Algebraic Geometry of Infinite Type
Now let $F$ be a local field (e.g. $F=\mathbb{F}_ q((\omega)), F/\mathbb{Q}_ p$ finite extension). Let $F\supseteq \mathcal{O}_ F\supseteq \mathfrak{m}_ F$ and $\kappa_ F:=\mathcal{O}_ F/\mathfrak{m}_ F$ the residue field. Set $k:=\overline{\kappa_ F}$ the algebraic closure of the residue field. Let $\varpi$ be a uniformizer of $\mathcal{O}_ F$.
🔗Isocrystals
Definition 1. Let $LG$ be the loop group of $F$ defined over $\kappa_ F$: $$\begin{split} LG:\mathrm{CAlg}_ {\kappa_ F}^{\mathrm{Perf}} &\rightarrow \mathrm{Grps}\ R&\mapsto G(W_ {\mathcal{O}}(R)[\frac{1}{\varpi}]), \end{split}$$ where $$W_ {\mathcal{O}}(R):=\varprojlim_ {n}(W(R)\otimes_ {\mathbb{Z}_ p}\mathcal{O}_ F/\varpi^n).$$
In particular, $LG(\kappa)=G(F)$.
Then one can prove that $LG=\varinjlim_ {n}X_ n$, where each $X_ n$ is a perfect affine scheme over $\kappa_ F$, though one can not require every $X_ n$ is a group scheme.
Let $\mathcal{G}/\mathcal{O}_ F$ be a smooth integral model of $G/F$. Then we can define
Definition 2. $$\begin{split} L^+\mathcal{G}: \mathrm{CAlg}^{\mathrm{Perf}}_ {\kappa_ F} &\rightarrow \mathrm{Grps}\ R&\mapsto \mathcal{G}(W_ {\mathcal{O}}(R)). \end{split}$$
Then $L^+\mathcal{G}$ is a closed in $LG$.
In fact, $L^+\mathcal{G}$ is represented by a group prefect affine scheme over $\kappa_ F$. For each $n$, let $$\begin{split} L^n\mathcal{G}: \mathrm{CAlg}^{\mathrm{Perf}}_ {\kappa_ F} &\rightarrow \mathrm{Grps}\ R&\mapsto \mathcal{G}(W_ {\mathcal{O}}(R)/\varpi^n). \end{split}$$ Then $L^+\mathcal{G}=\varprojlim_ {n}L^n\mathcal{G}$, where each $L^n\mathcal{G}$ is represented by an affine group perfect scheme.
Note that each $L^{n+1}\mathcal{G}\rightarrow L^n\mathcal{G}$ is a surjection, whose fibers are (perfection of ) affine spaces. We also observe that $L^1\mathcal{G}=\mathcal{G}\otimes_ {\mathcal{O}}\kappa_ F$ and $L^0\mathcal{G}={1}$.
Example 3. If $G/F$ extends to a reductive group scheme $\underline{G}/\mathcal{O}_ F$, then $\underline{G}(\mathcal{O}_ F)$ is called a hyperspecial subgroup of $G(F)$. We call $L^+\underline{G}$ hyperspecial.
Example 4. In fact, $G/F$ always admits a smooth affine integral model $\mathcal{I}$ called the Iwahori, such that
- $\mathcal{I}\otimes_ {\mathcal{O}_ F}\kappa_ F$ is connected solvable;
- $LG/L^+\mathcal{I}$ is ind-proper.
Remark 5. If $\underline{G}/\mathcal{O}_ F$ exists, then we have Cartesian square
Fact: If $\mathcal{G}/\mathcal{O}_ F$ is an affine flat integral model of $G/F$, then $LG/L^+\mathcal{G}$ is represented by an ind-scheme $$LG/L^+\mathcal{G}=\varinjlim{}X_ n$$ where $X_ n$ is of (perfectly) finite type over $\kappa_ F$.
We always have $$1\rightarrow (L^+\mathcal{I})^u\rightarrow L^+\mathcal{I}\rightarrow S\rightarrow 1$$ such that
- $L^+\mathcal{I}$ is pro-unipotent;
- $S$ is a torus;
- there exists a splitting $S\rightarrow L^+\mathcal{I}$.
Then motivated by the case of finite fields (or $L^1\mathcal{G}$), we want to consider the correspondence
Definition 6. We define $[LG/\mathrm{Ad}_ \sigma LG]_ {\textrm{'et}}$ to be the 'etale sheafification of $$R\mapsto LG(R)/\mathrm{Ad}_ {\sigma}LG(R)=G(W_ {\mathcal{O}}(R)[\frac{1}{\varpi}])/\mathrm{Ad}_ \sigma G(W_ {\mathcal{O}}(R)[\frac{1}{\varpi}]).$$
Definition 7. Let $\mathrm{Isoc}_ G^{\textrm{'et}}$ be the moduli stack, defined by $$R\mapsto{(\mathcal{E},\varphi)|\mathcal{E}\textrm{ is a }G\textrm{-torsor on }D_ R^\times:=\mathrm{Spec}(W_ {\mathcal{O}}(R)[\frac{1}{\varpi}]), \textrm{s.t. there exists} R\xrightarrow{\textrm{'etale}}R’, \mathcal{E}|_ {D_ {R’}^\times}\textrm{ is trivial}, \varphi:\mathcal{E}\xrightarrow{\cong}\sigma_ R^\ast\mathcal{E}}.$$
Lemma 8. There is an identification $[LG/\mathrm{Ad}_ \sigma LG]_ {\textrm{'et}}\cong \mathrm{Isoc}_ G^\textrm{'et}$.
Remark 9. It is not clear if every $G$-torsor on $D_ R^\times$ can be trivialized 'etale locally on $\mathrm{Spec} R$.
- There exists a counterexample when $\mathcal{O}=\kappa_ F[[\varpi]]$ and allowing $R$ to be imperfect.
- It is known that every $G$-torsor on $D_ R^\times$ can be trivialized after passing to suitable $h$-cover $R\rightarrow R’$.
In general, we have $$\mathrm{Isoc}_ G^{\textrm{pro'et}}\hookrightarrow\mathrm{Isoc}_ G^\textrm{'et}\hookrightarrow \mathrm{Isoc}_ G^h,$$ where $$\mathrm{Isoc}_ G^\textrm{pro'et}(R)=\{(\mathcal{E},\varphi)| \mathcal{E}\textrm{ is a }G\textrm{-torsor on }D_ R^\times:=\mathrm{Spec}(W_ {\mathcal{O}}(R)[\frac{1}{\varpi}]),\textrm{s.t. there exists} R\xrightarrow{\textrm{pro-'etale}}R’,\mathcal{E}|_ {D_ {R’}^\times}\textrm{ is trivial}, \varphi:\mathcal{E}\xrightarrow{\cong}\sigma_ R^\ast\mathcal{E}\},$$ and $$\mathrm{Isoc}_ G^\textrm{h}(R)={(\mathcal{E},\varphi)|\mathcal{E}\textrm{ is a }G\textrm{-torsor on }D_ R^\times:=\mathrm{Spec}(W_ {\mathcal{O}}(R)[\frac{1}{\varpi}]), \varphi:\mathcal{E}\xrightarrow{\cong}\sigma_ R^\ast\mathcal{E}},$$
Proposition 10. For any $K\hookrightarrow K’$ inclusion of algebraically closed fields, we have $$\mathrm{Isoc}^\textrm{'et}_ G(K)\xrightarrow{\cong}\mathrm{Isoc}^\textrm{'et}_ G(K’).$$
The field $L:=W_ \mathcal{O}(K)[\frac{1}{\varpi}]$ has cohomological dimension 1, and we define $$B(G):=|\mathrm{Isoc}^\textrm{pro'et}_ G(K)|=|\mathrm{Isoc}^\textrm{'et}_ G(K)|=|\mathrm{Isoc}^h_ G(K)|,$$ namely, $B(G)=G(L)/\mathrm{Ad}_ \sigma G(L)\cong G(\breve{F}/)/\mathrm{Ad}_ \sigma G(\breve{F})$.
Fact: Assume that $G$ is reductive, then
- There is an injection $$\begin{split} (\nu,\kappa_ G):B(G)&\hookrightarrow X_ *(T)_ \mathbb{Q}^{I_ F}\times \pi_ 1(G)_ {\Gamma_ {\widetilde{F}/F}},\ b&\mapsto (\nu_ b,\kappa_ G(b)), \end{split}$$ where $\Gamma_ {\widetilde{F}/F}$ denotes the finite quotient of $\Gamma_ F=\mathrm{Gal}(\overline{F}/F)$, such that the action of $\Gamma_ F$ on the root data of $G$ factors through the quotient.
- there exists a partial order: $b_ 1\leq b_ 2$ if and only if $\kappa_ G(b_ 1)=\kappa_ G(b_ 2)$ and $\nu_ {b_ 2}-\nu_ {b_ 1}$ is a riational positive combination of simple coroots.
- Minimal elements $B(G)_ {bsc}\cong \pi_ 1(G)_ {\Gamma_ {\widetilde{F}/F}}$, and $b\in B(G)$ satisfies $\nu_ {b}\in X_ *(Z_ G)$.
- For $b\in G(L)$, $\mathrm{Aut}(b)(\overline{\kappa_ F}):={g\in G(L):g^{-1}b\sigma(g)=b}$. More generally, $$\mathrm{Aut}(b): \mathrm{CAlg}^{\mathrm{pref}_ {\overline{\kappa_ F}}} \rightarrow \mathrm{Grps}, \quad R \mapsto \mathrm{Aut}_ {\mathrm{Isoc}(R)}(b)$$ and $$J_ b: F\textrm{-Alg} \rightarrow \mathrm{Grps}, \quad R \mapsto {g\in G(R\times_ F\breve{F}):g^{-1}b\sigma(g)=b}$$ In particular, $\mathrm{Aut}(b)(\overline{\kappa_ F})=J_ b(F)\hookrightarrow J_ b(\breve{F})$, is conjuagate to $Z_ G(\nu_ b)(\breve{F})$ a levi.
- As a corollary, $b$ is basic if and only if $J_ b(K)=G(\breve{F})$.
Proposition 11. If $G$ is quasi-split, then
- $J_ b$ is an inner form of a Levi of $G$.
- $\mathrm{Aut}(b)(\overline{\kappa_ F})=\underline{J_ b}(F)$ regarded as an ind-scheme over $\overline{\kappa_ F}$.
Assume that $G$ is quasi-split. Consider $$G\supseteq B\supseteq T\supseteq S\supseteq A,$$ where
- $B$ is a Borel,
- $T$ is a maximal torus,
- $S$ is a maximal $F^{\textrm{ur}}$-split torus,
- $A$ is a maximal $F$-split torus.
Then $$Z_ G(S)=T$$ and $$N_ G(S)=N_ G(T).$$
Definition 12. We define the extended affine Wely group $\widetilde{W}$ by $$\widetilde{W}:=N_ G(S)(\breve{F})/Z_ G(S)(\mathcal{O}_ {\breve{F}}).$$
Define $$W_ 0:=N_ G(S)(\breve{F})/Z_ G(S)(\breve{F}),$$ we have a short exact sequence (compatible with $\sigma$-action) $$1\rightarrow X_ *(T)_ {I_ F}\rightarrow \widetilde{W}\rightarrow W_ 0\rightarrow 1.$$ Observe that $H^1(Z_ G(S)(\mathcal{O}_ F),\mathrm{Gal})=0$ by Lang’s isogeny theorem and compatibility with taking inverse limit.
We similarly define $B(\widetilde{W}):=\widetilde{W}//\mathrm{Ad}_ \sigma \widetilde{W}$. We then have a diagram
Proposition 13. The natural map $B(N_ G(S))\rightarrow B(G)$ is a surjection.
For any $w$, there exists some integer $n$, such that $(w\sigma)^n:=w\cdot \sigma(w)\cdot \sigma^2(w)\cdot\cdots\cdot \sigma^{n-1}(w)\in X_ *(T)_ {I_ F}$.
Define $\widetilde{\nu}(w):=\frac{(w\sigma)^n}{n}$, which is independent of the choice of $n$.
Definition 14. An element in $\widetilde{W}$ is called $\sigma$-straight, if $$l(w)=\frac{l(w\sigma)^n}{n}=<{\nu}(w),2\rho>\in\mathbb{Z}_ {\geq0}$$ for $n>>0$, where $\nu(w)$ is translating $\widetilde{\nu}(w)$ to dominant chamber.
Definition 15. A $\sigma$-conjugacy class in $\widetilde{W}$ is called $\sigma$-straight, if it contains a $\sigma$-straight element.
Theorem 16. $B(\widetilde{W})_ {\textrm{str}}\xrightarrow{\cong}B(G)$.
Definition 17. We define local Shtukas $$\mathrm{Sht}^{\textrm{loc}}:=[LG/\mathrm{Ad}_ \sigma \mathcal{I}_ w]_ \textrm{'et},$$ where $\mathcal{I}_ w:=L^+\mathcal{I}$ denotes the Iwahori.
We can also describe local shtukas via moduli $$\mathrm{Sht}^{\textrm{loc}}(R)={(\mathcal{E},\phi):\mathcal{E}\textrm{ is an }\mathcal{I}_ w\textrm{-torsor on }D_ R:=\mathrm{Spec} W_ \mathcal{O}(R), \phi:\mathcal{E}|_ {D_ R^\times}\xrightarrow{\cong}\sigma^\ast\mathcal{E}|_ {D_ R^\times}}.$$ Then we have a natural map $$\mathrm{Sht}^{\textrm{loc}}=[LG/\mathrm{Ad}_ \sigma \mathcal{I}_ w]_ \textrm{'et}\rightarrow [LG/\mathrm{Ad}_ \sigma LG]_ \textrm{'et}=\mathrm{Isoc}_ G^\textrm{'et}.$$
Proposition 18. The natural map $ \mathrm{Sht}^{\textrm{loc}}\rightarrow \mathrm{Isoc}_ G^\textrm{'et}$ is representable by an ind-proper algebraic space.
Let $\mathcal{P}$ be a parahoric subgroup. Then we have $$\mathrm{Sht}^{\textrm{loc}}\rightarrow\mathrm{Sht}^{\textrm{loc}}_ \mathcal{P}\xrightarrow{\mathrm{Nt}}\mathrm{Isoc}^\textrm{'et},$$ where $$\mathrm{Sht}^{\textrm{loc}}_ \mathcal{P}:=[LG/\mathrm{Ad}_ \sigma L^+\mathcal{P}].$$ Fact: $\mathrm{Nt}$ is representable by ind-finitely presented ind-proper algebraic spaces:
Recall that $$|\mathrm{Isoc}^\textrm{'et}_ G(K)|=B(G)\hookrightarrow{(\nu,\kappa_ G)}X_ *(T)_ \mathbb{Q}^{I_ F}\times\pi_ 1(G)_ {\Gamma_ {\widetilde{F}/F}}.$$ We have a short exact sequence $$1\rightarrow W_ {\textrm{aff}}\rightarrow \widetilde{W}\xrightarrow{\widetilde{\kappa}}\pi_ 1(G)_ {I_ F}\rightarrow 1,$$ which admits a splitting $\pi_ 1(G)_ {I_ F}\rightarrow \widetilde{W}$.
Definition 19. For $b\in B(G)$, we define a subfunctor $$\mathrm{Isoc}_ {\leq b}^\textrm{'et}\hookrightarrow\mathrm{Isoc}^\textrm{'et}$$ by $$R\mapsto {(\mathcal{E},\phi)\in\mathrm{Isoc}^\textrm{'et}(R):\forall \textrm{ geometric point }x\rightarrow \mathrm{Spec} R, (\mathcal{E}_ x,\phi_ x)\leq b}.$$ We define $\mathrm{Isoc}_ b^\textrm{'et}$ by replacing $\leq b$ with $=b$.
Proposition 20. The immersions $\mathrm{Isoc}_ b^\textrm{'et}\xrightarrow{j_ b}\mathrm{Isoc}_ {\leq b}^\textrm{'et}\xrightarrow{i_ {\leq b}}\mathrm{Isoc}^\textrm{'et}$ satisfy
- $i_ {\leq b}$ is finitely presented closed immersion;
- $j_ b$ is affine and quasi-compact open immersion.
Theorem 21. $$\mathrm{Isoc}_ b^\textrm{'et}=[*/\underline{J_ b(F)}]_ {\textrm{pro'et}}.$$
For $b\in B(G)$, and pick $w_ b\in\widetilde{W}$ a $\sigma$-straight element whose $\sigma$-conjugacy class corresponds to $b$. Then $$\mathcal{I}_ b:=\mathrm{Aut}(\dot{w}_ b):R\mapsto {g\in \mathcal{I}_ w(R):\dot{w}_ b\sigma(g)\dot{w}_ b^{-1}=g}$$ is a subfunctor of $\mathcal{I}_ w$, and also a subfunctor of $$\underline{J_ b(F)}\cong\mathrm{Aut}(b):R\mapsto {g\in LG(R):b\sigma(g)b^{-1}=g}.$$
Proposition 22. We have $$\mathcal{I}_ w\backslash LG/\mathcal{I}_ w\cong\widetilde{W}.$$
Proposition 23. $$\mathrm{Sht}_ w^{\textrm{loc}}:={LG}_ w/\mathrm{Ad}_ \sigma \mathcal{I}_ w\cong \mathbf{B}_ {\textrm{prof'et}}\mathcal{I}_ b.$$ where prof'et is short for pro-finite-'etale.
Explicitly, $LG_ w=\mathcal{I}_ w\dot{w}\mathcal{I}_ w$ for $w\in\widetilde{W}$.
Example 24. Take $w=1$, then $\mathrm{Sht}_ 1^{\textrm{loc}}=[\mathcal{I}_ w/\mathrm{Ad}_ \sigma\mathcal{I}_ w]_ \textrm{'et}\cong [*/\mathcal{I}(\mathcal{O}_ F)]_ {\textrm{prof'et}}$.
Lemma 25. We have
Let $\mathcal{O}$ be a $\sigma$-conjugacy class in $\widetilde{W}$ and $\mathcal{O}_ {\textrm{min}}\in\mathcal{O}$ minimal length elements in $\mathcal{O}$.
Proposition 26. For every $w\in \mathcal{O}$, there exists a sequence $x_ 1,\dots, x_ r\in \widetilde{W}$, such that
- $l(x_ i)\leq 1$, ($l(x_ i)=0$ if and only if $x_ i \mathcal{I}_ w x_ i^{-1}=\mathcal{I}_ w$);
- $w_ i:=(x_ 1\cdots x_ i) w\sigma(x_ 1\cdots x_ i)^{-1},$ then $$l(w)\geq l(w_ 1)\geq l(w_ 2)\geq\cdots \geq l(w_ r)$$
- $w_ r\in\mathcal{O}_ {\textrm{min}}$.
Proposition 27. Let $w,v\in\mathcal{O}_ {\textrm{min}}$, there exists $x_ 1,\dots, x_ r\in\widetilde{W}$, and $w_ i:=x_ i w_ {i-1}\sigma(x_ i)^{-1}$, $w_ 0:=w$, such that
- $w_ r=v$;
- $l(w_ i)=l(w_ {i-1})$;
- either $l(w_ i\sigma(x_ i))=l(w_ i)+l(x_ i)$, or $l(w_ i)=l(w_ i\sigma(x_ i))+l(x_ i)$.
Proposition 28. There exists some $w\in \mathcal{O}_ {\textrm{min}}$ of the form $w=ux$, where
- $x$ is $\sigma$-straight,
- $x\sigma(W_ \mathcal{P})x^{-1}=W_ \mathcal{P}$, where $\mathcal{P}$ is some standard parahoric and $x$ is of minimal length in $W_ \mathcal{P} x$.
- $u\in W_ \mathcal{P}$.
Let $x,y\in\widetilde{W}$ and $l(x)\leq 1, l(y)\leq 1$. Now observe that if $l(yx)=l(x)+l(y)$, then the following data
In particular, $$\mathrm{Sht}_ {xw\sigma(x)^{-1}}\cong\mathrm{Sht}_ {w},$$ if
- $l(xw\sigma(x)^{-1})=l(w)$,
- either $l(w\sigma(x))=l(w)+l(x)$, or $l(w)=l(w\sigma(x))+l(x)$.
Corollary 29. For any two $w_ 1,w_ 2\in\mathcal{O}_ {\textrm{min}}$, we have that $$\mathrm{Sht}_ {w_ 1}^{\textrm{loc}}\cong\mathrm{Sht}_ {w_ 2}^{\textrm{loc}}.$$
Now take $w\in\mathcal{O}_ \textrm{min}$ as in Prop. 5.18, such that $w=ux$, where $u\in W_ \mathcal{P}$ and $x$ is $\sigma$-straight. Let $b\in B(G)$ the image of $x$ under $B(\widetilde{W})_ {\textrm{str}}\cong B(G)$.
Then We have the diagram
🔗More Sheaf Theory
Let $\Lambda=\overline{\mathbb{F}}_ \ell$, or $\overline{\mathbb{Z}}_ \ell,$ or $\overline{\mathbb{Q}}_ \ell$ be the coefficient ring. Let $k=\overline{k}$ be an algebraically closed field of characteristic $p\neq\ell$.
Recall that for $\Lambda=\mathbb{F}_ \ell$, $S$ a qcqs scheme, we have that $$\mathrm{Ind}D_ c(S_ \textrm{'et},\mathbb{F}_ \ell)\rightarrow D(S_ \textrm{'et}, \mathbb{F}_ \ell)\hookrightarrow D_ \textrm{'et}(S,\mathbb{F}_ \ell):=\varprojlim_ {n} D(S_ \textrm{'et},\mathbb{F}_ \ell)^{\geq n},$$ i.e. $ D_ \textrm{'et}(S,\mathbb{F}_ \ell)$ is the left completion of $D(S_ \textrm{'et},\mathbb{F}_ \ell)$ with respect to the natural t-structure and $D_ c(S_ \textrm{'et},\mathbb{F}_ \ell)=D_ {\textrm{cft}}(S_ \textrm{'et},\mathbb{F}_ \ell)$ is the smallest idempotent complete stable full subcategory of $D(S_ \textrm{'et},\mathbb{F}_ \ell)$ generated by $j_ !\Lambda_ U$, for $U\rightarrow S$ quasi-compact 'etale.
Notation: $\mathrm{Shv}^\ast(S,\mathbb{F}_ \ell):=\mathrm{Ind}\mathrm{Shv}_ c(S_ \textrm{'et},\mathbb{F}_ \ell)$. If $S$ is of finite type over $k$, then $$\mathrm{Shv}^\ast(S,\mathbb{F}_ \ell)\cong D(S_ \textrm{'et},\mathbb{F}_ \ell)\cong D_ \textrm{'et}(S,\mathbb{F}_ \ell).$$ In general, $$\mathrm{Shv}^\ast(S,\mathbb{F}_ \ell)^+\cong D(S_ \textrm{'et},\mathbb{F}_ \ell)^+\cong D_ \textrm{'et}(S,\mathbb{F}_ \ell)^+.$$
Remark 30. Note that $\mathrm{Shv}^\ast(S,\mathbb{F}_ \ell)$ is compactly generated, and hence dualizable; $D_ \textrm{'et}(S,\mathbb{F}_ \ell)$ satisfies pro-'etale descent.
Lemma 31. Assume that $S=\varprojlim_ {i} S_ i,$ where $S_ i$ are of finite type over $k$, and transition maps are eventually affine. Then $$\mathrm{Shv}^\ast(S,\mathbb{F}_ \ell)\cong\varinjlim_ {i}\mathrm{Shv}^\ast(S_ i,\mathbb{F}_ \ell)$$ with transtion maps given by $*$-pullbacks.
Definition 32. Let $S=\varprojlim_ {i}S_ i$, where $S_ i$ are of finite type over $k$, and transition maps are eventually affine. We define $$\mathrm{Shv}^\ast(S,\Lambda):=\varinjlim_ {i}\mathrm{Shv}^\ast(S_ i,\Lambda)$$ and $$\mathrm{Shv}^\ast(S_ i,\Lambda):=\mathrm{Ind}D_ {\textrm{cft}}((S_ i)_ \textrm{'et},\Lambda),$$ aka we define $*$-sheaf theory to be ind-completion of the derived category of constructible 'etale sheaves for a scheme of finite type over $k$ and use the left Kan extension to further define it for general qcqs schemes.
Example 33.
- If $S=\mathrm{Spec} k$, $k=\overline{k}$, then $$\mathrm{Shv}^\ast(S,\mathbb{F}_ \ell)\cong \mathrm{Shv}_ \textrm{'et}(S,\mathbb{F}_ \ell)\cong \mathrm{Shv}(S_ \textrm{'et},\mathbb{F}_ \ell)\cong\mathrm{Mod}_ {\mathbb{F}_ \ell}.$$
- $S=\mathrm{Spec} k(C)$, $C$ a curve over $k$, and $\Gamma:=\mathrm{Gal}(\overline{k(C)}/k(C))$. Then $$D(S_ \textrm{'et},\mathbb{F}_ \ell)\cong \mathrm{Rep}(\Gamma,\mathbb{F}_ \ell),$$ and we have $$\mathrm{Ind}\mathrm{Rep}_ {f.g.}(\Gamma,\mathbb{F}_ \ell)\twoheadrightarrow \mathrm{Rep}(\Gamma,\mathbb{F}_ \ell),$$ which is not an equivalence in general. We also have that $$\begin{split} \mathrm{Shv}^\ast(S,\mathbb{Z}_ \ell) &=\mathrm{Ind}\varinjlim_ {U\textrm{ open}} D_ {\textrm{cft}}(U,\mathbb{Z}_ \ell)\ &\cong \mathrm{Ind}\mathrm{Rep}_ {\textrm{un.a.e., f.g.}}(\Gamma,\mathbb{Z}_ \ell), \end{split}$$ where un$.$a$.$e$.$ is short for ``unramified almost everywhere’’. In this case, $$D(S_ \textrm{'et},\mathbb{Z}_ \ell)=\varprojlim_ {n}D(S_ \textrm{'et}, \mathbb{Z}/\ell^n\mathbb{Z})\cong \mathrm{Rep}_ {\textrm{cts}}(\Gamma,\mathbb{Z}_ \ell).$$
- Assume that $S$ is a profinite set over $k$. Then $$\mathrm{Shv}^\ast(S,\Lambda)=\varinjlim{}\mathrm{Shv}^\ast(S_ i,\Lambda).$$
Remark 34. The $*$-sheaf theory $\mathrm{Shv}^\ast$ does NOT satsify pro-'etale descent. For example, pro-'etale descent along $\overline{K}/K$ gives $\mathrm{Rep}(\Gamma,\mathbb{F}_ \ell)$ but not $\mathrm{IndRep}_ {f.g.}(\Gamma,\mathbb{F}_ \ell)$.
Theorem 35. The $*$-sheaf theory $\mathrm{Shv}^\ast$ satisfies 'etale descent. There exists a sheaf theory $$\mathrm{Corr}(\mathrm{Sch}^{\textrm{qcqs}}_ k)_ {\substack{H=f.p.\ V=\textrm{all}}}\rightarrow \mathrm{LinCat}_ \Lambda$$ lax symmetric monoidal functor. In particular, for a correspondence
Now we define $!$-sheaf theory by
If $\mathcal{C}$ is compactly generated, then $\mathcal{C}^\vee\cong \mathrm{Ind}(\mathcal{C}^{\omega})^{\textrm{op}}$ and $F^\circ= \mathrm{Ind}(F|_ {\mathcal{C}^\omega})^{\textrm{op}}$.
So for $f: S\rightarrow T$ a morphism of schemes of finite type over $k$, we have $$\mathrm{Shv}^\ast(S,\Lambda)=\mathrm{Ind}D_ {\textrm{cft}}(S_ \textrm{'et}, \Lambda),$$ $\mathrm{Shv}^!(S,\Lambda)=\mathrm{Ind}D_ {\textrm{cft}}(S_ \textrm{'et}, \Lambda)^{\textrm{op}}$, and then $(f_ *)^\circ$ is determined by $(f_ *)^\circ|_ {\mathrm{Shv}^!(S,\Lambda)^\omega}=(f_ *)^{\textrm{op}}$
Then using the right Kan extension, we define $!$-sheaf theory for prestaks
Let me explain the notations: For algebraic spaces,
- $\textrm{pro\’et}$: $\textrm{pro-\’etale}$ morphism (require eventually affine transition).
- $\textrm{Epro\’et}$: essentially $\textrm{pro-\’etale}$, i.e.
- IndEpro'et:$X=\varinjlim_ {i}X_ i\rightarrow Y$, where $X_ i\rightarrow Y$ is Epro'et, and $X=\varinjlim_ {i}X_ i$ has transition maps being closed immersion, ind-algspc.
For stacks, the definition is given using base change to algebraic spaces.
Some Facts:
- If $X$ is a scheme of finite presentation over $k$, then $$\mathrm{Shv}^!(X)=(\mathrm{Ind}D_ {\textrm{cft}}(X))^\vee\cong \mathrm{Ind}D_ {\textrm{cft}}(X)$$ via Verdier duality. If $X$ is a qcqs scheme, and $X=\varprojlim_ {i} X_ i$ with $X_ i$ affine and transition maps all affine, then $$\mathrm{Shv}^\ast(X)=\varinjlim_ {i}\mathrm{Shv}^\ast(X_ i)$$ with transition maps given by $*$-pullbacks and $$\mathrm{Shv}^!(X)=\varinjlim_ {i}\mathrm{Shv}^\ast(X_ i)$$ with $!$-pullbacks and in genreal $\mathrm{Shv}^!(X)\neq \mathrm{Shv}^\ast(X)$.
- $\mathrm{Shv}(X)=\varprojlim_ {\substack{S\textrm{ qcqs algebraic space over k}\ S\rightarrow X}}$ with transition maps given by $!$-pullbacks.
- $\mathrm{Shv}^!$ satisfies $h$-descent. Let $X\rightarrow Y$ be a $h$-cover (generated by 'etale surjections and proper surjections) and $X_ \bullet$ be the Cech nerve in $\mathrm{AlgSp}_ k$. Then $$\mathrm{Shv}^!(Y)\cong \varprojlim{}\mathrm{Shv}^!(X_ \bullet).$$
- Now let $X=\varprojlim_ {i}X_ i\rightarrow Y$ be pro-'et, $X_ i\rightarrow Y$ be 'etale, and $\mathcal{F}$ constructible sheaf, a compact object in $\mathrm{Shv}^!(X)$, then $\mathcal{F}$ comes from $\mathcal{F}_ i$ on $X_ i$.
Then $f_ *\mathcal{F}=\varinjlim_ {i}(g_ i)_ *\mathcal{F}_ i$ and $f_ *=(f^!)^R$, i.e. $(f^!,f_ *)$ is an adjoint pair of functors. - Now $f:X\xrightarrow[\textrm{pro-'et}]{f_ 1}X’\xrightarrow[\textrm{fp}]{f_ 2}Y$, then $f_ *=(f_ 2)_ *(f_ 1)_ *$.
- Assume that $X=\varinjlim_ {i}X_ i$ a colimit in $\mathrm{PreStk}_ k$ with proper f$.$p$.$ transition morphisms. Then $$\mathrm{Shv}^!(X)\cong\varprojlim_ {i}\mathrm{Shv}^!(X_ i)$$ with transition maps given by $!$-pullbacks. Note that for $f: X\rightarrow Y$ a proper morphism of finite presentation, we have an adjoint pair $(f_ \ast,f^!)$. Therefore, $$\mathrm{Shv}^!(X)\cong \varinjlim_ {i}\mathrm{Shv}^!(X_ i)$$ with transition maps being $*$-pushforwards. In particular, this is why we want to pass to the dual and use $!$-sheaf theory.
- For $f:X\rightarrow Y$ a ind-fp proper morphism, we have an adjoint pair $(f_ \ast,f^!)$. In addition, if $f:X\rightarrow Y$ is surjective, then we have
$$\mathrm{Shv}^!(Y)\cong \varprojlim{}\mathrm{Shv}^!(X_ \bullet)\cong \varinjlim{}\mathrm{Shv}^!(X_ \bullet),$$
with $X_ \bullet$ being the Cech nerve. The transition maps in $\varprojlim{}$ are given by $!$-pullbacks and the transition maps in $\varinjlim{}$ are given by $*$-pushforwards.
Corollary 36. $\mathrm{Shv}^!(\mathrm{Isoc}^\textrm{'et})=\mathrm{Shv}^!(\mathrm{Isoc}^h)\cong \varinjlim{}\mathrm{Shv}^!(\mathrm{Sht}^\textrm{loc}_ \bullet)$.
- Let $X=\varprojlim_ {i}X_ i$ be a qcqs algebraic space, $X_ i$ of finite type over $k$. Then
If $f_ {ij}$ are smooth, $f_ {ij}^!=f_ {ij}^\ast[\cdots]$ agrees with each other up to a suitable shift, then $$\mathrm{Shv}^!(X):=\mathrm{Shv}^\ast(X)^\vee\cong\mathrm{Shv}^\ast(X)$$ up to some choice.
🔗Properties of Morphisms
Recall that in the last subsection, we defined $!$-sheaf theory $$\mathrm{Corr}(\mathrm{PreStk})_ {V,\textrm{All}}\rightarrow \mathrm{LinCat}_ \Lambda,$$ enjoying the following properties:
- satisfies $h$-descent, ind-proper descent;
- pro-'etale descent fails in general;
- $\mathrm{Shv}^!(X)=\varprojlim_ {\textrm{qcqs algstk}S\rightarrow X}\mathrm{Shv}^!(S)$ with $!$-pullback as transition maps.
- If $S$ is of finite type over $k$, then $\mathrm{Shv}^!(S)=\mathrm{Ind}D_ {\textrm{ctf}}(S,\Lambda)^\textrm{op}$.
Proposition 37. Let $G$ be a profinite group, essentially pro-$p$ (i.e. a compact open is pro-$p$). Let $\Lambda=\overline{\mathbb{Z}}_ \ell,\overline{\mathbb{Q}}_ \ell,\overline{\mathbb{F}_ \ell}$, etc. be the coefficient ring, and $k=\overline{k}$ has characteristic $p>0$ and $p\neq \ell$ as before. Then $$\mathrm{Shv}^!(\mathbb{B}_ {\textrm{prof'et}}G,\Lambda)\cong \mathrm{Rep}^{\textrm{sm}}(G,\Lambda).$$
Proof.
For the Cech cover of $\rightarrow \mathbb{B}_ {\textrm{prof'et}}G$. So we have $$\mathrm{Shv}^!(\mathbb{B}_ {\textrm{prof'et}}G)\cong \varprojlim{}( \begin{tikzcd}[ampersand replacement=&] \mathrm{Shv}^!() \arrow[r, shift left] \arrow[r] & \mathrm{Shv}^!(G) \arrow[r] \arrow[r, shift left] \arrow[r, shift right] & \cdots \end{tikzcd} )$$ Now re-interpret each term in the limit as IndCoh on $G_ \Lambda$ and then obtain smooth representations of $G$.
Corollary 38. Let $G$ be a locally pro-finite group and $K\subseteq G$ be a pro-$p$ open subgroup. Then $$\mathrm{Shv}^!(\mathbb{B}_ {\textrm{prof'et}}G)\cong \mathrm{Rep}^{\textrm{sm}}(G,\Lambda).$$
Proof.
The natural map $\mathbb{B}_ {\textrm{prof'et}}K\rightarrow \mathbb{B}_ {\textrm{prof'et}}G$ is ind-finite morphism.
Corollary 39. $$\mathrm{Shv}^!(\mathrm{Isoc}^\textrm{'et}_ b)\cong \mathrm{Rep}^{\textrm{sm}}(J_ b(F),\Lambda).$$
Recall that for a morphism of finite presentation $f: X\rightarrow Y$ of qcqs schemes or algebraic spaces is called locally acyclic (LA) if for any $\overline{y}\rightarrow f(\overline{x})$,
we have that $$R\Gamma(X_ {(\overline{x})},\Lambda)\xrightarrow{\cong}\Gamma(X_ {\overline{x}}\times_ {Y_ {f(\overline{x})}}\overline{y},\Lambda).$$ We say $f$ is ULA if it is always $LA$ after any base change. Lemma 40. Surjective ULA maps are $h$-covers.
Proof.
Since it is a universally open map.
Basic properties:
- If $f:X\rightarrow Y$ is ULA, then $$f^!(-)\cong f^\ast(-)\otimes f^!\Lambda$$ for $\mathrm{Shv}^\ast$.
- Smooth base change: If $f$ is $\textrm{f}.\textrm{p}.$, and $g$ is ULA and the following diagram is Cartesian,
we have that $g^\ast f_ *\cong f_ \ast'g'^\ast$.
Definition 41. We say $f:X\rightarrow Y$ os cohomologically smooth if $f$ is ULA and $f^!\Lambda$ is an invertible object in $\mathrm{Shv}^!(X,\Lambda)$.
Example 42. If $f$ is smooth, then $f$ is cohomologically smooth.
Fact: $f^!\Lambda\cong\Lambda[\cdots]$ non-canonically.
Definition 43. We say $f:X\rightarrow Y$ is cohomologically unipotent if $f$ is cohomologically smooth and $f^\ast: \mathrm{Shv}^\ast(Y)\rightarrow \mathrm{Shv}^\ast(X)$ is fully faithful.
Example 44. If $X\rightarrow Y$ has affine space fibers, then $f$ is cohomologically unipotent.
Now passing to $\mathrm{Shv}^!$:
- If $f:X\rightarrow Y$ is cohomologically smooth, then $f^!$ admits a continuous right adjoint $f_ \flat$. We have non-canonical isomorphism $$f_ \flat\cong f_ *[\cdots].$$
- In this case, $f_ \flat$ satisfies smooth base change, i.e. $$g^!f_ \flat\cong f’_ \flat g’^!.$$
Definition 45. A morphism $f:X\rightarrow Y$ of qcqs algebraic spaces is called cohomologically pro-smooth if $X=\varprojlim_ {i}X_ i\rightarrow Y$ with $X_ i\rightarrow Y$ cohomologically smooth and $X_ i\rightarrow X_ j$ is cohomologically smooth and affine. If furthermore, $X_ i\rightarrow X_ j$ is surjective, then we call $f$ is strongly cohomologically smooth.
If $f$ is strongly coh smooth, then $f$ is an open morphism.
Definition 46. A morphism $f:X\rightarrow Y$ of qcqs algebraic spaces is called cohomologically pro-unipotent if $X=\varprojlim_ {i}X_ i\rightarrow Y$ with $X_ i\rightarrow Y$ cohomologically unipotent and $X_ i\rightarrow X_ j$ is cohomologically unipotent and affine.
In this case, $X_ i\rightarrow X_ j$ is automatically surjective, so $f$ is automatically strongly cohomologically smooth and hence open.
Example 47. $L^+G\rightarrow \mathrm{Spec} k$ is cohomologically pro-unipotent.
Proposition 48. If $f$ is cohomologically pro-smooth, then $f^!$ admits a continuous right adjoint $f_ \flat:\mathrm{Shv}^!(X)\rightarrow \mathrm{Shv}^!(Y)$, satisfying base change and projection formula.
Assume that $f: X=\varprojlim_ {i}X_ i\rightarrow Y$ abd $f_ i:X_ i\rightarrow Y$, then $$f_ \flat\mathcal{F}=\varinjlim_ {i}(f_ {i})_ \flat\mathcal{F}_ i$$ does not preserve constructible objects in general. But if $f$ is cohomologically pro-unipotent, then $f^!$ is fully faithful, and hence $f_ \flat$ preserves constructible objects.
Now we do some general sheaf theory. Let $\mathcal{D}: \mathrm{Corr}(\mathcal{C})_ {V,\textrm{all}}\rightarrow \mathrm{LinCat}_ \Lambda$ be a sheaf theory. For a correspondence
we have $$f_ +\circ g^\star: \mathcal{D}(Y)\rightarrow \mathcal{D}(X).$$ Let HR$\subseteq$All stable under base change, composition, and satisfies projection formula. Now let $f: X\rightarrow Y$ be in HR. We have adjoint pair $(f_ \star,f^\star)$. then we have a natural map $$\Lambda_ Y\rightarrow f_ \star f^\star\Lambda_ Y.$$
Proposition 49. If $ \Lambda_ Y\rightarrow f_ \star f^\star\Lambda_ Y$ admits a section, then $$\mathcal{D}(Y)\cong\varprojlim{}\mathcal{D}(X_ \bullet)$$ is an equivalence, where $X_ \bullet$ is the Cech nerve.
Proof.
Let $F:\Delta^+\rightarrow \mathrm{LinCat}_ \Lambda$ be the augmented simplicial diagram,
If
- the diagram
is right adjointable for every $[n]\rightarrow [m]$, - $F$ is conservative,
- $C_ {-1}$ admits limit of split cosimplicial objects and $F$ preserves such limits,
then $C_ {-1}\cong \varprojlim_ {i}{C_ i}$ by general descent results.
However, we know that
- $\mathcal{D}\rightarrow f_ \star f^\star\mathcal{F}\cong f_ \star \Lambda_ X\otimes \mathcal{F}$ by projection formula (using $f^\star\Lambda_ Y\cong\Lambda_ X$), which implies that $f^\star$ is conservative,
- $\mathcal{F}^\bullet$ split cosimplicial object in $\mathcal{D}(Y)$, then $f^\star(\mathcal{F}^\bullet)$ is split.
Therefore, we proved the proposition.
Now come to the situation we care about. Let $K=\varprojlim_ {i}K_ i$ be a pro-$p$ group and consider
where $\pi: E\rightarrow S$ is a $K$-torsor. Then $(\pi^!)^R=\pi_ *$ and we want to show that $$\omega_ S\rightarrow \pi_ *\omega_ E$$ admits a section. Note that $E=\varprojlim_ {i}E_ i\rightarrow S$ and $\pi_ *\omega_ E=\varinjlim_ {i}\pi_ *\omega_ {E_ i}$. Since $\pi$ is pro-finite, $$\pi_ *=\pi_ !=(\pi^!)^L.$$ Now $\omega_ S\rightarrow \pi_ *\omega_ {E_ i}\xrightarrow{t_ i}\omega_ S$ is given by multiplication by $\#K_ i$. Then $$\omega_ S\rightarrow \pi_ *\omega_ {E_ i}\xrightarrow{\frac{t_ i}{\#K_ i}}\omega_ S$$ gives the desired section.
🔗Placid Spaces
Definition 50. A morphism $f:X\rightarrow Y$ of qcqs algebraic spaces is called essentially cohomologically pro-smooth (pro-'etale, resp.) if $f$ factorizes as $X\xrightarrow{f’}X’\xrightarrow{f’‘}Y$ with $f’$ coh pro-smooth (pro-'etale, resp.) and $f’’$ is of finite presentation.
Definition 51. A qcqs algebraic space is called standard placid, if $X\rightarrow k$ is ess$.$ coh$.$ pro-smooth.
Example 52. If $Z/\mathcal{O}$ is smooth and of finite type, then $X=L^+Z$ is standard placid.
Definition 53. A generalized dualizing sheaf on a standard placid space $X$ is an object $\eta_ X\in\mathrm{Shv}^\ast(X)$, which is of the form $$\eta_ X=r^\ast\omega_ {X’}\otimes \mathcal{L},$$ where $r:X\rightarrow X’$ is coh pro-smooth, $X’$ is of finite presentation over $k$, and $\mathcal{L}$ is invertible.
Definition 54. A generalized constant sheaf on a standard placid space $X$ is an object $\Lambda_ X\in\mathrm{Shv}^!(X)$, which is of the form $$\eta_ X=r^!\Lambda_ {X’}\otimes \mathcal{L},$$ where $r:X\rightarrow X’$ is coh pro-smooth, $X’$ is of finite presentation over $k$, and $\mathcal{L}$ is invertible.
Proposition 55. Let $X$ be a standard placid space.
- Given any generalized dualizing sheaf $\eta_ X$, we have an equivalence $$\begin{split} \mathbb{D}_ X^\eta: \mathrm{Shv}^\ast_ c(X)^{\textrm{op}}&\xrightarrow{\cong}\mathrm{Shv}^\ast_ c(X)\ \mathcal{F}&\mapsto\underline{\mathrm{Hom}}(\mathcal{F},\eta_ X). \end{split}$$
- Assume that $f: X\rightarrow Y$ is coh pro-smooth and $\eta_ Y$ is a generalized dualizing sheaf on $Y$. Then $\eta_ X:=f^\ast\eta_ Y$ is a generalized dualizing sheaf on $X$ and $$f^\ast\mathbb{D}_ Y^\eta\cong \mathbb{D}_ X^\eta f^\ast.$$
- We can define a perverse t-structure on $\mathrm{Shv}^\ast_ {(c)}(X)$.
Proposition 56. Let $\Lambda_ X^\eta$ be a generalized constant sheaf on a standard placid space $X$. Then $$\begin{split} \mathrm{Shv}^!(X)\otimes \mathrm{Shv}^!(X)&\rightarrow \mathrm{Mod}_ \Lambda\ (\mathcal{F},\mathcal{G}) &\mapsto \mathrm{Hom}(\Lambda_ X^\eta,\mathcal{F}\otimes^!\mathcal{G}) \end{split}$$ defines $\mathbb{D}_ X^{\Lambda_ \eta}:\mathrm{Shv}^!(X)\xrightarrow{\cong}\mathrm{Shv}^!(X)^\vee$.
We have the diagram
Definition 57.
- An 'etale stack $X:\mathrm{CAlg}_ k^{\textrm{perf}}\rightarrow \mathrm{Ani}$ is called quasi-placid, if there exists a surjective representable (by qcqs) coh pro-smooth morphism $$U\rightarrow X$$ with $U$ standard placid space.
- $X$ is called placid if there exists atlas $U\rightarrow X$ such that
- presentable and strongly coh pro-smooth;
- of universal coh descent.
- $X$ is called very placid if one can choose $U\rightarrow X$ to ess coh pro-unipotent.
Example 58. $\mathbb{B}_ {\textrm{prof'et}}K$ is quasi-placid, for $K$ pro-p group.
Example 59. Artin stack of finite type is very placid.
Example 60. Affine Schubert $\mathrm{Gr}_ {\leq \lambda}/L^+G$ is very placid.
Example 61. $LG_ {\leq\lambda}/\mathrm{Ad}_ \sigma L^+G$ is very placid.
Proposition 62. If $X$ is quasi-placid, then there exists a generalized constant sheaf $\Lambda_ X^\eta\in \mathrm{Shv}^!(X)$, such that for atlas $U\xrightarrow{\varphi} X$, we have that $\varphi^!\Lambda_ X^\eta$ is a generalized constant sheaf on $U$. Furthermore, we also have $$\mathbb{D}_ X^\eta:\mathrm{Shv}^!(X)^\vee\xrightarrow{\cong}\mathrm{Shv}^!(X)$$ and we can define perverse t-structure.
An open question: let $Z$ be a scheme of finite type over $\mathcal{O}$, is $L^+Z$ quasi-placid?
🔗Sheaves on Isocrystals
Now we apply the general theory above to the situation we are studying.
Recall that $$\mathrm{Nt}:\mathrm{Sht}^{\textrm{loc}}:=LG/\mathrm{Ad}_ \sigma \mathcal{I}_ w\rightarrow \mathrm{Isoc}:=LG/\mathrm{Ad}_ \sigma LG.$$ We use $\mathrm{Hk}_ \bullet(\mathrm{Sht}^{\textrm{loc}})$ to denote the corresponding Cech nerve. Then we have that $$\mathrm{Shv}^!(\mathrm{Isoc})\cong\varinjlim{}\mathrm{Shv}^!(\mathrm{Hk}_ \bullet(\mathrm{Sht}^{\textrm{loc}}))$$ and also $$\mathrm{Shv}^!(\mathrm{Sht}^{\textrm{loc}})\cong\varinjlim_ {w}\mathrm{Shv}^!(\mathrm{Sht}^{\textrm{loc}}_ {\leq w}),$$ which is a filtered colimit induced by stratification.
By definition, $$\mathrm{Hk}_ 2(\mathrm{Sht}^{\textrm{loc}})=LG/\mathrm{Ad}_ \sigma \mathcal{I}_ w\times_ {LG/\mathrm{Ad}_ \sigma LG} LG/\mathrm{Ad}_ \sigma \mathcal{I}_ w$$ is classifying all
which is the same data as $\mathcal{E}_ 1^\sigma\dashrightarrow\mathcal{E}_ 2$, which are classified by the quotient stack $$(LG\times^{\mathcal{I}_ w} LG)/\mathrm{Ad}_ \sigma \mathcal{I}_ w.$$ In general, we have $$\mathrm{Hk}_ n(\mathrm{Sht}^{\textrm{loc}})=\frac{LG\times^{\mathcal{I}_ w}LG\times^{\mathcal{I}_ w}\cdots\times^{\mathcal{I}_ w}LG}{\mathrm{Ad}_ \sigma\mathcal{I}_ w}.$$ In particular, $\mathrm{Hk}_ n(\mathrm{Sht}^{\textrm{loc}})$ is ind-very placid. So, Isoc is sifted very placid, meaning that it is a geometric realization of ind-very placid, where by ind we mean a filtered colimit.
Theorem 63. The category $\mathrm{Shv}^!(\mathrm{Isoc})$ is compactly generated and self-dual.
The theorem is an easy consequence of the following lemma, since it boils down to compact generation of $\mathrm{Shv}^!(\mathrm{Sht}_ {\leq w}^{\textrm{loc}})$ thanks to the limit descriptions above, and Eg. 5.13.
Lemma 64. If $X$ is very placid, then $\mathrm{Shv}^!(X)$ is compactly generated and self-dual.
Proof.
Let $U\rightarrow X$ be essentially cohomologically pro-unipotent and $U$ is standard placid. Then $$\mathrm{Shv}^!(X)\cong\varprojlim{}\mathrm{Shv}^!(U\times_ X U\times_ X\cdots \times_ X U).$$ Passing to left adjoints (essentially using Bouthier’s results, $!$-pullback admits left adjoint), $\mathrm{Shv}^!(X)\cong \varinjlim{}\mathrm{Shv}^!(U_ \bullet)$ is compactly generated.
Remark 65. $\mathrm{Shv}^!(X)^\omega\subseteq \mathrm{Shv}^!_ c(X)$ may not be equal (even for $X=\mathbb{B} G$). It is also not clear yet if $\mathbb{D}_ X^\eta$ preserves $\mathrm{Shv}^!(X)^\omega$ (a priori only preserves $\mathrm{Shv}^!_ c(X)$). For example, consider $*\rightarrow X=\mathbb{B} H$. Then $\mathbb{D}_ X^\eta \pi_ !\Lambda=\pi_ *\Lambda$. Observe that $C_ c^\bullet(H)=\mathrm{End}(\pi_ !\Lambda)$ acts on $$\mathrm{Hom}(\pi_ !\Lambda,\pi_ *\Lambda)=C^\bullet(H)=\mathrm{Hom}(C_ c^\bullet(H),\Lambda)\cong C_ c^\bullet(H)[\cdots].$$
Now we have $$\mathbb{D}_ X:(\mathrm{Shv}^!(\mathrm{Isoc})^\omega)^{\textrm{op}}\xrightarrow{\cong}\mathrm{Shv}^!(\mathrm{Isoc})^\omega.$$
For $b\in B(G)$, $\mathrm{Isoc}_ {\leq b}\hookrightarrow \mathrm{Isoc}$ is closed immersion of finite presentation, and
Then we have $$\mathrm{Shv}^!(\mathrm{Isoc}_ {b}^\textrm{\'et})\cong\mathrm{Rep}^{\textrm{sm}}(J_ b(F),\Lambda).$$ We also have that $$\mathrm{Shv}^!(\mathrm{Isoc})=\varinjlim_ {b}\mathrm{Shv}^!(\mathrm{Isoc}_ {\leq b})$$ with transition maps given by $\ast$-pushforward and each $\mathrm{Shv}^!(\mathrm{Isoc}_ {\leq b})$ is compactly generated and self dual. Proposition 66. An object $\mathcal{F}\in \mathrm{Shv}^!(\mathrm{Isoc})$ is compact if and only if
- $\mathcal{F}\in\bigcup_ {\textrm{finite union}}\mathrm{Shv}^!(\mathrm{Isoc}_ {\leq b_ i})$ for some finite set ${b_ 1,\dots, b_ m}$.
- $j_ b^!\mathcal{F}\in\mathrm{Shv}^!(\mathrm{Isoc}_ b)^\omega$.
Proof.
For the ``only if’’ part: Since $j_ b^!$ admits continuous right adjoint $(j_ b)_ *$, we know that $j_ b^!$ preserves compact object.
For the ``if’’ part: $i_ {<b}^\ast$ exists as the left adjoint of $(i_ {<b})_ *$. Then we argue inductively:
- Excision sequence: if $i:Z\hookrightarrow X$ is the closed immersion of finite presentation, $j: U\hookrightarrow X$ is the complementary quasi-compact open embedding, then we will have exact triangles. Therefore, we have fiber sequences $$(j_ b)_ !j_ b^!\mathcal{F}\rightarrow \mathcal{F}\rightarrow (i_ {<b})_ *i_ {<b}^\ast\mathcal{F}$$ and $$(i_ {<b})_ *i_ {<b}^!\mathcal{F}\rightarrow\mathcal{F}\rightarrow (j_ b)_ *j_ b^!\mathcal{F}.$$
- Duality preserves compact objects, $\mathbb{D} f_ !=f_ *\mathbb{D}$, $\mathbb{D} f^!\cong f^\ast\mathbb{D}$, and hence, $(i_ {<b})_ *$ and $(j_ {<b})_ *$ preserve compact objects.
- Proved by induction, $i_ {<b}^!\mathcal{F}$ is compact.
Corollary 67. The category $\mathrm{Shv}^!(\mathrm{Isoc})$ is generated by $(j_ b)_ *\textrm{c-ind}_ K^{J_ b(F)}\Lambda$, where $K\subseteq J_ b(F)$ is a pro-$p$ open compact subgroup.
Lemma 68. Let $V$ be a representation of $J_ b(F)$. Then $$\mathbb{D}(j_ b)_ * V= (j_ \flat)_ !\mathbb{D} V,$$ where $\mathbb{D} V=\mathrm{Hom}(V,C_ c^\infty(J_ b(F)))$ is the Bernstein-Zelevinsky duality.
Proof.
We have a commutative diagram
and on representations Now note that a self dual $\mathbb{D}:\mathcal{C}\xrightarrow{\cong} \mathcal{C}^\vee$ is equivalent to self dual data $$\begin{split} u:\mathrm{Mod}_ \Lambda&\rightarrow \mathcal{C}\otimes\mathcal{C}\ e:\mathcal{C}\otimes\mathcal{C}&\rightarrow \mathrm{Mod}_ \Lambda. \end{split}$$ In our case, the unit is given by $$\mathrm{Mod}_ \Lambda=\mathrm{Shv}^!(\mathrm{pt})\xrightarrow{!\textrm{-pullback}}\mathrm{Shv}^!(\mathrm{Isoc})\xrightarrow{\Delta_ *}\mathrm{Shv}^!(\mathrm{Isoc}\times\mathrm{Isoc})\cong\mathrm{Shv}^!(\mathrm{Isoc})\otimes \mathrm{Shv}^!(\mathrm{Isoc}).$$ Othe torsor is trivialized. n the level of representations, $\mathbb{B} G(F)\rightarrow \mathbb{B} (G(F)\times G(F))$ corresponds to the module $C_ c^\infty(G(F))$ equipped with left and right $G(F)$-action.
- Let $X=\varprojlim_ {i}X_ i$ be a qcqs algebraic space, $X_ i$ of finite type over $k$. Then