The monoidal structure on monodromic sheaves

• The monoidal structure
• Functoriality of monodromic sheaves

Lemma 1. If $1 \to H_1 \to H_2 \to H_3 \to 1$ is a short exact sequence of connected linear algebraic groups, then $$ \pi_1^\mathrm{c}(H_1) \to \pi_1^\mathrm{c}(H_2) \to \pi_1^\mathrm{c}(H_3) \to 1 $$ is exact.

Last time we proved that the right map is surjective, and wanted to prove exactness in the middle. We just need to show that $$ \ker(\tilde{\pi} \colon \tilde{H}_ 2 \to \tilde{H}_ 3) $$ is connected. But if we look at $$ \tilde{H}_ 2 / \ker(\tilde{\pi})^0 \twoheadrightarrow \tilde{H}_ 2 / \ker(\tilde{\pi}) = \tilde{H}_ 3 $$ then the source is connected, so we must have that $\ker(\tilde{\pi})^0 = \ker(\tilde{\pi})$.

Anyways, we had for $H$ a connected algebraic group the ind-finite ind-scheme $$ \mathcal{R}_ {\pi_1^\mathrm{c}(H), \mathbb{G}_ m}(\Lambda) = \lbrace \text{continuous } \pi_1^\mathrm{c}(H) \to \Lambda^\times \rbrace $$ over $\mathbb{Z}_ \ell$. This can also be thought of as the space of $\Lambda$-linear character sheaves, where $\chi$ corresponds to $\mathrm{Ch}_ \chi$.

Now when $\Lambda$ is an algebraic extension of $\mathbb{F}_ \ell$ or $\mathbb{Q}_ \ell$, or an integral extension of $\mathbb{Z}_ \ell$, we want to look at the (∞-)category $\mathsf{Shv}(H, \Lambda)$ which is the unbounded derived category of $\Lambda$-étale sheaves on $H$. There are inclusions $$ \mathsf{Mod}_ \Lambda \to \mathsf{Shv}_ {\chi\mathrm{-mon}}(H, \Lambda) \subseteq \mathsf{Shv}_ \mathrm{mon} \subseteq \mathsf{Shv}(H, \Lambda), $$ where

• $\mathsf{Shv}_ {\chi\mathrm{-mon}}$ is the full $\Lambda$-linear (presentable, stable, tensored over $\mathsf{Mod}_ \Lambda$) subcategory generated by $\mathrm{Ch}_ \chi$,
• $\mathsf{Shv}_ \mathrm{mon}(H, \Lambda)$ is the full $\Lambda$-linear subcategory generated by $\lbrace \mathrm{Ch}_ {\chi^\prime} \rbrace$ here $\chi^\prime \colon \pi_1^\mathrm{c}(H) \to \Lambda^{\prime \times}$ for $\Lambda^\prime$ a finite $\Lambda$-algebra.

These are all compactly generated, with these character sheaves being the compact generators. The inclusions all have continuous $\Lambda$-linear right adjoints, as they send compact objects to compact objects. Here, the compact objects can be identified as $$ \mathsf{Shv}_ \mathrm{mon}(H, \Lambda)^\omega = \lbrace \mathscr{F} \in D_c^b(H, \Lambda) = \mathsf{Shv}(H, \Lambda)^\omega : \mathcal{H}^i \mathscr{F} \in \mathsf{Shv}_ \mathrm{mon}(H, \Lambda)^{\omega,\heartsuit} \rbrace, $$ where $\mathsf{Shv}_ \mathrm{mon}(H, \Lambda)^{\omega,\heartsuit}$ is the thick abelian subcategory generated by $\mathrm{Ch}_ {\chi^\prime}$. Then we see that there is an equivalence $$ \mathrm{Ch} \colon \mathsf{Coh}(\mathcal{R}_ {\pi_1^\mathrm{c}(H), \mathbb{G}_ m})^\heartsuit \xrightarrow{\cong} \mathsf{Shv}_ \mathrm{mon}(H, \Lambda)^{\omega, \heartsuit}; \quad \mathscr{O}_ \chi \mapsto \mathrm{Ch}_ \chi. $$

The monoidal structure

Note that the category $\mathsf{Shv}(H, \Lambda)$ has a natural monoidal structure given by $$ \mathscr{F} \star \mathscr{G} = m_\ast(\mathscr{F} \boxtimes \mathscr{G}), $$ where $m \colon H \times H \to H$ is the multiplication map. The unit is given by $\delta_1 = (\lbrace 1 \rbrace \to H)_ \ast \Lambda$.

Proposition 2. 1. The full subcategory $\mathsf{Shv}_ \mathrm{mon}(H) \subseteq \mathsf{Shv}(H)$ is a bimodule.

1. The subcategory $\mathsf{Shv}_ \mathrm{mon}(H)$ with the restricted monoidal structure has a monoidal unit $\tilde{\mathrm{Ch}}$.
2. The inclusion $\mathsf{Shv}_ \mathrm{mon}(H) \hookrightarrow \mathsf{Shv}(H)$ has a right adjoint $\mathrm{Av}^\mathrm{mon} \colon \mathsf{Shv}(H) \to \mathsf{Shv}_ \mathrm{mon}(H)$, and it moreover has a natural monoidal structure.

There are also similar statements for $\mathsf{Shv}_ {\chi\mathrm{-mon}}(H)$.

We start with some lemmas.

Lemma 3. We have an equivalence $$ \mathsf{Shv}_ \mathrm{mon}(H) \otimes \mathsf{Shv}_ \mathrm{mon}(H) \xrightarrow{\cong} \mathsf{Shv}_ \mathrm{mon}(H \times H). $$

Lemma 4. Let $\mathscr{F} \in \mathsf{Shv}(H)$ such that $m^\ast \mathscr{F} \cong \mathrm{Ch}_ \chi \boxtimes \mathscr{F}$ for some $\chi$. Then $\mathscr{F} \in \mathsf{Shv}_ \mathrm{mon}(H)$.

Lemma 5. For any $\mathscr{F} \in \mathsf{Shv}(H)$, we have $\mathrm{Ch}_ \chi \star \mathscr{F} \in \mathsf{Shv}_ \mathrm{mon}(H)$.

For (1), because $\star$ commutes with colimits, it suffices to show that if $\mathrm{Ch}_ \chi$ is a character sheaf and $\mathscr{F} \in \mathsf{Shv}(H)^\omega$ then $\mathrm{Ch}_ \chi \star \mathscr{F} \in \mathsf{Shv}_ \mathrm{mon}(H)$. Now this follows from the lemma.

Lemma 6. Let $\tilde{\mathrm{Ch}} = \mathrm{Av}^\mathrm{mon}(\delta_1)$. Then we have $\mathrm{Av}^\mathrm{mon}(\mathscr{F}) = \tilde{\mathrm{Ch}} \star \mathscr{F}$ for all $\mathscr{F} \in \mathsf{Shv}(H)$.

This proves (2), and the following is (3).

Lemma 7. For $\mathscr{F}, \mathscr{G} \in \mathsf{Shv}(H)$, there is a canonical isomorphism $$ \mathrm{Av}^\mathrm{mon}(\mathscr{F} \star \mathscr{G}) \cong \mathrm{Av}^\mathrm{mon}(\mathscr{F}) \star \mathrm{Av}^\mathrm{mon}(\mathscr{G}). $$

Example 8. Let us consider the case of $H = \mathbb{G}_ a / \mathbb{F}_ p$. Then for each $\phi \colon \mathbb{G}_ a(\mathbb{F}_ p) \to \Lambda^\times$, we have an Artin–Schreier sheaf $\mathrm{Ch}_ \phi$ on $\mathbb{G}_ a$. Then we have $$ \mathsf{Mod}_ \Lambda \cong \mathsf{Shv}_ {\phi\mathrm{-mon}}(\mathbb{G}_ a) \subseteq \mathsf{Shv}(\mathbb{G}_ a), $$ where $\mathrm{Ch}_ \phi = \mathrm{Ch}_ {\phi-\mathrm{mon}}$.

Example 9. The more interesting case is the torus, say $H = \mathbb{G}_ m$. In this case we have for $\chi \colon \mathbb{F}_ q^\times \to \Lambda^\times$ we have the Kummer local system $\mathrm{Ch}_ \chi$ on $\mathbb{G}_ m$. Then there is a functor $$ \mathsf{Mod}_ \Lambda \hookrightarrow \mathsf{Shv}_ {\chi\mathrm{-mon}}(\mathbb{G}_ m) $$ that is not an equivalence, so $\mathrm{Ch}_ \chi \neq \mathrm{Ch}_ {\chi\mathrm{-mon}}$. Even when $\chi$ is the trivial character, we have $$ \mathrm{Ch}_ {1\mathrm{-mon}} = \varinjlim_n \mathscr{L}_ n $$ where $\mathscr{L}_ n$ is the unipotent local system on $\mathbb{G}_ m$ with monodromy $$ \begin{pmatrix} 1 \br 1 & 1 \br & \ddots & \ddots \br & & 1 & 1 \end{pmatrix}. $$ In general, $\mathrm{Ch}_ {\chi\mathrm{-mon}} = \mathrm{Ch}_ \chi \otimes \mathrm{Ch}_ {1\mathrm{-mon}}$.

Functoriality of monodromic sheaves

Proposition 10. Let $f \colon H_1 \to H_2$ be a homomorphism of connected linear algebraic groups.

1. The functor $f^\ast \colon \mathsf{Shv}(H_2) \to \mathsf{Shv}(H_1)$ sends $\mathsf{Shv}_ \mathrm{mon}(H_2) \to \mathsf{Shv}_ \mathrm{mon}(H_1)$. Moreover, it admits a continuous right adjoint $f_\ast^\mathrm{mon} = \mathrm{Av}^\mathrm{mon} \circ f_\ast$.
2. If $f$ is surjective, we have $f_\ast^\mathrm{mon} = f_\ast$, and moreover $f_! = f_\ast$ up to shifts when restricted to $\mathsf{Shv}_ \mathrm{mon}(H_1)$.

The first part is clear, because character sheaves are sent to character sheaves. For the second part, we need to show that $$ H^i(f_\ast \mathrm{Ch}_ \chi) \in \mathsf{Shv}_ \mathrm{mon}(H_2)^{\omega, \heartsuit}. $$ By the lemma from last time, it is enough to show that $$ f^\ast \mathcal{H}^i(f_\ast \mathrm{Ch}_ \chi) \in \mathsf{Shv}_ \mathrm{mon}(H_1)^{\omega,\heartsuit}. $$ On the other hand, we have that $H_1 \times_{H_2} H_1 = H_1 \times \ker f$, so using base change we can identify $$ f^\ast \mathcal{H}^i(f_\ast \mathrm{Ch}_ \chi) = \mathrm{Ch}_ \chi \otimes H^i(\mathrm{Ch}_ \chi \vert_{\ker f}), \quad f^\ast \mathcal{H}^i(f_! \mathrm{Ch}_ \chi) = \mathrm{Ch}_ \chi \otimes H_c^i(\mathrm{Ch}_ \chi \vert_{\ker f}). $$ Now it is clear from the first equality that $f^\ast \mathcal{H}^i(f_\ast \mathrm{Ch}_ \chi)$ is monodromic.

We now need to show that $f_! = f_\ast$ up to shifts, when restricted to monodromic sheaves. We can treat the cases when $\ker f$ is finite and $\ker f$ is connected separately. When $\ker f$ is finite, we always have $f_\ast = f_!$ so there is nothing to do. When $\ker f$ is connected, let $B$ be the Borel of $\ker f$ and factorize $$ H_1 \to H_1 / B \to H_2. $$ Then we are reduced to showing that $$ R\Gamma_c(B, \mathrm{Ch}_ \chi) = R\Gamma(B, \mathrm{Ch}_ \chi)[d]. $$ This can be reduced to the case of $\mathbb{G}_ a$ and $\mathbb{G}_ m$.

Proposition 11. Let $H$ be a torus. Then there exists a t-exact monoidal equivalence $$ \mathrm{Ch} \colon \mathsf{Ind}(\mathsf{Coh}(\mathcal{R}_ {I_F^t,\hat{H}})) \xrightarrow{\cong} \mathsf{Shv}_ \mathrm{mon}(H). $$ If $f \colon H_1 \to H_2$ corresponds to $\hat{f} \colon \hat{H}_ 2 \to \hat{H}_ 1$, then $f^\ast$ corresponds to $\hat{f}_ \ast^\mathrm{IndCoh}$ and $f_\ast^\mathrm{mon}$ corresponds to $\hat{f}^{\mathrm{IndCoh}!}$.