Note on linear relations in Galois cohomology and {é}tale $K$-theory of curves

Abstract

In this paper we investigate a local to global principle for étale K-theory of curves. More precisely, we show that the result obtained by G.Banaszak and the author in [BK13] describing the sufficient condition for the local to global principle to hold is the best possible (i.e this condition is also necessary). We also give examples of curves that fulfil the assumptions imposed on the Jacobian of the curve. Finally, we prove the dynamical version of the local to global principle for étale K-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S.Barańczuk in [B17]. We show that all our results remain valid for Quillen K-theory of X if the Bass and Quillen-Lichtenbaum conjectures hold true for X .