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.

In complex geometry, the most interesting class of complex manifolds is probably the Kahler class. In the non-archimedean world, say over a fixed p-adic base field $K$, the analogue of a compact complex manifold is a smooth proper rigid analytic space. In some ways, these are already surprisingly “close” to being Kahler – in particular, the Hodge-de Rham spectral sequence of such a space always degenerates at $E_1$. However, Hodge symmetry can definitely fail. A standard example is the non-archimedean Hopf surface $X = \mathbf{A}^2_{K} \smallsetminus \{ (0,0) \} / p^{\mathbf{Z}}$ where $p^n$ acts through diagonal multiplication. By a fun direct calculation, one checks that $H^0(X,\Omega^1_X)=0$ and $H^1(X,\mathcal{O}_X) = K$, so Hodge symmetry fails in degree one.

We now see a natural question: is there is some non-archimedean analogue of the Kahler condition which restores Hodge symmetry? Two years ago, Shizhang Li hit upon the following candiate condition:

A smooth proper rigid space $X$ satisfies (*) if it admits a formal model $\mathfrak{X}$ over $\mathcal{O}_K$ whose special fiber is projective (as opposed to merely proper).

Using fantastic ideas due to Shizhang, we managed to prove the following suggestive result.

Theorem. Let $X$ be a smooth proper rigid space satisfying (*). Then $h^{1,0}(X) = h^{0,1}(X)$.

Of course, one can then guess that (*) implies Hodge symmetry in all degrees. This speculation seems to have caught the imagination of others in the field, but until recently I personally regarded it as not much more than wishful thinking. However, my perspective completely changed a month ago, when I learned from Shizhang that, according to Robert Friedman, the archimedean analogue of “(*) implies Hodge symmetry” is a theorem! More precisely, we have the following result:

Theorem. Let $D$ be the complex disk, with $D^\times =D \smallsetminus \{0 \}$ the punctured disk. Let $f:Y \to D$ be a proper map of complex analytic spaces. Suppose that $f^{-1}(D^\times) \to D^\times$ is a submersion, and that the central fiber $Y_0=f^{-1}(0)$ is the analytification of a projective (and not necessarily smooth) algebraic variety. Then for all $t \in D^\times$ with $|t| \ll 1$, the fiber $Y_t$ satisfies Hodge symmetry and Hodge-de Rham degeneration.

Of course, the analogy is that $\mathfrak{X} \to \mathrm{Spf} \mathcal{O}_K$ is analogous to $Y \to D$, and $X$ is analogous to the “nearby” fibers $Y_t$ with $0<|t| \ll 1$.

The proof of this theorem uses the full power of mixed Hodge theory. In fact the claim about Hodge-de Rham degeneration is exactly Corollary 11.24 in the book of Peters-Steenbrink. Hodge symmetry is even more subtle, and the argument for this doesn’t seem to be written down anywhere; Friedman explained it to Shizhang, who explained it to me, but the details entailed such a horrible explosion of gradings, filtrations, and multi-indices that I can’t hope to reproduce it here.

Anyway, I’m now completely convinced that Shizhang’s condition (*) implies Hodge symmetry in all degrees, and that this is truly the “right” p-adic analogue of the Kahler condition.