GAGA for Henselian schemes

Abstract

The global analogue of a Henselian local ring is a Henselian pair-a ring R and an ideal I which satisfy a condition resembling Hensel's lemma regarding lifting coprime factorizations of monic polynomials over R/I to factorizations over R. The geometric counterpart is the notion of a Henselian scheme, which can serve as a substitute for formal schemes in applications such as deformation theory. In this paper we prove a GAGA-style cohomology comparison result for Henselian schemes in positive characteristic, making use of a "Henselian \'etale" topology defined in previous work in order to leverage exactness of finite pushforward for abelian sheaves in the \'etale topology of schemes. We will also discuss algebraizability of coherent sheaves on the Henselization of a proper scheme, proving (without a positive characteristic restriction) algebraizability for coherent subsheaves. We can then deduce a Henselian version of Chow's theorem on algebraization and the algebraizability of maps between Henselizations of proper schemes.