematics Subject Classification: 11G25, 11G45, 14F42

Keywords and Phrases: Kato homology, Bloch-Ogus theory, niveau spectral sequence, arithmetic homology, higher class field theory

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