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 $D_{mot}(\mathrm{Spec}\,K)$ .

3 thoughts on “A trick and the decomposition theorem”

Zhiyu says:

October 5, 2020 at 1:56 am

A nice trick by considering the generic part (similar to the proof of 5.3.11 of [BBD])! Theory of weights can be used to show pure perverse sheaf F on X_0 is semisimple after base change to X (but this may not be true over X_0), and Gabber’s purity implies IC sheaves are pure, moreover proper pushforward preserve purity (the reason is like Weil II), so we have the decomposition theorem ([BBD] 5.3.8). Moreover, every simple perverse sheaf is pure ([BBD] 5.3.5).

But do we have an example where the decomposition doesn’t hold over K? For smooth proper map, it’s canonical by the trick of Lefschetz operator (as in Deligne’s thesis?). And I remember in Springer resolution we also use some invariant property of the decomposition to endow the G-action on each piece, maybe it also holds for semismall map? For K finitely generated over its prime filed, it seems we also have the general decomposition theorem over K, see the introduction theorem 1.8 and section 5 of https://arxiv.org/abs/1402.1292.

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arithmetica says:

October 6, 2020 at 1:55 pm

Hi Zhiyu, I might be misunderstanding your question. For any proper morphism $f$ of finite type $K$ -schemes, I think $Rf_{\ast} IC_{X,\mathbf{Q}_{\ell}}$ always splits as the direct sum of its shifted perverse cohomology sheaves, by some combination of the decomposition theorem / relative hard Lefschetz / the splitting criteria in Deligne’s thesis / Chow’s lemma if $f$ is not projective. Are you asking whether each $\phantom{}^{\mathfrak{p}}\mathcal{H}^n(Rf_{\ast}IC_{X,\mathbf{Q}_{\ell}})$ is semisimple?

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1. Zhiyu says:
  
  October 6, 2020 at 2:53 pm
  
  I am interested in a counterexample in “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)”, by the cited paper above it seems this is always true (so each perverse sheaf {^p}H^n is a direct sum of IC sheaves).
  
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