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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)