The Stacks project

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. $\square$

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05PA. Beware of the difference between the letter 'O' and the digit '0'.