Unramified Class Field Theory of Arithmetical Surfaces

A finiteness theorem for zero-cycles overp-adic fields

“…‫ޚ‬ induces an isomorphism CH 0 .X s /=n ' ‫=ޚ‬n. When F is finite, A 0 .X s / is a finite group computed by higher class field theory [KS83]. When moreover X s is separably rationally connected, that computation shows A 0 .X s / D 0.…”

“…On the other hand, the situation in the case that k is a field of arithmetic nature presents a striking contrast to the case k D ‫.ރ‬ If k is finite, Ker. V / has been explicitly determined by geometric class field theory [KS83,Prop. 9].…”

Arithmetic homology and an integral version of Kato's conjecture

“…Finiteness of CH 0 (X) is a result of class field theory [20]. b) ⇒ c) is clear, and c) ⇒ a) follows from…”

“…A closed embedding Z → X with open complement U gives rise to a short exact sequence of complexes (20), hence a long exact sequence…”

Algebraic K-Theory and Twisted Reciprocity Laws

“…The argument is then the same as Suslin's. For * = 0, the theorem still holds, simply because Ker(deg) is torsion (Kato-Saito, [15]) and K 0 (k) = Z. 2…”