Lemma 38.22.1. Let S be the spectrum of an Artinian ring. For any scheme X over S, and any quasi-coherent \mathcal{O}_ X-module there exists a universal flattening. In fact the universal flattening is given by a closed immersion S' \to S, and hence is a flattening stratification for \mathcal{F} as well.
38.22 Flattening stratification over an Artinian ring
A flatting stratification exists when the base scheme is the spectrum of an Artinian ring.
Proof. Choose an affine open covering X = \bigcup U_ i. Then F_{flat} is the product of the functors associated to each of the pairs (U_ i, \mathcal{F}|_{U_ i}). Hence it suffices to prove the result for each (U_ i, \mathcal{F}|_{U_ i}). In the affine case the lemma follows immediately from More on Algebra, Lemma 15.17.2. \square
Comments (0)