## 38.22 Flattening stratification over an Artinian ring

A flatting stratification exists when the base scheme is the spectrum of an Artinian ring.

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.

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. $\square$

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