Adic Sheafiness of $A_{\text{inf}}$ Witt Vectors over Perfectoid Rings

TL;DR

This paper proves that the adic sheafiness of Witt vectors over perfectoid rings ensures the acyclicity of the structure sheaf and establishes an equivalence between vector bundles and finite projective modules, advancing perfectoid geometry.

Contribution

It extends the sheafiness results to general stably uniform Banach rings in the context of perfectoid rings using prism theory.

Findings

The structure sheaf is acyclic on the adic spectrum.

Equivalence between vector bundles and finite projective modules.

Generalization of sheafiness arguments to broader classes of Banach rings.

Abstract

For $(R, R^{+})$ an analytic perfectoid ring in char $p$, let $A_{\text{inf}}(R^{+})$ be the ring of Witt vectors with the induced topology from $(R, R^{+})$. We prove that $\text{Spa}(A_{\text{inf}}(R^{+}),A_{\text{inf}}(R^{+}))$ is sheafy and its structure sheaf is acyclic. We first show $A_{\text{inf}}(R^{+})$ is a stably uniform Banach ring using elements from the theory of prisms. The "stably uniform implies sheafy" argument is applied to Tate Huber rings in Buzzard-Verberkmoes(2015) and is generalized to analytic Huber rings in Kedlaya(2019). Here we show that the "stably uniform implies sheafy" argument in Kedlaya(2019) can be applied to general stably uniform Banach rings whose underlying topological ring is a Huber ring. Finally we show the equivalence of categories of vector bundles over $\text{Spa}(A_{\text{inf}}(R^{+}),A_{\text{inf}}(R^{+}))$ and finite projective modules over $A_{\text{inf}}(R^{+})$.