Descending finite projective modules from a Novikov ring
Dongryul Kim
TL;DR
The paper establishes that descent data for finite projective modules along a Novikov-ring diagram with exponents in a submonoid $\Gamma\subset\mathbb{R}$ are always effective, yielding an equivalence between finite projective $A$-modules and Novikov descent data. The proof passes through Novikov isocrystals, showing that descent data map fully faithfully into isocrystals and that, after handling reduced and then general rings, the data are effective. Extending from $\Gamma=\mathbb{R}$ to arbitrary $\Gamma$ completes the descent result in full generality. A key application in perfectoid geometry shows that for perfect $\mathbb{F}_p$-algebras $A$, vector bundles on $\mathrm{Spec}A$ descend to vector bundles on $\mathrm{Spd}A$, and finite étale structures correspond accordingly, connecting analytic and algebraic categories in a precise, tensorial way.
Abstract
We prove a descent result for finite projective modules, motivated by a question in perfectoid geometry. Given a commutative ring $A$, we formulate a descent problem for descending a finite projective module over the Novikov ring with coefficients in $A$ to a finite projective module over $A$. The main theorem of this paper is that all such descent data are effective. As an application, we prove for every perfect $\mathbb{F}_p$-algebra $A$, a vector bundle on $\operatorname{Spd} A$ always descends to a vector bundle on $\operatorname{Spec} A$.