We use Scholze’s framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper, we provide a detailed analysis of local systems on diamonds for the étale, pro-étale, and the $v$-topology and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb {C}_p$, this enables us to prove a category equivalence between $\mathbb {C}_p$-local systems with integral models, and modules under the $v$-structure sheaf that modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover. 2020 MSC: 14G45, 14G22, 11G25.

中文翻译：

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金刚石和 p-Adic 矢量束的局部系统

我们使用 Scholze 的菱形框架来获得关于 $p$-adic 矢量束和本地系统之间对应关系的新见解。在希格斯场消失的情况下，这种对应出现在 $p$-adic Simpson 理论的上下文中。在本文中，我们对 étale、pro-étale 和 $v$ 拓扑的钻石局部系统进行了详细分析，并研究了所有三种拓扑的结构滑轮。应用于 $\mathbb {C}_p$ 上的有限类型的适当进数空间，这使我们能够证明具有积分模型的 $\mathbb {C}_p$-local 系统和 $v$- 下的模块之间的类别等价性对每个 $p^n$ 取模的结构层可以在适当的覆盖上简单化。$v$-拓扑的灵活性以及局部系统积分模型的下降结果使我们能够证明模块类别中的平凡化条件可以在任何正常的适当覆盖上检查。这一结果导致 Deninger 和第二作者将平行传输理论扩展为在适当的法线覆盖上具有数值平坦缩减的向量束。2020MSC：14G45、14G22、11G25。