Remarks on Fargues-Scholze, part 2

Today I want to talk about section VII.3 in the manuscript. Here they define and study a functor f_{\natural} on solid sheaves which is left adjoint to the usual pullback functor. But why is this called relative homology?

To explain this name, you have to remember that in the usual formalism of etale cohomology, homology is realized as the compactly supported cohomology of the dualizing complex. Symbolically, if X is a variety with structure map f:X \to \mathrm{Spec} k, then the homology of X is given by Rf_! Rf^! \Lambda. Now, you might ask whether this works in families: if f:X \to Y is some map of varieties, maybe I can find some complex on Y whose stalk at y realizes the homology of X_y? If the constant sheaf is f-ULA, then formation of Rf^! \Lambda commutes with any base change, and exactly the same formula works, but in general there is no naive sheaf with this property.

The punchline now is that f_{\natural} \Lambda does have this property: when the constant sheaf is f-ULA (e.g. if Y is a point) it agrees with Rf_! Rf^! \Lambda by Proposition VII.5.2, and its formation commutes with arbitrary base change, so it really does give a complex on Y whose stalks realize the homology of the fibers of f. The only twist is that f_{\natural}\Lambda is a solid sheaf in general, not a classical etale sheaf.

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.

sheafiness; perversity

\bullet One of the main annoyances in the theory of adic spaces is that, for a given Huber pair (A,A^+), the structure presheaf on \mathrm{Spa}(A,A^{+}) is not always a sheaf. One usually remedies this by restricting attention to various classes of Huber rings, e.g. strongly Noetherian Tate rings, perfectoid Tate rings, Noetherian adic rings, etc. However, the following class of rings doesn’t seem to be addressed in the literature:

Definition. Let A be an adic ring with finitely generated ideal of definition I \subset A. We say A is strongly Noetherian outside I if, for all n \geq 0, the scheme \mathrm{Spec}\,A\left\langle X_1,\dots,X_n \right\rangle \smallsetminus V(IA\left\langle X_1,\dots,X_n \right\rangle ) is Noetherian.

Here A\left\langle X_1,\dots,X_n \right\rangle denotes the usual ring of restricted power series. Note that if A is a Tate ring and (A_0,(\varpi)) is any couple of definition, then A is strongly Noetherian if and only if A_0 is strongly Noetherian outside (\varpi). I should also point out that the condition of being strongly Noetherian outside I is already considered in a very interesting paper of Fujiwara-Gabber-Kato; they use the terminology “topologically universally rigid-Noetherian”, but I prefer my terminology on account of the previous sentence. Anyway, the following conjecture seems reasonable:

Conjecture. If A is strongly Noetherian outside I, the structure presheaf on \mathrm{Spa}(A,A) is a sheaf.

This implies that any strongly Noetherian Tate ring is sheafy (which of course is already known), but it also implies e.g. that if A is topologically finitely presented over \mathcal{O}_K for some nonarchimedean field K, then A is sheafy. Sheafiness in the latter situation is known when K is discretely valued, but to the best of my knowledge it’s open for general K.

I’m sure this conjecture is within reach, and maybe it’s an easy exercise for experts on sheafiness (*cough* Kiran *cough*). Note that FGK already proved some interesting consequences of this definition, which are probably relevant to proving this conjecture. Precisely, they show that if A is strongly Noetherian outside I, then:
1. The I-power-torsion submodule of any finitely generated A-module is killed by a power of I.
2. If N \subset M is any inclusion of A-modules, with M finitely generated, then the subspace topology on N induced by the I-adic topology on M coincides with the I-adic topology on N.

\bullet Let j:U \to X be some immersion of varieties over a separably closed field. Everyone knows that the intermediate extension functor j_{!\ast} on perverse sheaves (say with coefficients in \Lambda = \mathbf{Q}_{\ell}) is pretty great: it’s totally canonical, it commutes with Verdier duality, it preserves irreducibility, it preserves monic and epic maps of perverse sheaves, etc.

Recently I noticed that if f: Z \to X is any map of varieties, with X smooth and Z irreducible, there’s still a natural functor f^{\ast !}:\mathrm{Perv}(X) \to \mathrm{Perv}(Z) which commutes with Verdier duality. To define this functor, note that for any f and any \mathcal{F} \in D^b_c(X), there is a natural map f^\ast \mathcal{F} \otimes Rf^! \Lambda \to Rf^! \mathcal{F}, obtained by adjunction from the composite map Rf_!( f^\ast \mathcal{F} \otimes Rf^! \Lambda) \cong \mathcal{F} \otimes Rf_! Rf^! \Lambda \to \mathcal{F} (the first isomorphism here is the projection formula). Since X is smooth, the dualizing complex \omega_X is just \Lambda[2 \dim X], so then Rf^! \Lambda = Rf^!\omega_X[-2 \dim X] = \omega_Z[-2 \dim X]. Thus we get a natural map \alpha: f^\ast \mathcal{F} \otimes \omega_Z[-2 \dim X] \to Rf^! \mathcal{F}.

Next, note that the complex \omega_Z is concentrated in degrees [-2 \dim Z,0], and in the lowest of these degrees it’s just the constant sheaf, i.e. \tau^{ \leq -2 \dim Z} \omega_Z \cong \Lambda[2 \dim Z]. In particular, there is a canonical map \Lambda[2 \dim Z] \to \omega_Z. Shifting by -2 \dim X and tensoring with f^\ast \mathcal{F} gives a map \beta: f^\ast \mathcal{F}[2 \dim Z - 2 \dim X] \to f^\ast \mathcal{F} \otimes \omega_Z[-2 \dim X]. Putting things together, we get a natural map \alpha \circ \beta: f^\ast \mathcal{F}[2 \dim Z - 2 \dim X] \to Rf^! \mathcal{F}. Set c= \dim X - \dim Z, so after shifting this becomes a natural map

\gamma=\gamma_{\mathcal{F}}: f^{\ast}\mathcal{F}[-c] \to Rf^!\mathcal{F}[c].

This shifting has the advantage that Verdier duality exchanges the functors f^{\ast}[-c] and Rf^![c] on D^b_c, and one can check that the Verdier dual of \gamma_{\mathcal{F}} identifies with \gamma_{\mathbf{D}\mathcal{F}}.

Definition. The functor f^{\ast !}: \mathrm{Perv}(X) \to \mathrm{Perv}(Z) sends any \mathcal{F} to the image of the map ^p\mathcal{H}^0(\gamma): ^p\mathcal{H}^0(f^{\ast} \mathcal{F}[-c]) \to ^p\mathcal{H}^0(Rf^!\mathcal{F}[c]).

Here of course ^p\mathcal{H}^0(-) denotes the zeroth perverse cohomology sheaf.

Exercise. Show that f^{\ast !}(\Lambda[\dim X]) \cong \mathcal{IC}_Z.

It might be interesting to compute this functor in some other examples. Note that it can be quite stupid: if f is a closed immersion (with c > 0) and \mathcal{F} \in \mathrm{Perv}(X) is already supported on Z, then f^{ \ast !} \mathcal{F} = 0. On the other hand, if f is smooth and surjective, then f^{\ast !} \cong f^{\ast}[-c] \cong Rf^![c] is faithful.