Michael McQuillan: 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 algebrisation theorems for formal deformations. En passant, and contrary to what’s asserted in EGA, III.5.4.6, we give an example of an adic Noetherian formal scheme whose nil radical is not coherent, and wholly resolve the question posed therein on the relation between algebraisability of reduced and non-reduced objects.
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