Surjectivity and flatness over DVR's (after Moret-Bailly)

TL;DR

This paper investigates conditions for the existence of certain morphisms and flat surjectivity in schemes over DVRs, extending understanding of scheme morphisms in algebraic geometry.

Contribution

It provides new criteria for the existence of morphisms and flat surjectivity over schemes, especially when the base is a DVR spectrum.

Findings

Conditions for the existence of a morphism g such that f∘g is the canonical morphism.

Situations where the morphism f is flat and surjective.

Results are primarily focused on schemes over the spectrum of a DVR.

Abstract

We study morphisms of schemes $f : X \to S$ which are locally of finite type. We present conditions under which there exists a morphism $g : S'\to X$ of $S$--schemes such that $f \circ g $ is the canonical morphism $S'\to S$. Furthermore, we exhibit situations in which $f$ is flat surjective. Our results are mostly concerned with $S$ being the spectrum of a DVR.