Descending finite projective modules from a Novikov ring

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 $\operatorname{Spec} A$.