## Euler characteristics and perverse sheaves

Let $X$ be a variety over a separably closed field $k$, and let $A$ be some object in $D^b_c(X,\mathbf{Q}_{\ell})$. Laumon proved the beautiful result that the usual and compactly supported Euler characteristics $\chi(X,A)$ and $\chi_c(X,A)$ are always equal. Recently while trying to do something else, I found a quick proof of Laumon’s result, as well as a relative version, and I want to sketch this here.

Pick an open immersion into a compactification $j:X \to X'$; after a blowup, we can assume that $Z=X' - X$ is an effective Cartier divisor. Write $i:Z \to X'$ for the inclusion of the boundary. By the usual triangle $j_!A \to Rj_*A \to i_*i^* Rj_*A \to$, we reduce to showing that $\chi(X',i_*i^* Rj_*A)=0$. Filtering $A$ by its perverse cohomology sheaves, we reduce further to the case where also $A$ is perverse. Cover $X'$ by open affines $X_n'$ such that $Z_n= Z \cap X_n'$ is the divisor of a function $f_n$. By an easy Mayer-Vietoras argument, it’s now enough to show that for every open $U$ contained in some $X_n'$, $\chi(U,(i_* i^{\ast}Rj_{\ast}A)|U) = 0$.

But now we win: for any choice of such $U \subset X_n'$, there is an exact triangle $R\psi_{f_n}(A|U \cap X) \to R\psi_{f_n}(A|U \cap X) \to (i_* i^{\ast}Rj_{\ast}A)|U \to$ in $D^b_c(U,\mathbf{Q}_{\ell})$ where $R\psi_{f_n}:\mathrm{Perv}(U \cap X) \to \mathrm{Perv}( U \cap Z)$ is the unipotent nearby cycles functor associated with $f_n$, and the first arrow is the logarithm of the unipotent part of the monodromy. Since $\chi(U, -)$ is additive in exact triangles and the first two terms agree, we’re done.

A closer reading of this argument shows that you actually get the following stronger statement: for any $A$, the class $[i_*i^* Rj_*A] \in K_0\mathrm{Perv}(X')$ is identically zero. From here it’s easy to get a relative version of Laumon’s result.

Theorem. Let $f:X \to Y$ be any map of $k$-varieties. Then for any $A\in D^b_c(X,\mathbf{Q}_\ell)$, there is an equality $[Rf_! A]=[Rf_\ast A]$ in $K_0\mathrm{Perv}(Y)$.

## Geometry of the B_dR affine Grassmannian

As many readers of this blog already know, one key result in modern p-adic geometry is Scholze’s theorem that the $B_{\mathrm{dR}}$-affine Grassmannian is an ind-spatial diamond. The proof of this given in the Berkeley notes is a bit tricky and technical: it uses covering by infinite-dimensional objects in a crucial way, as well as an abstract Artin-type representability criterion.  So I’m very pleased to report that Bence Hevesi has given a beautiful new proof of this theorem in his Bonn master’s thesis. Bence’s proof avoids representability criteria or coverings by huge objects. Instead, his idea is to reduce to $\mathrm{GL}_n$ and then construct explicit charts for closed Schubert cells, using moduli of local shtukas at infinite level. You can read Bence’s outstanding thesis here.

## Better than excellent

MH once pointed out the “linguistic trap” Grothendieck created when he defined the notion of an excellent ring: “Suppose somebody finds an even better class of rings? Then what?”

It turns out there IS an even better class of rings/schemes, which occurs naturally in some contexts.

Definition. A scheme $X$ is marvelous if it is Noetherian and excellent, and if $\dim \mathcal{O}_{Y,y} = \dim Y$ for every irreducible component $Y \subset X$ and every closed point $y \in Y$. A ring $A$ is marvelous if $\mathrm{Spec}(A)$ is marvelous.

You can easily check that any marvelous scheme is finite-dimensional. Moreover, it turns out that a Noetherian quasi-excellent scheme is marvelous if and only if the function $x \in |X| \mapsto \dim \overline{ \{ x \} }$ is a true dimension function for $X$ (in a certain technical sense). This function is of course the most naive and clean possibility for a dimension function on any given scheme, but it doesn’t always have the right properties.

Unfortunately, marvelous schemes are so marvelous that, unlike excellent schemes, they aren’t stable under many natural operations, not even under passing to an open subscheme! In fact, $X$ is marvelous if it is covered by marvelous open affines, but the converse fails. You can check that a scheme as simple as $\mathrm{Spec}\mathbf{Z}_p[x]$ isn’t marvelous, even though $\mathrm{Spec}\mathbf{Z}_p$ is marvelous. So regular excellent schemes aren’t always marvelous, and adjoining a polynomial variable can kill marvelousity. I briefly entertained the hope that any Jacobson excellent scheme is marvelous, but this fails too (the scheme $S$ considered in EGAIV3 (10.7.3) is a counterexample).

It’s not all bad news, though:

1. anything of finite type over $\mathbf{Z}$ or a field is marvelous,
2. any excellent local ring is marvelous,
3. any ring of finite type over an affinoid $K$-algebra in the sense of rigid geometry is marvelous,
4. any scheme proper over a marvelous scheme is marvelous; more generally, if $X$ is marvelous and $f: Y \to X$ is a finite type morphism which sends closed points to closed points, then $Y$ is marvelous,
5. if $A$ is a marvelous domain, then the dimension formula holds: $\dim (A/\mathfrak{p}) + \mathrm{ht}\,\mathfrak{p} = \dim A$ for all prime ideals $\mathfrak{p} \subset A$. (Recall that the dimension formula can fail, even for excellent regular domains.)

You might be wondering why I would care about such a stupid and delicate property. The reason is the following. Fix any marvelous scheme $X$ and any $n$ invertible on $X$. Then there is a canonical potential dualizing complex $\omega_{X} \in D^{b}_{c}(X,\mathbf{Z}/n)$ (in the sense of Gabber) which restricts to $\mathbf{Z}/n[2\dim ](\dim)$ on the regular locus of $X$. Here $\dim$ is the (locally constant) dimension of the regular locus, so this numerology is the same as in the case of varieties. Moreover, for any prime $\ell$ invertible on $X$, there is a good theory of $\ell$-adic perverse sheaves on $X$ with the same numerology as in the case of varieties; in particular, the IC complex restricts to $\mathbf{Q}_{\ell}[\dim]$ on the regular locus. (See sections 2.2 and 2.4 of Morel’s paper for more. Note in particular the hypothesis on $X$ in the first sentence of section 2.2: it is exactly the condition that $X$ is marvelous.) This discussion all applies, in particular, when $X=\mathrm{Spec}(A)$ for any $K$-affinoid ring $A$. This turns out to be an important ingredient in my forthcoming paper with Bhargav…

(One more comment: Most real-life examples of marvelous schemes, e.g. examples 1. and 3. above, are also Jacobson. It might be more reasonable to consider the class of marvelous Jacobson schemes, because these are permanent under finite type maps. But on the other hand we lose excellent local rings when we do this.)

## A trick and the decomposition theorem

In this post I’ll talk about a really fun trick Bhargav explained to me yesterday.

Let $K$ be a field with separable closure $C$. Algebraic variety over $K$ means separated $K$-scheme of finite type. Let $\ell$ be a prime invertible in $K$. Perverse sheaf means perverse $\mathbf{Q}_\ell$-sheaf.

If $f:X \to Y$ is a proper map of algebraic varieties over $K$, the decomposition theorem tells you that after base extension to $C$ there is a direct sum decomposition

$Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus_i IC_{Z_i}(\mathcal{L}_i)[n_i]\,\,\,\,\,\,\,\,(\dagger)$

in $D^b_c(Y_C,\mathbf{Q}_\ell)$. Here $Z_i \subset Y_{C}$ is some finite set of closed subvarieties, and $\mathcal{L}_i$ is a lisse $\mathbf{Q}_\ell$-sheaf on the smooth locus of $Z_i$. (My convention is that $IC_{Z}(\mathcal{L}) = j_{!\ast} (\mathcal{L}[\dim Z])$ where $j:Z^{sm} \to X$ is the natural map, so $IC_{X,\mathbf{Q}_\ell} = IC_{X}(\mathbf{Q}_\ell)$. Some people have different conventions for shifts here.)

The decomposition $(\dagger)$ is non-canonical. In particular, it is not $\mathrm{Aut}(C/K)$-equivariant, so it has no reason to descend to an analogous direct sum decomposition of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Indeed, typically there is no such decomposition! However, as Bhargav explained to me, one can still descend a certain piece of $(\dagger)$ to $D^b_c(Y,\mathbf{Q}_\ell)$ when $f$ is projective. This turns out to be good enough for some interesting applications.

To present Bhargav’s trick, let me make a definition. (What follows is a slight reinterpretation of what Bhargav told me, all mistakes are entirely due to me.)

Definition. Let $\mathcal{F}$ be a perverse sheaf on a variety $X$. Let $j:U \to X$ be the inclusion of the maximal dense open subvariety such that $j^\ast \mathcal{F}$ is a (shifted) lisse sheaf. Then we define the generic part of $\mathcal{F}$ as the perverse sheaf $\mathcal{F}^{gen} = j_{!\ast} j^{\ast} \mathcal{F}$.

Note that $\mathcal{F}^{gen}$ is zero if and only if $\mathcal{F}$ is supported on a nowhere-dense closed subvariety. Also, in general there is no map between $\mathcal{F}^{gen}$ and $\mathcal{F}$. However, in some cases $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$:

Proposition. Let $\mathcal{F}$ be a perverse sheaf on a $K$-variety $X$, and suppose that the pullback of $\mathcal{F}$ to $X_{C}$ is a direct sum of IC sheaves. Then $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$.

Proof. Let $j:U \to X$ be as in the definition of the generic part of $\mathcal{F}$, with closed complement $Z \subset X$. Our assumptions together with the definition of the generic part guarantee that $\mathcal{F}|X_C \simeq \mathcal{F}^{gen}|X_C \bigoplus \oplus_i IC_{Z_i}(\mathcal{L}_i)$ for some closed subvarieties $Z_i \subset X_C$ contained in $Z_C$.

Now look at the natural maps $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F} \overset{\alpha}{\to} \mathcal{F} \overset{\beta}{\to} \phantom{}^{\mathfrak{p}}j_{\ast} j^{\ast} \mathcal{F}$. Set $\mathcal{G} = \mathrm{im}\,\alpha$ and $\mathcal{H} = \mathrm{im}\,\beta$. Since $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F}$ does not admit any nonzero quotient supported on $Z$, the composite map $\mathcal{G}|X_C \hookrightarrow \mathcal{F}|X_C \to \oplus_i IC_{Z_i}(\mathcal{L}_i)$ is zero.  Thus $\alpha$ factors over an inclusion $\mathcal{G}|X_C \subset \mathcal{F}^{gen}|X_C$. Moreover, $\mathcal{G}$ has the same generic part as $\mathcal{F}$. This is enough to imply that $\mathcal{G} = \mathcal{F}^{gen}$, so we have a natural inclusion $\mathcal{F}^{gen} \simeq \mathcal{G} \subset \mathcal{F}$. A dual argument shows that $\beta$ factors over a surjection $\mathcal{F} \twoheadrightarrow \mathcal{H} \simeq \mathcal{F}^{gen}$. It is now easy to see that the composite map $\mathcal{F}^{gen} \hookrightarrow \mathcal{F} \twoheadrightarrow \mathcal{F}^{gen}$ is an isomorphism, so $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$. $\square$

Corollary 0. Let $f:X \to Y$ be a projective map of $K$-varieties. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})^{gen}$ is a direct summand of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$.

Proof. The decomposition theorem and the relative hard Lefschetz theorem give a decomposition $Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus \phantom{}^{\mathfrak{p}}\mathcal{H}^i(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})[-i]$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})$ is a direct sum of IC sheaves after pullback to $Y_C$, so we can apply the previous proposition. $\square$

Corollary 1. Let $f:X \to Y$ be a projective alteration of $K$-varieties with $X$ smooth. Then $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $Rf_{\ast}\mathbf{Q}_{\ell}[\dim X]$.

Proof. Check that $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}\mathbf{Q}_{\ell}[\dim X])^{gen}$ by playing with trace maps. Now apply the previous corollary. $\square$

Corollary 2. Let $K/\mathbf{Q}_p$ be a finite extension. Then for any $K$-variety $X$, the $p$-adic intersection cohomology $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a de Rham $G_K$-representation.

Proof. Let $X' \to X$ be a resolution of singularities. The previous corollary shows that $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a direct summand of $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ compatibly with the $G_K$-actions. Since $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ is de Rham and the de Rham condition is stable under passing to summands, we get the result. $\square$

Note that we can’t prove this corollary by applying the decomposition theorem directly out of the box: the decomposition theorem does immediately give you a split injection $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p) \to H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$, but this map is not guaranteed a priori to be $G_K$-equivariant!

Corollary 3. Let $K$ be a finite extension of $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$. If $H^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all smooth projective $K$-varieties $X$, then $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all proper $K$-varieties $X$. In particular, the weight-monodromy conjecture holds for the $\ell$-adic intersection cohomology of all proper $K$-varieties for $K/\mathbf{F}_p((t))$ finite.

Proof. Entirely analogous to the previous proof. $\square$

It would be interesting to know whether Corollary 1 has a “motivic” incarnation. Here I will pretend to understand motives for a minute. Suppose we have an assignment $X \mapsto D_{mot}(X)$ on quasi-projective $K$-varieties, where $D_{mot}(X)$ is a suitable triangulated category of constructible motivic sheaves on $X$ with $\mathbf{Q}$-coefficients. This should come with the formalism of (at least) the four operations $f^{\ast}_{mot}, Rf_{mot\ast}, \otimes, R\mathcal{H}\mathrm{om}$, and with faithful exact $\ell$-adic realization functors $\mathcal{R}_{\ell}: D_{mot}(X) \to D^b_c(X,\mathbf{Q}_{\ell})$ compatible with the four operations. I think this has all been constructed by Ayoub, maybe with some tiny additional hypothesis on $K$? Let $\mathbf{Q}_{X} \in D_{mot}(X)$ denote the symmetric monoidal unit. It then makes sense to ask:

Question. In the setting of Corollary 1, is there an idempotent $e \in \mathrm{End}_{D_{mot}(Y)}(Rf_{mot \ast} \mathbf{Q}_{X}[\dim X])$ such that $\mathcal{R}_{\ell}( e Rf_{mot \ast} \mathbf{Q}_{X}[\dim X]) \simeq IC_{Y,\mathbf{Q}_{\ell}}$ for all $\ell$?

This would imply that the split injections $IH^{\ast}(Y_{\overline{K}},\mathbf{Q}_\ell) \to H^{\ast}(X_{\overline{K}},\mathbf{Q}_\ell)$ provided by Corollary 1 can be chosen “independently of $\ell$”, i.e. that they are the $\ell$-adic realizations of some split injection in $D_{mot}(\mathrm{Spec}\,K)$.

## Rampage six weeks in

When we first discussed the idea of running a weekly online seminar on p-adic geometry, I figured there would be around 30 or so participants at each talk. It’s a bit crazy, then, that the audience for the first six RAMpAGe talks has ranged from 80 to over 225 people. A huge thanks to everyone for their participation and interest! We have many more excellent speakers in store for you! If you have any feedback for us, please don’t hesitate to get in contact.

Two more points:

1. We are doing our best to post notes and videos (at the speakers’ discretion) for each talk on the seminar website linked above. Hopefully we will post notes for every talk!
2. Bogdan Zavyalov planned to mention the sad case of Azat Miftakhov at the end of his talk. Due to the extended mathematical discussion, this ended up not happening, so I am mentioning it here instead. Please please go here and read more.

## Diversion

The following excerpt from Wikipedia made me laugh out loud:

The large power output of the Sun is mainly due to the huge size and density of its core (compared to Earth and objects on Earth), with only a fairly small amount of power being generated per cubic metre. Theoretical models of the Sun’s interior indicate a maximum power density, or energy production, of approximately 276.5 watts per cubic metre at the center of the core,[76] which is about the same rate of power production as takes place in reptile metabolism or a compost pile.

Takeaway: If the sun were a giant ball of lizards, nothing would change.

## Brain teaser: mysterious moduli and local Langlands

Fix an integer $n>1$. Let $X$ denote the moduli space of triples $(\mathcal{E}_1, \mathcal{E}_2,f)$ where $\mathcal{E}_i$ is a vector bundle of rank $n$ on the Fargues-Fontaine curve which is trivial at all geometric points, and $f: \mathcal{E}_1 \oplus \mathcal{E}_2 \to \mathcal{O}(1/2n)$ is an injection which is an isomorphism outside the closed Cartier divisor at infinity.

Brain teaser a. Prove that $X$ is a locally spatial diamond over $\breve{\mathbf{Q}}_p$ with a Weil descent datum to $\mathbf{Q}_p$.

Now, let $D$ be the division algebra over $\mathbf{Q}_p$ of invariant $1/2n$, and let $\tau$ be an irreducible representation of $D^\times$ whose local (inverse) Jacquet-Langlands correspondent is supercuspidal. Note that $D^\times$ acts on $X$ by its natural identification with $\mathrm{Aut}(\mathcal{O}(1/2n))$.

Brain teaser b. Prove that the geometric etale cohomology of $X$ satisfies the following:

$R\Gamma_c(X_{\mathbf{C}_p},\overline{\mathbf{Q}_\ell})\otimes_{D^\times} \tau \cong \varphi_{\tau}[1-2n](\tfrac{1-2n}{2})$ if $\tau$ is orthogonal, and $R\Gamma_c(X_{\mathbf{C}_p},\overline{\mathbf{Q}_\ell})\otimes_{D^\times} \tau \cong 0$ if $\tau$ is not orthogonal.

Here $\varphi_\tau$ denotes the Langlands parameter of $\tau$.

It is probably not fair to call these brain teasers. Anyway, here is one big hint: the infinite-level Lubin-Tate space for $\mathrm{GL}_{2n}$ is naturally a $\mathrm{GL}_n(\mathbf{Q}_p)^2$-torsor over $X$, by trivializing the bundles $\mathcal{E}_i$.

## Brain teaser: generic perversity on fibers

Inspired by Shizhang’s Rampage talk last week, here is a brain teaser. Feel free to post your solution in the comments!

Let $f:X \to Y$ be any map of irreducible complex varieties, and let $\mathcal{F}$ be a perverse sheaf on $X$. Prove that there is a dense open subset $U \subset Y$ such that for any closed point $y \in U$, the shifted restriction $(\mathcal{F}|X_y)[-\dim Y]$ is a perverse sheaf on the fiber $X_y$.

## Rampage!

I’m excited to announce a new weekly online-only research seminar on p-adic geometry and related topics, organized by Arthur-César Le Bras, Jared Weinstein, and myself. We will “meet” on Zoom on Thursdays at 16:00 UTC (that’s 9 am in California, noon in Boston, 5 pm in London, 6 pm in Bonn…). Bhargav Bhatt will give the first talk on June 18.

Please follow the instructions at the seminar website here to get the Zoom link. We will also keep an up-to-date schedule on researchseminars.org here.

All credit to Jared for the name!

## Things to do during a pandemic

• Try and fail to buy toilet paper at Edeka.
• Make ketchup from scratch.
• Watch 30 episodes of Iron Chef.
• Annoy your wife by following her around the apartment, or by listening to old blues songs too loudly.