Remarks on Fargues-Scholze

The Fargues-Scholze geometrization paper is available! In this post, and probably some future posts also, I’ll make some random comments on this paper. These won’t mean anything unless you’ve read (at the very least) the beautifully written introduction to the paper. To be clear, I have nothing of substance to say about the “big picture” – these will be purely technical remarks.

First of all, at the bottom of p. 324, one finds the slightly cryptic claim that although there are no general functors in the setting (for a locally closed immersion), one can define functors in the setting, where is the inclusion of any Harder-Narasimhan stratum into . This is stated without proof. However, if you read carefully, you will notice that these lisse functors are actually used in the paper, in the proof of Proposition VII.7.6. So maybe it’s worth saying something about how to construct them.

To build in the setting, factor as the composition . Here is the open substack of bundles which are “more semistable” than . It will also be convenient to write . Note that is a closed immersion, and is an open immersion, so clearly preserves . The subtlety is in making sense of , since then we can write as usual.

For , we need the local chart and its punctured version . Recall that these charts also come with compatible maps and . Then for any , the correct definition turns out to be

.

The point here is that in the lisse world, the only pushforward functors which come for free are the functors for cohomologically smooth maps . Since and are cohomologically smooth – one of the hardest theorems in the paper! – the above construction preserves . Moreover, it’s easy to check that the formula above has the right properties. Indeed, the *-restriction of the RHS of (1) to is just , by Proposition VII.7.2, while its complementary restriction to clearly vanishes.

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