Flattening and algebrisation

TL;DR

This paper introduces a functorial flattening technique for coherent sheaves on algebraic spaces, extending classical theorems and enabling near-optimal algebraisation of formal deformations.

Contribution

It develops a functorial flattening method for coherent sheaves, generalizing Raynaud and Hironaka's results, and applies it to formal algebraic spaces and Artin stacks.

Findings

Constructed a functorial ideal called the flatifier for flattening sheaves.

Extended flattening theorems to formal algebraic spaces and Artin stacks.

Provided a counterexample to a claim in EGA-3 and proved an equivalence related to algebraisability.

Abstract

To, say, a proper algebraic or holomorphic space $X/S$, and a coherent sheaf ${\mathcal F}$ on $X$ we identify a functorial ideal, the fitted flatifier, blowing up sequentially in which leads to a flattening of the proper transform of ${\mathcal F}$. As such, this is a variant on theorems of Raynaud \& Hironaka, but it's functorial nature allows its application to a flattening theorem for formal algebraic spaces or Artin champs, where we apply it to prove close to optimal algebraisation theorems for formal deformations. En passant, contrary to what is asserted in EGA-3 Remarque 5.4.6, we give an example of an adic Noetherian formal scheme whose nil radical is not coherent and establish the equivalence conjectured therein between arbitrary algebraisability and that of the reduction.