Lemma 61.5.2. Let $X = \mathop{\mathrm{Spec}}(A)$ as above. Given any finite stratification $X = \coprod T_ i$ by constructible subsets, there exists a finite subset $E \subset A$ such that the stratification (61.5.1.2) refines $X = \coprod T_ i$.

**Proof.**
We may write $T_ i = \bigcup _ j U_{i, j} \cap V_{i, j}^ c$ as a finite union for some $U_{i, j}$ and $V_{i, j}$ quasi-compact open in $X$. Then we may write $U_{i, j} = \bigcup D(f_{i, j, k})$ and $V_{i, j} = \bigcup D(g_{i, j, l})$. Then we set $E = \{ f_{i, j, k}\} \cup \{ g_{i, j, l}\} $. This does the job, because the stratification (61.5.1.2) is the one whose strata are labeled by the vanishing pattern of the elements of $E$ which clearly refines the given stratification.
$\square$

## 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Your email address will not be published.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).

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.

## Comments (0)