L'invariant de Suslin en caractéristique positive

Abstract

RÉSUMÉPour une k-algèbre simple centrale A d'indice inversible dans k, Suslin a défini un invariant cohomologique de SK1 (A) ‘Sus2’. Dans ce texte, nous généralisons cet invariant à toute k-algèbre simple centrale par un relèvement de la caractéristique positive à la caractéristique 0. Pour pouvoir définir cet invariant, on a besoin des groupes de cohomologie des différentielles logarithmiques de Kato [Kat1].

Comparing Invariants of SK1

“…This injection also continues to the relative cohomology groups; i.e. there exists an injection H i+1 n,A ⊗r (k) → H i+1 n,B ⊗r K (K) for any integer r and A and B as above [Wou,Prop. 4.10].…”

“…To obtain a cycle module we have to tweak it a little bit. For this paper we do not need a cycle module, so we rather work with this functor of graded groups to ease the discussion (see [Wou,§4.1 (d)] for more details -see also Remark 2.6 infra).…”

“…So we have an inclusion Br(k) → Br(K); given a central simple algebra A over k, we get a lifted Azumaya algebra B over R and an associated central simple algebra B K over K. Because of the definition, ind k (A) = ind K (B K ) and by a theorem of Platonov we get SK 1 (A)(k) ∼ = SK 1 (B K )(K) [Pla,Thm. 3.12] -see also [Wou,Cor. 3.3].…”

“…The injection ι is defined in a similar way as one can get an injection from the splitting (2.5). It can be proved that i * behaves well by going to the relative cohomology groups H 4 4,A (k ) and H 4 4,B K (K ) [Wou,Prop. 4.10].…”

“…Even more, SL 1 (B K ) is part of B R and the isomorphism SK 1 (B K )(K ) ∼ = SK 1 (A)(k ) is induced by the residue map on SL 1 (B K ) [Pla,Cor. 3.13] -see also [Wou,Cor. 3.3].…”