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

Abstract

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In \cite{bk13} G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for {\'e}tale $K$-theory of a curve . This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases this result is the best possible i.e if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for {\'e}tale $K$-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Bara{\'n}czuk in \cite{b17}. We show that all our results remain valid for Quillen $K$-theory of ${\cal X}$ if the Bass and Quillen-Lichtenbaum conjectures hold true for ${\cal X}.$