A generalization of class formation by using hypercohomology

Class Formations and Higher Dimensional Local Class Field Theory

“…We work with generalized class formations as introduced by Koya in [Ko1]. He uses them to prove the reciprocity law for 2-local fields.…”

An analogue of Brumer’s criterion for hypercohomology groups and class formations

“…Let Z(2) be the Lichtenbaum complex for K. (For basic properties and proofs, see [10], [11] and [12].) We know that the pair (Gal(K s /K), Z(2) [2]) forms a class formation [8].…”

“…Assume that the pair (G, A • ) forms a class formation. See [8] about such class formation including complexes. In this case, as in [1], we can make more explicit the conditions to determine scd p G. (1) For each r = 0, .…”

“…The author developed in [8] the theory of class formations, including complexes and hypercohomology groups. Class formations can be applied to higher-dimensional local fields and, of course, to the classical class field theory of usual local and global fields.…”

Generalized class formations and higher class field theory

“…By using the complex Z(2) defined by Lichtenbaum, Koya proved that for 2-dimensional local field K the pair (G K , Z(2)) is a class formation and deduced the reciprocity map for K (see [Ko1]). Once the existence of the Z(n) with the properties (b), (d), (e) and (f) above is established, his proof would work for arbitrary higher dimensional local fields as well (i.e.…”