## An appreciation

Like many people, I’ve had rather mixed experiences with referees. On one occasion, I had a paper rejected by a referee who explicitly admitted in their report that they weren’t qualified to evaluate it.  More recently, though, my experiences have been very positive:

• The referee for my paper with Kiran was extremely careful, and provided us with many detailed comments which (hopefully) have lead to a huge increase in the correctness and readability of the paper. Time from submission to receipt of referee report: 8 months
• The referee for my paper with Bhargav found many small typos, as well as one or two minor inaccuracies. Time from submission to receipt of referee report: 5 1/2 months
• The referee for my paper with Tasho and Jared found many small typos and inaccuracies, together with one more significant conceptual blunder. Time from submission to receipt of referee report: 6 weeks (!)

New versions of these papers should all be available soon. In the mean time, if any of the referees mentioned above are reading this, please accept my gratitude for your hard work!

## Several things part 3

• Here’s a suggestive hallucination, useful to me for keeping some things straight, but maybe not useful to anyone else:
Let $G$ be a p-adic reductive group, $C$ an algebraically closed field of characteristic zero (or maybe just of characteristic $\neq p$). The set $X_G$ of isom. classes of smooth irreducible $C$-representations of $G(\mathbf{Q}_p)$ really wants to be an algebraic variety, but it’s not. However, $X_G$ has a canonical best approximation by an (ind-)algebraic variety, the Bernstein variety $Z_G$. The canonical map $X_G \to Z_G$ is “quasifinite and birational”.  The algebraic functions on $X_G$ are given by trace forms, i.e. by functions of the form $\pi \mapsto \mathrm{tr}(f| \pi)$ for some arbitrary $f \in C_c(G,C)$. There is also a canonical second-best approximation of $X_G$ by an algebraic variety, the spectral Bernstein variety $Z_{G}^{\mathrm{spec}}$, i.e. the coarse quotient of the stack parametrizing ($G$-relevant) L-parameters $W_{\mathbf{Q}_p} \to \phantom{}^L G$. It is second-best in the sense that there is a quasifinite map $Z_G \to Z_{G}^{\mathrm{spec}}$. The composite map $X_G \to Z_{G}^{\mathrm{spec}}$ sends $\pi$ to its semisimple L-parameter.

Should I post more hallucinations like this?

• I have to admit that I struggle psychologically with things related to foundations, especially subtleties arising from “big” constructions and the usual prophylactics involving universes or cutoff cardinals or whatever. For one thing, I don’t really care. But more significantly, the idea that ZFC (or something like it) should be accepted as the “standard foundations” of mathematics is absolutely revolting and nonsensical to me. The fact that everything in ZFC is a set makes it a complete non-starter for me as a reflection of how mathematics really operates. In some sense, I don’t really believe in “naked” sets.
$\phantom{}$
Anyway, I was never able to articulate my thoughts about this stuff very precisely. It was thus something of a revelation when I read this article at the Xena project, and realized that type theory is what I’ve been craving all along. I also strongly recommend this article by Todd Trimble which articulates my problems with ZFC much more eloquently than I can. (I don’t really understand ETCS yet, but it also seems like it would satisfy me.)
• Is the twitter account @GeoMoChi08 a parody? I would dearly love to know what’s going on with this account (and with @math_jin).

## Comparing local Langlands correspondences

At least six people have independently asked me some variant of the question:

What are the prospects for showing that the Fargues-Scholze construction of L-parameters is compatible with other constructions of the local Langlands correspondence?

In this post I’ll briefly lay out the answer as I see it.

For reductive groups $G$ over finite extensions $F/\mathbf{Q}_p$, the situation is complicated, since the status of LLC is complicated.

1. $\mathrm{GL}_n$ and $D_{1/n}^{\times}$. Compatibility for these groups is known and already proved in Fargues-Scholze, and follows from the realization of local Langlands and local Jacquet-Langlands in the cohomology of the Lubin-Tate tower.
2. Any inner form of $\mathrm{GL}_n$. Compatibility here is Theorem 1.0.3 in H.-Kaletha-Weinstein.
3. $\mathrm{SL}_n$ and inner forms. Compatibility should follow from the previous two points, but I guess it’s not completely trivial. Someone should write it down.
4. $\mathrm{GSp}_4$ and $\mathrm{Sp}_4$, and their unique inner forms. Compatibility for these groups has been proved by my student Linus Hamann. His preprint should be available very soon, and I’ll write a detailed blog post about it at that time. The arguments here rely on a number of special features of the group $\mathrm{GSp}_4$.
5. Split $\mathrm{SO}_{2n+1}$ and closely related groups. Partial results here are definitely possible by extending Hamann’s arguments, but it’s not clear to me whether complete results can be expected. I’ll say more about this when I write about Hamann’s paper.
6. Unitary groups. Partial results should be possible by combining some aspects of Hamann’s methods with recent works of Nguyen and Bertoloni-Meli–Nguyen.
7. $\mathrm{GSp}_{2n}$ and $\mathrm{Sp}_{2n}$ and their inner forms, $n>2$. This seems out of reach.
8. Even special orthogonal groups. I’m frankly confused about what’s going on here. Is there even an unambiguous LLC? In any case, this also seems hard.
9. Exceptional groups. There’s no “other” LLC here. Go home. (OK, for $G_2$ there’s a very cool recent paper of Harris-Khare-Thorne.)
10. General groups splitting over a tame extension, $p$ not too small. Here Kaletha has given a general construction which attaches a supercuspidal L-packet to any supercuspidal L-parameter. Compatibility of this construction with Fargues-Scholze might be approachable by purely local methods, but it seems to require substantial new ideas. An extremely weak partial result – constancy on Kaletha’s packets of the FS map from reps to L-parameters – is probably within reach, using the main results in H.-Kaletha-Weinstein.

The key point in many of the above situations is the following. Let’s say a group $G$ is accessible if it admits a geometric conjugacy class of minuscule cocharacters $\mu$ such that
1. The pair $(G,\mu)$ is totally Hodge-Newton reducible in the sense of Chen-Fargues-Shen.
2. Any L-parameter $\varphi: W_F \to \phantom{}^L G$ can be recovered up to isomorphism from the composition $r_{\mu} \circ \varphi$. (In practice one asks for slightly weaker versions of this.)
3. The local Shimura varieties attached to the local Shimura datum $(G,\mu,b)$ (with $b \in B(G,\mu)$ the unique basic element) uniformize the basic locus in a global Shimura variety of abelian type.

For groups satisfying this condition, there is hope. Very roughly, condition 2. implies that the FS construction is incarnated in the cohomology of a single local Shimura variety, whose cohomology can also be tightly related to the cohomology of a global Shimura variety using conditions 1. and 3. One then needs to know enough about the cohomology of these global Shimura varieties, namely that it realizes the “other” LLC you care about. Of course, this short outline veils substantial technical difficulties.

It turns out that $\mathrm{GL}_{n}$, $\mathrm{GU}_n$, $\mathrm{GSp}_4$, and $\mathrm{SO}_{2n+1}$ are all accessible, and this accounts for the definitive results in scenarios 1.-4. above and my optimism in scenarios 5.-6. On the other hand, $\mathrm{GSp}_{2n}$ is not accessible for $n>2$, and neither is $\mathrm{SO}_{2n}$ for $n>3$, and no exceptional groups are accessible. Hence my pessimism in scenarios 7.-9.

For reductive groups over finite extensions $F/\mathbf{F}_{p}((t))$, the situation is completely different. Here Genestier-Lafforgue have constructed a local Langlands correspondence for all groups, uniquely characterized by its compatibility with V. Lafforgue’s construction of global Langlands parameters. It is an extremely attractive problem to compare the Genestier-Lafforgue LLC with the Fargues-Scholze LLC. This should absolutely be within reach! After all, both constructions are realized in the cohomology of moduli spaces of shtukas, so the only “real” task should be to physically relate the moduli spaces of shtukas used by GL with those used by FS. This is probably not trivial: the spaces used by FS are local and totally canonical, while those used by GL seem to depend on a globalization and some auxiliary choices in a messy way. Nevertheless, I’d be surprised if this comparison is still an open problem two years from now.

## H.-Kaletha-Weinstein study guide and FAQ

Tasho Kaletha, Jared Weinstein and I have just posted a joint paper, On the Kottwitz conjecture for local shtuka spaces. This is a heavily revised and rewritten version of a preprint by Tasho and Jared which has been available since 2017 (henceforth “KW”).

Is the main result here the same as in KW?
More or less, yes. The main theorem confirms the Kottwitz conjecture for the cohomology of moduli spaces of local shtukas, ignoring the Weil group action and allowing for an “error term” consisting of a non-elliptic virtual representation.

Is the rough idea of the proof still the same as in KW?
Yes: the idea is still to compute the cohomology by applying a suitable Lefschetz-Verdier trace formula, and explicitly computing the “local terms” at all of the elliptic fixed points.

OK, what’s new, then?
Well, basically all of the details are different. First and foremost, the discussion of the relevant Lefschetz-Verdier trace formula for v-stacks has been completely rewritten, using Lu-Zheng’s magical formalism of symmetric monoidal 2-categories of cohomological correspondences. This leads to huge conceptual simplifications, both in the statements and the proofs. Actually, we go beyond the formalism of Lu-Zheng, introducing certain 2-categories of “based cohomological correspondences”. This formalism play a crucial role in the proof of Theorem 4.5.3, which is a key ingredient in the proof of the main theorem. It’s also hard to imagine proving Theorem 4.6.1 without this formalism.

Secondly, a key calculation of local terms on the $B_{\mathrm{dR}}$-affine Grassmannian, Theorem 5.1.3, now has a completely different and correct (!) proof, via a degeneration to a similar calculation on the Witt vector affine Grassmannian. This degeneration argument relies on the observation that the local terms appearing in the Lefschetz-Verdier trace formula are compatible with any base change, Proposition 5.3.1; this seems to be a new observation even for schemes. (My first proof of Proposition 5.3.1 involved checking the commutativity of 500 diagrams, but eventually I realized that it follows immediately from the Lu-Zheng formalism!) Once we’ve degenerated to the Witt vector affine Grassmannian, we make an intriguing global-to-local argument using the trace formula and known properties of the weight functors in geometric Satake, and relying crucially on a recent result of Varshavsky.

There are some other notable improvements:

• All assumptions in KW of the form “assume that some representation admits an invariant $\overline{\mathbf{Z}_{\ell}}$-lattice” have now been eliminated. This is far from formal, since we definitely don’t have a 6-functor formalism with $\overline{\mathbf{Q}_{\ell}}$-coefficients at hand. Instead, the idea is to reduce “by a continuity argument” to the case where things do admit lattices. The key ingredient here is Theorem 6.5.4, formerly a conjecture of Taylor, which should have lots of other applications.
• The final part of the main theorem, giving practical criteria for the error term to vanish, is new.
• Theorem 1.0.3 is new.

On a practical note, the Fargues-Scholze paper is now finished and available, so we can properly build on the powerful machinery and results in that paper.

Finally, the margins are no longer gigantic.

Sure. The main Theorem 1.0.2/6.5.1 in the paper is essentially a combination of two separate results, Theorem 6.5.2 and Theorem 3.2.9. Theorem 3.2.9 is conditional on the refined local Langlands correspondence, and the material needed to formulate and prove this result is developed in Chapter 3 and Appendix A. The bulk of the paper (sections 4-6 and Appendices B-C) is devoted to building up enough material to prove Theorem 6.5.2.

Each of sections 3, 4, and 5 is fairly self-contained, although section 5 depends slightly on section 4. Section 6 builds heavily on sections 4 and 5, and also on [FS21]. Section 5 is actually devoted to proving a single result, Theorem 5.1.3 (and its reinterpretation Theorem 5.1.4), and can be treated as a black box.

One thing you could do is read the introduction, then jump to Proposition 6.4.7 and work your way backwards as needed until all the notation makes sense. This proposition is the technical heart of the paper. The key inputs into the proof of Proposition 6.4.7 are Theorem 5.1.4, Theorem 4.5.3, and Corollary 4.3.8. The use of “the trace formula” in this whole argument is actually somewhat hidden: it is entirely encoded in the equality sign labelled “Cor. 4.3.8” at the bottom of p. 66.

From Proposition 6.4.7, Theorem 6.4.9 (a weak form of Theorem 6.5.2) follows quickly, and then Theorem 6.5.2 follows from Theorem 6.4.9 by a continuity argument which relies crucially on Theorem 6.5.4.

This sounds cool, but you didn’t punt any key ingredients into a separate and currently nonexistent paper, did you?
I’m glad you asked. In working out the trace formula formalism in enough generality to give a conceptual proof of the main theorem, it became clear at some point that we needed the existence and expected properties of the functors $Rf_!$ and $Rf^!$ in etale cohomology for “smooth-locally nice” maps $f$ between Artin v-stacks. For maps which are actually “nice”, and in particular non-stacky, these functors were constructed by Scholze, but extending them to the setting of stacky maps is not formal. Already in the analogous setting of schemes versus Artin stacks, these functors for Artin stacks were only constructed recently by Liu-Zheng, making heavy use of $\infty$-categories. In a forthcoming paper, Jared and I together with my postdoc Dan Gulotta will explain how to construct $Rf_!$ and $Rf^!$ for smooth-locally nice maps of Artin v-stacks. The arguments will very closely follow the arguments of Liu-Zheng, with the exception of one key non-formal piece of input (which took me three years of intermittent work).

Why do you only prove an explicit formula for the virtual character of $\mathrm{Mant}_{b,\mu}(\rho)$ restricted to elliptic elements of $G(F)$? What’s so special about elliptic elements in this context?
The fixed points of elliptic elements lie in the “admissible locus” inside the relevant closed Schubert cell. This is discussed in detail in the introduction.

Fine, but maybe there’s still an obvious explicit formula for the virtual character of $\mathrm{Mant}_{b,\mu}(\rho)$ at any strongly regular semisimple element.
Here’s an example to illustrate why I think this is a hard problem. I’ll be slightly heuristic about the details; if you want more precision, please leave a comment.

Let’s take $G= \mathrm{GL}_2$ and $\mu=(1,0)$, so we’re in the Lubin-Tate/Drinfeld setting with $G_b(F)=D^\times$ the units in the quaternion algebra over $F$. Let $\rho$ be the trivial representation of $D^\times$. Then $\mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1}$ as a virtual representation of $G(F)$, by an old calculation of Schneider-Stuhler. Note that $\mathrm{St}+\mathbf{1}$ is a principal series representation, hence non-elliptic, so the virtual character of $\mathrm{St}-\mathbf{1}$ on elliptic elements of $G(F)$ is the constant function $-2$. This matches perfectly with the fact that any elliptic $g\in G(F)$ has two fixed points in $\mathbf{P}^1$, both contained in $\Omega^{1}$, and the “naive” local terms of the relevant sheaf $j_!\mathbf{Z}_{\ell}[1]$ at both these points are $-1$. Here $j:\Omega^1 \to \mathbf{P}^1$ is the evident open immersion.

On the other hand, if $g \in G(F)$ is regular semisimple and nonelliptic, then it’s conjugate to some $t=\mathrm{diag}(t_1,t_2)$ with $t_1 \neq t_2 \in F^\times$. In this case there are still two fixed points, but they both lie in the “boundary” $\mathbf{P}^1-\Omega^{1}$. Since $j_!\mathbf{Z}_{\ell}[1]$ restricted to the boundary is identically zero, its naive local terms at both fixed points are zero. Nevertheless, the trace formula still works, so the contribution of the true local terms at these two points must add up to the virtual character of $\mathrm{St}-\mathbf{1}$ evaluated at $t$. This character value can be computed explicitly by van Dijk’s formula, and turns out to be $-2+\tfrac{|t_1| + |t_2|}{|t_1-t_2|}$. So this slightly strange expression needs to emerge from the sum of these two local terms.

Putting these observations together, we see that in this example, the true local terms do not equal the naive local terms at the non-elliptic fixed points. Morally, the analysis in Section 3 of the paper can and should be interpreted as an unravelling of naive local terms (at elliptic fixed points). Hopefully this makes the difficulty clear.

Can you give an example where the error term in Theorem 1.0.2 is nonzero?
Sure, look at the two-dimensional Lubin-Tate local Shimura datum and take $\rho$ trivial again, so $\rho \in \Pi_{\phi}(G_b)$ with $\phi$ the Steinberg parameter. Then $\mathrm{Mant}_{b,\mu}(\rho) = \mathrm{St}-\mathbf{1}$ as discussed in the previous answer, while the sum on the right-hand side in Theorem 1.0.2 is just $2 \mathrm{St}$. So in this case $\mathrm{err}= -(\mathrm{St}+\mathbf{1})$ is, up to sign, a reducible principal series representation.

It would be very interesting to formulate an extension of the Kottwitz conjecture which covered all discrete L-parameters. Since the Kottwitz conjecture is best regarded as a reflection of the Hecke eigensheaf property in Fargues’s conjecture, this is closely related to understanding Fargues’s conjecture for general discrete parameters.

It seems like the Kottwitz conjecture should also make sense for local shtukas over Laurent series fields. Why doesn’t the paper cover this case too?
Good question! The short answer is that all of the p-adic geometry should work out uniformly over Laurent series fields and over finite extensions of $\mathbf{Q}_p$, and indeed Fargues-Scholze handles these two situations uniformly. However, the literature on local Langlands and harmonic analysis for reductive groups over positive characteristic local fields case is pretty thin. This is the main reason we stuck to the mixed characteristic case. For more on the state of the literature, take a look at section 1.6 in Kal16a.

Do the methods of this paper give any information about $\mathrm{Mant}_{b,\mu}$ for non-basic b?
Yes: If $b$ is non-basic, or $b$ is basic and $\rho$ is parabolically induced, our methods can be applied to prove that $\mathrm{Mant}_{b,\mu}(\rho)$ is always a virtual combination of representations induced from proper parabolic subgroups of $G$. This is a soft form of the Harris-Viehmann conjecture. The details of this argument will be given elsewhere (in the sense of Deligne-Mumford).

Any surprising subtleties?
Lemma 6.5.5 was a bracing experience. My first “proof” was absolute nonsense, due to some serious misconceptions I had about harmonic analysis on p-adic groups – big thanks to JT for setting me straight. My second proof didn’t work because I cited a result in a famous article (>100 mathscinet citations) whose proof turns out to be fallacious – big thanks to JFD for setting me straight. Tasho and I devised the current proof, which in hindsight is actually a very natural argument. (I was hoping, apparently unreasonably, for a shortcut.)

Closing thoughts:
-At least for me, the proof of Theorem 6.5.4 is a very striking illustration of the raw Wagnerian power of the machinery developed in [FS21]. This result was totally out of reach a few years ago, but with the machinery of [FS21] in hand, the proof is less than a page! Note that this argument crucially uses the $D_{\mathrm{lis}}$ and Hecke operator formalism of [FS21] with general coefficient rings.
-Let me again emphasize how heavily the paper depends on the recent preprints [LZ20], [Var20], and (of course) [FS21].
-The paper depends, more significantly than most, on a choice of isomorphism $\mathbf{C} \simeq \overline{\mathbf{Q}_{\ell}}$. It’s naturally of interest to wonder how everything depends on this choice. Imai has formulated a conjectural answer to this question. Of course, “independence of $\ell$” results in the etale cohomology of diamonds are probably very hard.

## Several more things

Next year
I’m very pleased that several outstanding young p-adic geometers are coming to Bonn soon: Haoyang Guo, Emanuel Reinecke, and Bogdan Zavyalov will all begin postdoc positions at the MPIM in September, and Ian Gleason will also begin a postdoc at the university in the fall. The group of p-adic people here was already pretty strong, but now it’ll just be off the charts.  And as a bonus, JB will be here starting in July!

Next week
Starting next week, I’m giving three lectures in the Arithmetic Monday seminar on local Shimura varieties and their cohomology. The goal here is to give a gentle introduction to local Langlands and local Jacquet-Langlands correspondences, and the idea that they should be realized in the cohomology of local Shimura varieties (whatever those are). I’ll try to keep everything down to earth and example-based, and I won’t really prove anything. Nevertheless, I hope these lectures might be useful to people starting out in this area.

## Several things

If you look at any detailed introductory text on adic spaces (e.g. the notes of Conrad, Morel, Wedhorn, etc.) you’ll find lots and lots and lots of preliminary material on valuation theory. On the one hand, this isn’t so crazy, since adic spaces are built from valuation spectra, and you have to eat your spinach before you get to have cake. On the other hand, I think this is pretty unfortunate, since valuation theory is incredibly boring and dry, and (more importantly) when you actually work with analytic adic spaces in real life, you never need to worry about most of this material. How often have I worried about horizontal specializations versus vertical specializations, etc.? Essentially never.

Open problems
Suppose you want to study representation theory of p-adic reductive groups with coefficients in some Noetherian ring $R$ with $p \in R^\times$. You might be surprised to learn that the following basic results are all unknown in general (as Jean-Francois Dat kindly pointed out to me), even when $R$ is a DVR:
-parabolic induction preserves finitely generated objects,
-products of cuspidal objects are cuspidal,
-the category is Noetherian,
I believe the state of the art here is a paper of Dat from 2009, which explains the interrelations between these problems and solves them for many groups. Does anyone have in mind a strategy to solve these problems completely? I would love to know.

## The dangers of naming things after people

I spent part of last weekend reading Alice Silverberg’s blog, which is simultaneously depressing and hilarious. Everyone should read it, but you probably won’t enjoy reading it unless you enjoy Coen brothers movies or the short stories of Kafka. Anyway, the following thoughts have been in my head for a few months, but I decided to record them here after reading this.

The cohomology of non-basic local Shimura varieties is described by the “Harris-Viehmann conjecture”, which is formally stated as Conjecture 8.4 in Rapoport-Viehmann’s paper. This story started with a daring and beautiful conjecture of Harris (conjecture 5.2 here), whose formulation however turned out to be slightly incorrect in general, cf. Example 8.3 in RV. The conjecture was then modified by Viehmann, and Rapoport named this modified conjecture the Harris-Viehmann conjecture (footnote 5 in RV).

Unfortunately, Conjecture 8.4 in RV is still not correct as stated: the Weil group action on the summands appearing on the right-hand side needs to be modified by certain half-integral Tate twists. As far as I know, Alexander Bertoloni-Meli is the only person who has publicly pointed out the need for this modification, and Conjecture 3.2.1 in his very cool paper is the only correct formulation of the Harris-Viehmann conjecture in print.

Since the need for these Tate twists was overlooked by a lot of very smart people, it only seems fair to me that Alexander should get credit for his contribution here. The obvious way to do this would be to refer to the Harris–Viehmann–Bertoloni-Meli conjecture, or the Bertoloni-Meli–Harris–Viehmann conjecture. You could pick the second option if you’re a stickler for alphabetical name orders in mathematics, or the first option if you feel (as I do) that Harris’s contribution here deserves priority.

But it gets even worse, because Harris also formulated another conjecture along similar lines (conjecture 5.4 in his article linked above), which has gotten somewhat less attention but which is nevertheless extremely interesting.* It turns out that one can formulate a unified conjecture which encompasses both Harris’s conjecture 5.4 and the Harris–Viehmann–Bertoloni-Meli conjecture. What should it be called? The Harris–Viehmann–Bertoloni-Meli—Harris conjecture? I guess not.

*Here’s a comment from MH: “I was (and am) much more attached to this conjecture than to the one that is called the Harris-Viehmann conjecture, because it required some work to find the right formalism (the parabolics that transfer between inner forms), whereas the other conjecture (independently of the incorrect formulation in my paper) was just the obvious extension of Boyer’s result.”

## Unappreciative

Some things I appreciate but which I used to not appreciate: The Bernstein center, $\infty$-categories, etale cohomology, L-packets, perverse sheaves.

Some things I still don’t appreciate: Monads, endoscopy, rigid cohomology, A-packets, Springer theory.

## 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.