The six functors for Zariski-constructible sheaves in rigid geometry

In this post I want to talk about my recent paper with Bhargav Bhatt, which you can find here. This paper was a lot of fun to write, and I hope the toolkit we built will be useful for other researchers in this area. In this post I want to make some random remarks on this paper, which probably won’t mean anything if you don’t go read the real introduction to the paper first.

One funny point is that the proof of Theorem 1.6 leans on Theorem 1.7 fairly heavily, but in fact you can prove Theorem 1.6 without appealing to Theorem 1.7, at the price of much more intricate arguments. This was actually the state of the manuscript until mid-December, when we finally figured out how to prove Theorem 1.7.

Another funny point is that the discussion of the “standard” / “constructible” t-structure on D^{(b)}_{zc}(X,\mathbf{Z}_{\ell}) turned out to be surprisingly subtle, cf. Theorem 3.39. Note that D^{(b)}_{zc}(X,\mathbf{Z}_{\ell}) is by definition a full subcategory of D(X_v,\mathbf{Z}_{\ell}), and the latter carries an obvious t-structure. Nevertheless, we weren’t able to settle the question of whether these t-structures are compatible:

Question. Do the cohomological functors ^c \mathcal{H}^n(-) on D^{(b)}_{zc}(X,\mathbf{Z}_{\ell}) produced by Theorem 3.39 agree with the usual cohomology sheaves on D(X_v,\mathbf{Z}_{\ell})?

I would be extremely interested to know the answer to this.

One thing missing from the paper is any discussion of ULA sheaves. (See Fargues-Scholze for the foundations of ULA sheaves in p-adic geometry. In what follows I take \ell \neq p, but the case \ell = p should actually also be OK.) The first basic point to make is that for any rigid space X/K, any object A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell}) is ULA for the structure map X \to \mathrm{Spa}K. Sketch: The claim is local on X, so we can assume X is quasicompact. By Proposition 3.6 and stability of ULA sheaves under proper pushforward, we reduce to the special case where A = \mathbf{F}_{\ell} is constant. By an argument with resolution of singularities, we now reduce further to the case where A is constant and X is smooth, which is handled in Fargues-Scholze. Identical remarks apply with \mathbf{Z}_{\ell}-coefficients, or with general \mathbf{Z}/n coefficients (but then only for objects of “finite tor-dimension”).

This is already enough to show that the Lefschetz trace formula works as expected for proper rigid spaces in characteristic zero. More precisely, suppose given a correspondence c=(c_1,c_2): C \to X \times X of proper rigid spaces over an algebraically closed field, and a cohomological correspondence u: c_1^{\ast}A \to Rc_2^{!}A on some A \in D^b_{zc}(X,\mathbf{Z}_{\ell}). Then the usual recipe to define local terms applies, and the expected equality \mathrm{tr}(u|R\Gamma(X,A)) = \sum_{\beta \in \pi_0 \mathrm{Fix}(c)} \mathrm{loc}_{\beta}(u,A) holds true. (Note that R\Gamma(X,A) is a perfect \mathbf{Z}_{\ell}-complex by Theorem 3.35.(3).)  This can be proved by imitating the unpleasant arguments with giant diagrams in SGA5, or by the amazing categorical magic of Lu-Zheng. Of course, the local terms \mathrm{loc}_{\beta}(u,A) are just as mysterious as in the case of schemes.

It’s also natural to guess that an analogue of Deligne’s generic ULA theorem holds in this setting. Let me formally state this as a conjecture.

Conjecture. Let f:X \to Y be a proper map of characteristic zero rigid spaces, and let A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell}) be any given object. Then there is a dense Zariski-open subset of Y over which A is f-ULA.

This should be within reach, but I didn’t think about it very much.

Finally, I want to highlight the open problems mentioned in Remark 4.10, Remark 4.14 and Section 4.5. Conjecture 4.16 is probably (for whatever reason) my favorite open problem right now. Actually, I don’t even know how to prove that IH^{\ast}(X_C,\mathbf{Q}_p) is Hodge-Tate, or the even weaker statement that it has integral Hodge-Tate-Sen weights.

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