Note on linear relations in Galois cohomology and étale K-theory of curves

P. Krasoń
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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 ${\cal X}.$
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关于伽罗瓦上同调中的线性关系和曲线的 k -理论
本文研究了一类阿贝尔变种的Tate模中带系数数域伽罗瓦上同调的一个局部到全局原理。在\cite{bk13}中,G. Banaszak和作者得到了 {}$K$ -曲线理论的局部{变}全局原理成立的充分条件。通过对伽罗瓦上同调中相应问题的分析,实际上已经建立了这个条件。我们证明,在某些情况下,这个结果是最好的可能,即如果这个条件不成立,我们得到反例。我们还给出了一些曲线及其雅可比矩阵的例子。最后,我们证明了曲线的 $K$ -理论的局部{变}全局原理的动态版本。最近S. Barańczuk在\cite{b17}中考虑了modell - weil型群的动态局域到全局原理。我们证明,如果Bass和Quillen- lichtenbaum的猜想成立,那么我们所有的结果仍然适用于${\cal X}$的Quillen $K$ -理论 ${\cal X}.$
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