Bloch–Kato conjecture on Galois symbols

Let kk be a field, let ii be a non-negative integer, and let nn be a positive integer prime to the characteristic ch(k)\operatorname{ch}(k) of kk. The Galois symbol map is

hk,ni:Ki(k)/nHi(k,Z/n(i))h_{k,n}^i:K_i(k)/n\longrightarrow H^i(k,\mathbb{Z}/n(i))

Bloch–Kato conjecture. The map hk,nih_{k,n}^i is bijective. This conjecture identifies Milnor KK-theory modulo nn with the corresponding Galois cohomology group; its resolution is a fundamental result in the theory of Galois symbols and norm residue isomorphisms.

Sources & referencesView supporting material

Primary source

Rin Sugiyama, “On the kernel of the reciprocity map of simple normal crossing varieties over finite fields”, arXiv:0911.5065 (2009).

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