ScalingStacks

5.3.3. Action

We define the closed morphism of (B,A)(B,A)-bimodules u:E2⊗BA→E1⊗BAu:E_{2}\otimes_{B}A\to E_{1}\otimes_{B}A as the adjoint to the multiplication map E1∨​E2⊗BA→AE_{1}^{\vee}E_{2}\otimes_{B}A\to A. We define EE as the cone of uu.

We define a morphism of (B,A)(B,A)-bimodules v:E2⊗BE→E1⊗BEv:E_{2}\otimes_{B}E\to E_{1}\otimes_{B}E by

v11:E22⊗BA→τ2⊗1E22⊗BA→E2η1∙E2​E1​E1∨​E2⊗BA→σ∙E1​E2​E1∨​E2⊗BA→E1​E2​mult.E1​E2⊗BAv_{11}:E_{2}^{2}\otimes_{B}A\xrightarrow{\tau_{2}\otimes 1}E_{2}^{2}\otimes_{B}A\xrightarrow{E_{2}\eta_{1}\bullet}E_{2}E_{1}E_{1}^{\vee}E_{2}\otimes_{B}A\xrightarrow{\sigma\bullet}E_{1}E_{2}E_{1}^{\vee}E_{2}\otimes_{B}A\xrightarrow{E_{1}E_{2}\mathrm{mult.}}E_{1}E_{2}\otimes_{B}A
v12:E2​E1⊗BA→σ⊗1E1​E2⊗BAv_{12}:E_{2}E_{1}\otimes_{B}A\xrightarrow{\sigma\otimes 1}E_{1}E_{2}\otimes_{B}A
v21=0v_{21}=0
v22:E2​E1⊗BA→σ⊗1E1​E2⊗BA→E1η1∙E12​E1∨​E2⊗BA→τ1∙E12​E1∨​E2⊗BA→E12​mult.E12⊗BAv_{22}:E_{2}E_{1}\otimes_{B}A\xrightarrow{\sigma\otimes 1}E_{1}E_{2}\otimes_{B}A\xrightarrow{E_{1}\eta_{1}\bullet}E_{1}^{2}E_{1}^{\vee}E_{2}\otimes_{B}A\xrightarrow{\tau_{1}\bullet}E_{1}^{2}E_{1}^{\vee}E_{2}\otimes_{B}A\xrightarrow{E_{1}^{2}\mathrm{mult.}}E_{1}^{2}\otimes_{B}A

Note that the morphism vv corresponds, by adjunction, to the morphism w:E1∨​E2⊗BE→Ew:E_{1}^{\vee}E_{2}\otimes_{B}E\to E defined as follows

w11:E1∨​E22⊗BA→E1∨​τ2⊗1E1∨​E22⊗BA→λ​E2⊗1E2​E1∨​E2⊗BA→E2​mult.E2⊗BAw_{11}:E_{1}^{\vee}E_{2}^{2}\otimes_{B}A\xrightarrow{E_{1}^{\vee}\tau_{2}\otimes 1}E_{1}^{\vee}E_{2}^{2}\otimes_{B}A\xrightarrow{\lambda E_{2}\otimes 1}E_{2}E_{1}^{\vee}E_{2}\otimes_{B}A\xrightarrow{E_{2}\mathrm{mult.}}E_{2}\otimes_{B}A
w12:E1∨​E2​E1⊗BA→E1∨σ∙E1∨​E1​E2⊗BA→ε1∙E2⊗BA,w21=0w_{12}:E_{1}^{\vee}E_{2}E_{1}\otimes_{B}A\xrightarrow{E_{1}^{\vee}\sigma\bullet}E_{1}^{\vee}E_{1}E_{2}\otimes_{B}A\xrightarrow{\varepsilon_{1}\bullet}E_{2}\otimes_{B}A,\ w_{21}=0
w22:E1∨​E2​E1⊗BA→E1∨​σ⊗1E1∨​E1​E2⊗BA→ρ1∙E1​E1∨​E2⊗BA→E1​mult.E1⊗BA.w_{22}:E_{1}^{\vee}E_{2}E_{1}\otimes_{B}A\xrightarrow{E_{1}^{\vee}\sigma\otimes 1}E_{1}^{\vee}E_{1}E_{2}\otimes_{B}A\xrightarrow{\rho_{1}\bullet}E_{1}E_{1}^{\vee}E_{2}\otimes_{B}A\xrightarrow{E_{1}\mathrm{mult.}}E_{1}\otimes_{B}A.
0P6X

Lemma 5.3.4. The pair [E,v][E,v] gives EE a structure of differential (A,A)(A,A)-bimodule via Lemma 5.3.2. Furthermore, there is an isomorphism of functors Φℰ→∼(E⊗A−)Φ:Δσ(B−diff)→A−diff\Phi{\mathcal{E}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(E\otimes_{A}-)\Phi:\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)\to A\operatorname{\!-diff}\nolimits.

0P6Y

Proof. The vanishing of d​(v)11d(v)_{11} and d​(v)22d(v)_{22} follows from d⁡(τ1)=idd(\tau_{1})=\operatorname{id}\nolimits and d⁡(τ2)=idd(\tau_{2})=\operatorname{id}\nolimits. The vanishing of d​(v)12d(v)_{12} is clear. Finally, the vanishing of d​(v)21d(v)_{21} follows from the commutativity of the diagram (5.3.3). Since d⁡(v)=0d(v)=0, we have obtained a structure of differential (TB​(E1∨​E2),A)(T_{B}(E_{1}^{\vee}E_{2}),A)-bimodule on EE.

The object of Δσ​(B​−diff)\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits) corresponding to AA via Lemma 5.3.2 is (A,u)(A,u). We have ℰ⁡(A,u)=(E,v){\mathcal{E}}(A,u)=(E,v), where ℰ{\mathcal{E}} is the endofunctor defining the 22-representation on Δσ​(B​−diff)\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits). Since (E,v)(E,v) is an object of Δσ​(B​−diff)\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits), it follows that the action of TB​(E1∨​E2)T_{B}(E_{1}^{\vee}E_{2}) on EE factors through an action of AA. So, EE has a structure of differential (A,A)(A,A)-bimodule and we have an isomorphism of functors Φℰ→∼(E⊗A−)Φ:Δσ(B−diff)→A−diff\Phi{\mathcal{E}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(E\otimes_{A}-)\Phi:\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)\to A\operatorname{\!-diff}\nolimits. ∎

0P6Z

Remark 5.3.5. The maps vv and ww are described graphically as:

[Uncaptioned image]

We assume now that σ\sigma is invertible. We define τ\tau an endomorphism of (B,A)(B,A)-bimodules of E22⊗BA⊕E2​E1⊗BA⊕E1​E2⊗BA⊕E12⊗BAE_{2}^{2}\otimes_{B}A\oplus E_{2}E_{1}\otimes_{B}A\oplus E_{1}E_{2}\otimes_{B}A\oplus E_{1}^{2}\otimes_{B}A by

(5.3.4) τ=(τ2⊗100000σ−1⊗100000000τ1⊗1).\tau=\left(\begin{matrix}\tau_{2}\otimes 1&0&0&0\\ 0&0&\sigma^{-1}\otimes 1&0\\ 0&0&0&0\\ 0&0&0&\tau_{1}\otimes 1\end{matrix}\right).
0P70

Proposition 5.3.6. The pair (E,τ)(E,\tau) defines a 22-representation on AA and Φ\Phi induces a isomorphism of 22-representations Δσ​(B​−diff)→∼(Δσ​B)​−diff\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(\Delta_{\sigma}B)\operatorname{\!-diff}\nolimits. If E2E_{2} is right finite, then EE is right finite.

0P71

Proof. The fact that τ\tau defines an endomorphism of (A,A)(A,A)-bimodules of E2E^{2} satisfying the appropriate relations follows from the fact that it agrees with the endomorphism of ℰ2{\mathcal{E}}^{2} defining the 22-representation on Δσ​(B​−diff)\Delta_{\sigma}(B\operatorname{\!-diff}\nolimits). We deduce that (E,τ)(E,\tau) is a 22-representation on AA and Φ\Phi is a morphism of 22-representations.

Note that EE is finitely generated and projective as a (non-differential) AoppA^{\operatorname{opp}\nolimits}-module if E1E_{1} and E2E_{2} are finitely generated and projective BoppB^{\operatorname{opp}\nolimits}-modules. ∎

0P72

Remark 5.3.7. Consider three 22-representations (Ei,τi)1≤i≤3(E_{i},\tau_{i})_{1\leq i\leq 3} on a differential algebra BB together with closed morphisms σi​j:Ei​Ej→∼Ej​Ei\sigma_{ij}:E_{i}E_{j}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E_{j}E_{i} for i≠ji\neq j satisfying (4.3.5). Assume E1E_{1} and E2E_{2} are right finite. We will construct a triple tensor product 22-representation.

Define

λi​j:Ei∨​Ej→Ei∨​Ej​ηiEi∨​Ej​Ei​Ei∨→Ei∨​σj​i​Ei∨Ei∨​Ei​Ej​Ei∨→εi​Ej​Ei∨Ej​Ei∨\lambda_{ij}:E_{i}^{\vee}E_{j}\xrightarrow{E_{i}^{\vee}E_{j}\eta_{i}}E_{i}^{\vee}E_{j}E_{i}E_{i}^{\vee}\xrightarrow{E_{i}^{\vee}\sigma_{ji}E_{i}^{\vee}}E_{i}^{\vee}E_{i}E_{j}E_{i}^{\vee}\xrightarrow{\varepsilon_{i}E_{j}E_{i}^{\vee}}E_{j}E_{i}^{\vee}

and denote by σi​j∨:Ei∨​Ej∨→Ej∨​Ei∨\sigma_{ij}^{\vee}:E_{i}^{\vee}E_{j}^{\vee}\to E_{j}^{\vee}E_{i}^{\vee} the map adjoint to σi​j\sigma_{ij}.

Let A′=TB​(E1∨​E2⊕E2∨​E3⊕E1∨​E3)A^{\prime}=T_{B}(E_{1}^{\vee}E_{2}\oplus E_{2}^{\vee}E_{3}\oplus E_{1}^{\vee}E_{3}). There is a derivation ∂\partial of A′A^{\prime} whose restriction to B⊕E1∨​E2⊕E2∨​E3B\oplus E_{1}^{\vee}E_{2}\oplus E_{2}^{\vee}E_{3} is 00 and whose restriction to E1∨​E3E_{1}^{\vee}E_{3} is

∂:E1∨​E3→E1∨​η2​E3(E1∨​E2)​(E2∨​E3).\partial:E_{1}^{\vee}E_{3}\xrightarrow{E_{1}^{\vee}\eta_{2}E_{3}}(E_{1}^{\vee}E_{2})(E_{2}^{\vee}E_{3}).

Define A′′A^{\prime\prime} to be the differential algebra with underlying algebra A′A^{\prime} and with differential ∂+dA′\partial+d_{A^{\prime}}.

Let EE be the set of quadruples (i,j,k,l)(i,j,k,l) with i,j,k,l∈{1,2,3}i,j,k,l\in\{1,2,3\}, j−l≥i−k>0j-l\geq i-k>0 and (i,j,k,l)≠(2,3,1,2)(i,j,k,l)\neq(2,3,1,2). Given such a quadruple, we define

fi​j​k​l:El∨​Ek∨​Ei​Ej→σl​k∨​Ei​EjEk∨​El∨​Ei​Ej→Ek​λl​i​Ej(Ek∨​Ei)​(El∨​Ej)f_{ijkl}:E_{l}^{\vee}E_{k}^{\vee}E_{i}E_{j}\xrightarrow{\sigma_{lk}^{\vee}E_{i}E_{j}}E_{k}^{\vee}E_{l}^{\vee}E_{i}E_{j}\xrightarrow{E_{k}\lambda_{li}E_{j}}(E_{k}^{\vee}E_{i})(E_{l}^{\vee}E_{j})
gi​j​k​l:El∨​Ek∨​Ei​Ej→El∨​Ek∨​σi​jEl∨​Ek∨​Ej​Ei→El​λk​j​Ei(El∨​Ej)​(Ek∨​Ei)g_{ijkl}:E_{l}^{\vee}E_{k}^{\vee}E_{i}E_{j}\xrightarrow{E_{l}^{\vee}E_{k}^{\vee}\sigma_{ij}}E_{l}^{\vee}E_{k}^{\vee}E_{j}E_{i}\xrightarrow{E_{l}\lambda_{kj}E_{i}}(E_{l}^{\vee}E_{j})(E_{k}^{\vee}E_{i})
h3221:E1∨​E2∨​E3​E2→E1∨​λ23​E2E1∨​E3​E2∨​E2→E1∨​E3​ε2E1∨​E3h_{3221}:E_{1}^{\vee}E_{2}^{\vee}E_{3}E_{2}\xrightarrow{E_{1}^{\vee}\lambda_{23}E_{2}}E_{1}^{\vee}E_{3}E_{2}^{\vee}E_{2}\xrightarrow{E_{1}^{\vee}E_{3}\varepsilon_{2}}E_{1}^{\vee}E_{3}

where we put σr​r=τr\sigma_{rr}=\tau_{r} and σr​r∨=τr\sigma_{rr}^{\vee}=\tau_{r}. We define I′′I^{\prime\prime} to be the two-sided ideal generated by the images of fi​j​k​l+gi​j​k​l+δj​k​h3221f_{ijkl}+g_{ijkl}+\delta_{jk}h_{3221} for (i,j,k,l)∈E(i,j,k,l)\in E. We put A=A′′/I′′A=A^{\prime\prime}/I^{\prime\prime}.

As in §5.3.2, we have an isomorphism of differential categories Δ123​(B​−diff)→∼A​−diff\Delta_{123}(B\operatorname{\!-diff}\nolimits)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}A\operatorname{\!-diff}\nolimits (cf §4.3.5).

We obtain a bimodule 22-representation on AA as in §4.3.5. We define the differential (B,A)(B,A)-bimodule

E=    E3⊗BA⊕E2⊗BA⊕E1⊗BA   π31        π32        π21         E={\lx@xy@svg{\hbox{\raise 2.5pt\hbox{\kern 71.13152pt\hbox{{\hbox{\kern-64.13002pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise-2.66666pt\hbox{$\textstyle{E_{3}\otimes_{B}A\oplus E_{2}\otimes_{B}A\oplus E_{1}\otimes_{B}A}$}}}}}\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-8.99098pt\raise 37.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{31}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 71.13152pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern-42.33748pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\ \pi_{32}}$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 5.69052pt\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\ignorespaces{}{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{}{}{{}}{}\ignorespaces\ignorespaces\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{{}{}{}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern 25.32281pt\raise 19.54272pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.00694pt\hbox{$\scriptstyle{\pi_{21}\ \ }$}}}\kern 3.0pt}}}}}}\ignorespaces{}{}{}{{}}{\hbox{\kern 68.28625pt\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}}}}}

where

πi​j:Ei⊗BA→ηj​idEj​Ej∨​Ei⊗BA→Ej​multEj⊗BA.\pi_{ij}:E_{i}\otimes_{B}A\xrightarrow{\eta_{j}\operatorname{id}\nolimits}E_{j}E_{j}^{\vee}E_{i}\otimes_{B}A\xrightarrow{E_{j}\mathrm{mult}}E_{j}\otimes_{B}A.

We extend the left action of BB to an action of AA by letting the action maps

Ei∨​Ej⊗BE→EE_{i}^{\vee}E_{j}\otimes_{B}E\to E

for i<ji<j be given by

E1∨​E32⊗BA⊕E1∨​E3​E2⊗BA⊕E1∨​E3​E1⊗BA\textstyle{E_{1}^{\vee}E_{3}^{2}\otimes_{B}A\oplus E_{1}^{\vee}E_{3}E_{2}\otimes_{B}A\oplus E_{1}^{\vee}E_{3}E_{1}\otimes_{B}A}E3⊗BA⊕E2⊗BA⊕E1⊗BA\textstyle{E_{3}\otimes_{B}A\oplus E_{2}\otimes_{B}A\oplus E_{1}\otimes_{B}A}E3​ mult∘λ13​E3∘E1∨​τ3\scriptstyle{E_{3}\text{ mult}\circ\lambda_{13}E_{3}\circ E_{1}^{\vee}\tau_{3}}E1​mult∘ρ1​E3∘E1∨​σ31\scriptstyle{E_{1}\mathrm{mult}\circ\rho_{1}E_{3}\circ E_{1}^{\vee}\sigma_{31}}E3​ε1∘λ13​E1\scriptstyle{E_{3}\varepsilon_{1}\circ\lambda_{13}E_{1}\ \ }
E2∨​E32⊗BA⊕E2∨​E3​E2⊗BA⊕E2∨​E3​E1⊗BA\textstyle{E_{2}^{\vee}E_{3}^{2}\otimes_{B}A\oplus E_{2}^{\vee}E_{3}E_{2}\otimes_{B}A\oplus E_{2}^{\vee}E_{3}E_{1}\otimes_{B}A}E3⊗BA⊕E2⊗BA⊕E1⊗BA\textstyle{E_{3}\otimes_{B}A\oplus E_{2}\otimes_{B}A\oplus E_{1}\otimes_{B}A}E3​ mult∘λ23​E3∘E2∨​τ3\scriptstyle{E_{3}\text{ mult}\circ\lambda_{23}E_{3}\circ E_{2}^{\vee}\tau_{3}}E2​mult∘ρ2​E3∘E2∨​σ32\scriptstyle{\!\!E_{2}\mathrm{mult}\circ\rho_{2}E_{3}\circ E_{2}^{\vee}\sigma_{32}}E3​ε2∘λ23​E2\scriptstyle{E_{3}\varepsilon_{2}\circ\lambda_{23}E_{2}\!}E1​mult∘λ21​E3∘E2∨​σ31\scriptstyle{E_{1}\mathrm{mult}\circ\lambda_{21}E_{3}\circ E_{2}^{\vee}\sigma_{31}}
E1∨​E2​E3⊗BA⊕E1∨​E22⊗BA⊕E1∨​E2​E1⊗BA\textstyle{E_{1}^{\vee}E_{2}E_{3}\otimes_{B}A\oplus E_{1}^{\vee}E_{2}^{2}\otimes_{B}A\oplus E_{1}^{\vee}E_{2}E_{1}\otimes_{B}A}E3⊗BA⊕E2⊗BA⊕E1⊗BA\textstyle{E_{3}\otimes_{B}A\oplus E_{2}\otimes_{B}A\oplus E_{1}\otimes_{B}A}E3​mult∘λ13​E2∘E1∨​σ23\scriptstyle{E_{3}\mathrm{mult}\circ\lambda_{13}E_{2}\circ E_{1}^{\vee}\sigma_{23}}E2​ mult∘λ12​E2∘E1∨​τ2\scriptstyle{E_{2}\text{ mult}\circ\lambda_{12}E_{2}\circ E_{1}^{\vee}\tau_{2}\!\!}E2​ε1∘λ12​E1\scriptstyle{E_{2}\varepsilon_{1}\circ\lambda_{12}E_{1}\!}E1​mult∘ρ1​E2∘E1∨​σ21\scriptstyle{E_{1}\mathrm{mult}\circ\rho_{1}E_{2}\circ E_{1}^{\vee}\sigma_{21}}

Finally, we define the endomorphism τ\tau of E2E^{2} as in (4.3.6).

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

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2