Flattening and algebrisation

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.