Local systems on diamonds and $p$-adic vector bundles

Abstract

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, proétale, and the v-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over C p this enables us to prove a category equivalence between C p -local systems with integral models, and modules under the v-structure sheaf which 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. for suggesting to look into the v-topology. Moreover, we would like to thank Christopher Deninger for his comments, Torsten Wedhorn for useful hints on adic spaces and Matti Würthen for many helpful conversations.

The p-adic Corlette–Simpson correspondence for abeloids

“…The category v-bundles has previously been studied by the second and third author [26]: For example, they show that pro-finite-étaleness and related properties of vector bundles can be checked on proper covers.…”

“…In the case of K = C p , the category of pro-finite-étale v-vector bundles was studied in [41, §3.1] and [26] (more precisely, pro-finite-étale v-vector bundles are equivalent to vector bundles with properly trivializable reduction modulo all p n in the sense of loc cit).…”

“…If K = C p , then these categories are also equivalent to the category of C p -local systems on X v which possess an integral model, as defined in [26,Definition 3.23].…”

“…) is naturally equivalent to the category of K -local systems on X v which are constant on X . If K = C p one checks easily that this is indeed precisely the category of C p -local systems with integral model (for one direction use [26,Corollary 3.21]).…”

The $p$-adic Corlette-Simpson correspondence for abeloids

“…then these categories are also equivalent to the category of C p -local systems on X v which possess an integral model, as defined in[MW20, Definition 3.23].…”