Let be a connected reductive group over , and let be a -valued (geometric) conjugacy class of minuscule cocharacters, with reflex field . In their Annals paper, Caraiani and Scholze defined a very interesting stratification of the flag variety (regarded as an adic space over ) into strata , where runs over the Kottwitz set . Let me roughly recall how this goes: any (geometric) point determines a canonical modification of the trivial -bundle on the Fargues-Fontaine curve, meromorphic at and with “mermorphy ” in the usual sense. On the other hand, Fargues proved that -bundles on the curve are classified up to isomorphism by , and then Caraiani-Scholze and Rapoport proved that -meromorphic modifications of the trivial bundle are exactly classified by the subset (CS proved that only these elements can occur; R proved that all of these elements occur). The Newton stratification just records which element of this set parametrizes the bundle .

The individual strata are pretty weird. For example, if and , then and the open stratum is just the usual Drinfeld space , but the other strata are of the form , where is the evident parabolic in and the action on is via the natural map . Qualitatively, this says that they’re unions of profinitely many copies of lower-dimensional Drinfeld spaces. In particular, the non-open strata are not rigid analytic spaces. There are also examples of strata which don’t have any classical rigid analytic points. However, the ‘s are always perfectly well-defined from the topological or diamond point of view.

Anyway, I’m getting to the following thing, which settles a question left open by Caraiani-Scholze.

**Theorem. ***Topologically, the Newton stratification of is a true stratification: the closure of any stratum is a union of strata.*

The idea is as follows. After base-changing from to the completed maximal unramified extension (which is a harmless move), there is a canonical map sending to the isomorphism class of . Here denotes the stack of -bundles on the Fargues-Fontaine curve, regarded as a stack on the category of perfectoid spaces over . This stack is stratified by locally closed substacks defined in the obvious way, and by construction the Newton stratification is just the pullback of this stratification along . Now, by Fargues’s theorem we get an identification , so it is completely trivial to see that the stratification of is a true stratification (at the level of topological spaces). We then conclude by the following observation:

**Proposition. ***The map is universally open.*

The idea is to observe that factors as a composition of two maps . Here the first map is a -torsor by construction, so it’s universally open by e.g. Lemma 10.13 here. More subtly, the second map is also universally open. Why? Because it is *cohomologically* *smooth *in the sense of Definition 23.8 here; universal openness then follows by Proposition 23.11 in the same document.

For the cohomological smoothness claim, take any affinoid perfectoid space with a map , corresponding to some bundle . After some thought, one works out the fiber product “explicitly”: it parametrizes untilts of over together with isomorphism classes of -meromorphic modifications supported along the section induced by our preferred untilt, with the property that is trivial at every geometric point of . Without the final condition, we get a larger functor which etale-locally on is isomorphic to . (To get the latter description, note that etale-locally on we can trivialize on the formal completion of the curve along , and then use Beauville-Laszlo to interpret the remaining data as a suitably restricted modification of the trivial -torsor on . This is a Schubert cell in a Grassmannian. Then use Caraiani-Scholze’s results on the Bialynicki-Birula map.) Anyway anyway, after a little more fiddling around the point is basically that the projection is cohomologically smooth because it’s the base change of a smooth map of rigid spaces. By Kedlaya-Liu plus epsilon, the natural map is an open immersion, so is cohomologically smooth. Since was arbitrary, this is enough.