Aniruddh Venkatesan

math.AG — Algebraic Geometry

Bootstrapping GAGA Theorems

Contributed by Aniruddh Venkatesan

Let $X$ be a noetherian scheme with affine diagonal, and let $\lambda : \mathcal{X} \to X$ be a morphism of locally ringed spaces from a qcqs locally strongly noetherian analytic adic space. We prove that an equivalence on coherent sheaves, together with comparison of their cohomology, extends to every proper $X$-scheme. As an application, we obtain GAGA for proper schemes over the Fargues--Fontaine curve in equal and mixed characteristic.