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 i_! functors in the D_{lis} setting (for i a locally closed immersion), one can define functors i^{b}_! in the D_{lis} setting, where i^b: \mathrm{Bun}_{G}^b \to \mathrm{Bun}_G is the inclusion of any Harder-Narasimhan stratum into \mathrm{Bun}_G. This is stated without proof. However, if you read carefully, you will notice that these lisse i^{b}_! 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 i^{b}_! in the D_{lis} setting, factor i^b as the composition \mathrm{Bun}_{G}^b \overset{i}{\to}  \mathrm{Bun}_{G}^{ \leq b} \overset{j}{\to} \mathrm{Bun}_G. Here \mathrm{Bun}_{G}^{ \leq b} is the open substack of bundles which are “more semistable” than \mathcal{E}_b. It will also be convenient to write \mathrm{Bun}_{G}^{ < b} = \mathrm{Bun}_{G}^{ \leq b} - \mathrm{Bun}_{G}^{ b}. Note that i is a closed immersion, and j is an open immersion, so j_! = j_{\natural} clearly preserves D_{lis}. The subtlety is in making sense of i_!, since then we can write i_{!}^{b} = j_! i_! as usual.

For i_!, we need the local chart \pi_{b}: \mathcal{M}_b \to \mathrm{Bun}_{G}^{\leq b} and its punctured version \pi_{b}^\circ : \mathcal{M}_{b}^\circ = \mathcal{M}_{b} \times_{\mathrm{Bun}_{G}^{\leq b}} \mathrm{Bun}_{G}^{< b} \to \mathrm{Bun}_{G}^{\leq b}. Recall that these charts also come with compatible maps q_b: \mathcal{M}_b \to [\ast / G_b(E)] and q_b^{\circ}: \mathcal{M}_{b}^{\circ} \to [\ast / G_b(E)]. Then for any A \in D_{lis}(\mathrm{Bun}_{G}^{b},\Lambda) \cong D_{lis}([\ast / G_b(E)],\Lambda), the correct definition turns out to be

i_! A = \mathrm{Cone}(\pi_{b \natural}^{\circ} q_{b}^{\circ \ast}A \to \pi_{b \natural} q_{b}^{\ast}A)\,\,\,\,(1).

The point here is that in the lisse world, the only pushforward functors which come for free are the functors f_{\natural} for cohomologically smooth maps f. Since \pi_{b} and \pi_{b}^{\circ} are cohomologically smooth – one of the hardest theorems in the paper! – the above construction preserves D_{lis}. Moreover, it’s easy to check that the formula above has the right properties. Indeed, the *-restriction of the RHS of (1) to \mathrm{Bun}_{G}^{b} is just A, by Proposition VII.7.2, while its complementary restriction to \mathrm{Bun}_{G}^{ < b} clearly vanishes.

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