We investigate the Galois cohomology of finitely generated maximal pro-$p$ quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with $p$-Zassenhaus filtration on the group. This paper also contains a survey of Koszul property in Galois cohomology and its relation with absolute Galois groups.

中文翻译：

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伽罗瓦上同调中的 Koszul 代数和二次对偶

我们研究了绝对伽罗瓦群的有限生成的最大 pro-$p$ 商的伽罗瓦上同调。假设这些群的众所周知的推测描述，我们证明伽罗瓦上同调具有 PBW 性质。因此，特别是它是 Koszul 代数。在这种情况下，这肯定地回答了 Positselski 的猜想。我们还提供了关于毕达哥拉斯域的类似无条件结果。此外，我们建立了一些结果，将伽罗瓦上同调的二次对偶与群上的 $p$-Zassenhaus 过滤相关联。本文还包括对伽罗瓦上同调中的 Koszul 性质及其与绝对伽罗瓦群的关系的调查。