ScalingStacks

0P5Z

Proposition 4.3.9. There is a differential functor Δ​Φ:Δσ​𝒲→Δσ′​𝒲′\Delta\Phi:\Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime} given by (m,π)↦(Φ⁡(m),φ1​(m)∘Φ⁡(π)∘φ2​(m)−1)(m,\pi)\mapsto(\Phi(m),\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}(m)^{-1}).

There is a closed isomorphism of functors

φ=(φ2φ1):Δ​Φ​E→∼E′​Δ​Φ.\varphi=\left(\begin{matrix}\varphi_{2}\\ &\varphi_{1}\end{matrix}\right):\Delta\Phi E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\Delta\Phi.

If σ\sigma and σ′\sigma^{\prime} are invertible, then (Δ​Φ,φ)(\Delta\Phi,\varphi) defines a morphism of 22-representations Δσ​𝒲→Δσ′​𝒲′\Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}.

0P60

Proof. Let (m,π)(m,\pi) be an object of Δσ​𝒲\Delta_{\sigma}{\mathcal{W}}. Let π′=φ1​(m)∘Φ⁡(π)∘φ2−1​(m)\pi^{\prime}=\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}^{-1}(m), an element of Z​Hom𝒲′¯i⁡(E2′​Φ​(m),E1′​Φ​(m))Z\operatorname{Hom}\nolimits_{\overline{{\mathcal{W}}^{\prime}}^{i}}(E^{\prime}_{2}\Phi(m),E^{\prime}_{1}\Phi(m)).

We have

(E1′​π′)∘(σ′​Φ​(m))∘(E2′​π′)=(E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime})=
=(E1′​φ1​(m))∘(E1′​Φ​π)∘(E1′​φ2−1​(m))∘(σ′​Φ​(m))∘(E2′​φ1​(m))∘(E2′​Φ​π)∘(E2′​φ2−1​(m))\displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(E^{\prime}_{1}\Phi\pi)\circ(E^{\prime}_{1}\varphi_{2}^{-1}(m))\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\varphi_{1}(m))\circ(E^{\prime}_{2}\Phi\pi)\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))
=(E1′​φ1​(m))∘(E1′​Φ​π)∘(φ1​(E2​(m)))∘(Φ​σ​(m))∘(φ2−1​(E1​(m)))∘(E2′​Φ​π)∘(E2′​φ2−1​(m))\displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(E^{\prime}_{1}\Phi\pi)\circ(\varphi_{1}(E_{2}(m)))\circ(\Phi\sigma(m))\circ(\varphi_{2}^{-1}(E_{1}(m)))\circ(E^{\prime}_{2}\Phi\pi)\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))
=(E1′​φ1​(m))∘(φ1​(E1​(m)))∘Φ⁡((E1​π)∘σ⁡(m)∘(E2​π))∘(φ2−1​(E2​(m)))∘(E2′​φ2−1​(m))\displaystyle=(E^{\prime}_{1}\varphi_{1}(m))\circ(\varphi_{1}(E_{1}(m)))\circ\Phi\bigl((E_{1}\pi)\circ\sigma(m)\circ(E_{2}\pi)\bigr)\circ(\varphi_{2}^{-1}(E_{2}(m)))\circ(E^{\prime}_{2}\varphi_{2}^{-1}(m))

It follows that

(E1′​π′)∘(σ′​Φ​(m))∘(E2′​π′)∘(τ2′​Φ​(m))=(τ1′​Φ​(m))∘(E1′​π′)∘(σ′​Φ​(m))∘(E2′​π′),(E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime})\circ(\tau^{\prime}_{2}\Phi(m))=(\tau^{\prime}_{1}\Phi(m))\circ(E^{\prime}_{1}\pi^{\prime})\circ(\sigma^{\prime}\Phi(m))\circ(E^{\prime}_{2}\pi^{\prime}),

hence (Φ⁡(m),π′)(\Phi(m),\pi^{\prime}) is an object of Δσ′​𝒲′\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}. We put Δ​Φ​(m,π)=(Φ⁡(m),π′)\Delta\Phi(m,\pi)=(\Phi(m),\pi^{\prime}).

Let f∈HomΔσ​𝒲⁡((m,π),(m~,π~))f\in\operatorname{Hom}\nolimits_{\Delta_{\sigma}{\mathcal{W}}}((m,\pi),(\tilde{m},\tilde{\pi})). We have a commutative diagram

E2′​Φ​(m)\textstyle{E^{\prime}_{2}\Phi(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}φ2−1​(m)\scriptstyle{\varphi_{2}^{-1}(m)}E2′​Φ​(f)\scriptstyle{E^{\prime}_{2}\Phi(f)}Φ​E2​(m)\textstyle{\Phi E_{2}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Φ​π\scriptstyle{\Phi\pi}Φ​E2​(f)\scriptstyle{\Phi E_{2}(f)}Φ​E1​(m)\textstyle{\Phi E_{1}(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}φ1​(m)\scriptstyle{\varphi_{1}(m)}Φ​E1​(f)\scriptstyle{\Phi E_{1}(f)}E1′​Φ​(m)\textstyle{E^{\prime}_{1}\Phi(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}E1′​Φ​(f)\scriptstyle{E^{\prime}_{1}\Phi(f)}E2′​Φ​(m~)\textstyle{E^{\prime}_{2}\Phi(\tilde{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}φ2−1​(m~)\scriptstyle{\varphi_{2}^{-1}(\tilde{m})}Φ​E2​(m~)\textstyle{\Phi E_{2}(\tilde{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Φ​π~\scriptstyle{\Phi\tilde{\pi}}Φ​E1​(m~)\textstyle{\Phi E_{1}(\tilde{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}φ1​(m~)\scriptstyle{\varphi_{1}(\tilde{m})}E1′​Φ​(m~)\textstyle{E^{\prime}_{1}\Phi(\tilde{m})}

and it follows that Φ⁡(f)∈HomΔσ′​𝒲′⁡(Δ​Φ​(m,π),Δ​Φ​(m~,π~))\Phi(f)\in\operatorname{Hom}\nolimits_{\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}}(\Delta\Phi(m,\pi),\Delta\Phi(\tilde{m},\tilde{\pi})). We put (Δ​Φ)​(f)=Φ​(f)(\Delta\Phi)(f)=\Phi(f). This makes Δ​Φ\Delta\Phi into a differential functor Δσ​𝒲→Δσ′​𝒲′\Delta_{\sigma}{\mathcal{W}}\to\Delta_{\sigma^{\prime}}{\mathcal{W}}^{\prime}.

We have

(Δ​Φ)​(E⁡(m,π))=(    Φ⁡(E2​(m))⊕Φ⁡(E1​(m))   Φ⁡(π)         ,β),(\Delta\Phi)(E(m,\pi))=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 50.11348pt\hbox{{\hbox{\kern-50.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.5pt\hbox{$\textstyle{\Phi(E_{2}(m))\oplus\Phi(E_{1}(m))}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-11.31735pt\raise 21.03578pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\Phi(\pi)}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 19.91682pt\raise 8.53578pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}},\beta),
β=(φ1​E2∘Φ​σ∘Φ​E2​π∘Φ​τ2∘φ2−1​E2φ1​E2∘Φ​σ∘φ2−1​E10φ1​E1∘Φ​τ1∘Φ​E1​π∘Φ​σ∘φ2−1​E1)\beta=\left(\begin{matrix}\varphi_{1}E_{2}\circ\Phi\sigma\circ\Phi E_{2}\pi\circ\Phi\tau_{2}\circ\varphi_{2}^{-1}E_{2}&\varphi_{1}E_{2}\circ\Phi\sigma\circ\varphi_{2}^{-1}E_{1}\\ 0&\varphi_{1}E_{1}\circ\Phi\tau_{1}\circ\Phi E_{1}\pi\circ\Phi\sigma\circ\varphi_{2}^{-1}E_{1}\end{matrix}\right)

and

E′​((Δ​Φ)​(m,π))=(    E2′​(Φ⁡(m))⊕E1′​(Φ⁡(m))   φ1​(m)∘Φ⁡(π)∘φ2​(m)−1         ,β′),E^{\prime}((\Delta\Phi)(m,\pi))=({\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 50.11348pt\hbox{{\hbox{\kern-50.11348pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.50891pt\hbox{$\textstyle{E^{\prime}_{2}(\Phi(m))\oplus E^{\prime}_{1}(\Phi(m))}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-41.0553pt\raise 21.53079pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.24501pt\hbox{$\scriptstyle{\varphi_{1}(m)\circ\Phi(\pi)\circ\varphi_{2}(m)^{-1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 19.91682pt\raise 8.53578pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces}}}}},\beta^{\prime}),
OPENβ′=(σ′​Φ∘E2′​(φ1∘Φ​π∘φ2−1)∘τ2′​Φσ′​Φ0τ1′​Φ∘E1′​(φ1∘Φ​π∘φ2−1)∘σ′​Φ))\beta^{\prime}=\left(\begin{matrix}\sigma^{\prime}\Phi\circ E^{\prime}_{2}(\varphi_{1}\circ\Phi\pi\circ\varphi_{2}^{-1})\circ\tau^{\prime}_{2}\Phi&\sigma^{\prime}\Phi\\ 0&\tau^{\prime}_{1}\Phi\circ E^{\prime}_{1}(\varphi_{1}\circ\Phi\pi\circ\varphi_{2}^{-1})\circ\sigma^{\prime}\Phi\end{matrix}\right))

We have

β′​(E2′​φ200E2′​φ1)=(E1′​φ200E1′​φ1)​β,\beta^{\prime}\left(\begin{matrix}E^{\prime}_{2}\varphi_{2}&0\\ 0&E^{\prime}_{2}\varphi_{1}\end{matrix}\right)=\left(\begin{matrix}E^{\prime}_{1}\varphi_{2}&0\\ 0&E^{\prime}_{1}\varphi_{1}\end{matrix}\right)\beta,

hence (φ2​(m)φ1​(m))\left(\begin{matrix}\varphi_{2}(m)\\ &\varphi_{1}(m)\end{matrix}\right) defines a closed isomorphism Δ​Φ​(E⁡(m,π))→∼E′​(Δ​Φ​(m,π))\Delta\Phi(E(m,\pi))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}(\Delta\Phi(m,\pi)). The naturality of φ1\varphi_{1} and φ2\varphi_{2} implies immediately that of φ\varphi.

We have τi′​Φ∘Ei′​φi∘φi​Ei=Ei′​φi∘φi​Ei∘Φ​τi\tau^{\prime}_{i}\Phi\circ E^{\prime}_{i}\varphi_{i}\circ\varphi_{i}E_{i}=E^{\prime}_{i}\varphi_{i}\circ\varphi_{i}E_{i}\circ\Phi\tau_{i} for i∈{1,2}i\in\{1,2\}. Together with (4.3.4), it follows that τ′​(Δ​Φ)∘E′​φ∘φ​E=E′​φ∘φ​E∘(Δ​Φ)​τ\tau^{\prime}(\Delta\Phi)\circ E^{\prime}\varphi\circ\varphi E=E^{\prime}\varphi\circ\varphi E\circ(\Delta\Phi)\tau, hence (Δ​Φ,φ)(\Delta\Phi,\varphi) defines a morphism of 22-representations. ∎

Original mathematics by the credited authors. Source collection and HTML conversion remain in progress.

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2