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Almost Vector Bundles over Perfectoid Spaces

Yuntong Cui, Guo Li, Shuhan Jiang, Jiahong Yu

TL;DR

This work defines almost vector bundles within almost mathematics on perfectoid spaces and proves a $v$-descent theorem linking $X_v^{+,a}$ and $X_{ ext{an}}^{+,a}$. It then establishes a $v$-local structure result via spherical completeness, showing local freeness on suitable $v$-covers, and connects these bundles to arc-descent by relating them to locally free modules over $ ext{O}_{Perf^{ ext{tf}}_R}$. The results integrate integral coefficients with analytic and arc-topologies, providing a robust descent framework for vector bundle-like objects in $p$-adic geometry. The approach blends almost mathematics, perfectoid theory, and arc-descent to yield principled structure theorems with potential implications for $p$-adic Hodge theory and related areas.

Abstract

In this paper, we define vector bundles within the framework of almost mathematics (referred to as almost vector bundles) and establish the $v$-descent theorem together with a structure theorem for these bundles over perfectoid spaces. The proof yields several interesting intermediate results.

Almost Vector Bundles over Perfectoid Spaces

TL;DR

This work defines almost vector bundles within almost mathematics on perfectoid spaces and proves a -descent theorem linking and . It then establishes a -local structure result via spherical completeness, showing local freeness on suitable -covers, and connects these bundles to arc-descent by relating them to locally free modules over . The results integrate integral coefficients with analytic and arc-topologies, providing a robust descent framework for vector bundle-like objects in -adic geometry. The approach blends almost mathematics, perfectoid theory, and arc-descent to yield principled structure theorems with potential implications for -adic Hodge theory and related areas.

Abstract

In this paper, we define vector bundles within the framework of almost mathematics (referred to as almost vector bundles) and establish the -descent theorem together with a structure theorem for these bundles over perfectoid spaces. The proof yields several interesting intermediate results.
Paper Structure (13 sections, 35 theorems, 63 equations)

This paper contains 13 sections, 35 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Link with previous works
  4. Acknowledgments
  5. Preliminaries on Adic Spaces
  6. Almost Mathematics and Almost Vector Bundles
  7. Foundations of almost mathematics
  8. Almost vector bundles, locally almost free sheaves and locally free sheaves
  9. $V$-descent of Almost Vector Bundles
  10. $V$-local Structure of Almost Vector Bundles
  11. Properties of spherical completeness
  12. Spherical completeness and the structure of almost vector bundles
  13. $Arc$-vector Bundles

Key Result

Theorem 1

The restriction functor induces an equivalence between the category of $v$-vector bundles on $\mathrm{Spec}(A)$ and the category of Zariski vector bundles on $\mathrm{Spec}(A)$.

Theorems & Definitions (79)

  • Theorem : bhatt2017projectivity
  • Definition 1.1: Definition \ref{['def']}
  • Theorem 1.2: Theorem \ref{['SAAVB']}
  • Theorem 1.3: Theorem \ref{['Thm: SC over v']}
  • Theorem 1.4: Theorem \ref{['perfectoid']}
  • Remark 1.5
  • Corollary 1.6: Lemma \ref{['Lem: extension']}
  • Theorem 1.7: Section \ref{['section:prove arc=v']}
  • Theorem 1.8: kedlaya2019relative
  • Theorem 1.9: heuer2024gtorsorsperfectoidspaces
  • ...and 69 more