Descending finite projective modules from a Novikov ring

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 Fₚ-algebra A, a vector bundle on Spd A always descends to a vector bundle on Spec A.