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 [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for étale 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 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 [S. Barańczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206–211]. 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.

中文翻译：

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关于伽罗瓦上同调和曲线的étale K-理论中的线性关系的注释

在本文中，我们研究了在阿贝尔变体的 Tate 模块中具有系数的数域的伽罗瓦上同调的局部到全局原理。在 [G. Banaszak 和 P. Krasoń，关于 étale K 曲线群中的局部到全局原则，J. K-理论应用程序。代数几何。白杨。 12(2013) 183–201]，G. Banaszak 和作者获得了 étale 的局部到全局原则有效性的充分条件ķ- 曲线理论。这个条件实际上是通过对伽罗瓦上同调中相应问题的分析而建立的。我们表明，在某些情况下，这个结果是最好的，即如果这个条件不成立，我们将获得反例。我们还给出了一些曲线及其雅可比行列式的例子。最后，我们证明了 étale 的局部到全局原理的动态版本ķ- 曲线理论。S. Barańczuk 在 [S. Barańczuk，关于 Mordell-Weil 类型群的动态局部-全局原则，世博会。数学。 35(2) (2017) 206–211]。我们证明我们所有的结果对 Quillen 仍然有效ķ- 理论𝒳如果 Bass 和 Quillen-Lichtenbaum 猜想对𝒳.