ScalingStacks

5.1. Differential algebras

5.1.1. 22-representations

Let AA be a differential algebra.

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Definition 5.1.1. A 22-representation on AA is the data of a differential (A,A)(A,A)-bimodule EE and of an endomorphism τ\tau of the (A,A)(A,A)-bimodule E⊗AEE\otimes_{A}E such that

τ2=0,d⁡(τ)=id⁡ and ​(E⊗τ)∘(τ⊗E)∘(E⊗τ)=(τ⊗E)∘(E⊗τ)∘(τ⊗E).\tau^{2}=0,\ d(\tau)=\operatorname{id}\nolimits\text{ and }(E\otimes\tau)\circ(\tau\otimes E)\circ(E\otimes\tau)=(\tau\otimes E)\circ(E\otimes\tau)\circ(\tau\otimes E).

We say that the 22-representation is right finite if EE is finitely generated and projective as a (non-differential) AoppA^{\operatorname{opp}\nolimits}-module.

Consider a 22-representation on AA. Note that E⊗A−E\otimes_{A}- is a differential endofunctor of A​−diffA\operatorname{\!-diff}\nolimits, and τ\tau defines an endomorphism of (E⊗A−)2(E\otimes_{A}-)^{2}. This gives a structure of 22-representation on A​−diffA\operatorname{\!-diff}\nolimits. It restricts to a 22-representation on (A¯)i(\bar{A})^{i} if EE is strictly perfect as a differential AA-module.

Note that there is a morphism of differential algebras

Hn→EndA⊗Aopp⁡(En),Ti↦En−i−1⊗τ⊗Ei−1.H_{n}\to\operatorname{End}\nolimits_{A\otimes A^{\operatorname{opp}\nolimits}}(E^{n}),\ T_{i}\mapsto E^{n-i-1}\otimes\tau\otimes E^{i-1}.

Let A′A^{\prime} be another differential algebra with a 22-representation (E′,τ′)(E^{\prime},\tau^{\prime}). We define a morphism of 22-representations from (A,E,τ)(A,E,\tau) to (A′,E′,τ′)(A^{\prime},E^{\prime},\tau^{\prime}) to be an (A′,A)(A^{\prime},A)-bimodule PP together with a closed isomorphism of (A′,A)(A^{\prime},A)-bimodules φ:P⊗AE→∼E′⊗A′P\varphi:P\otimes_{A}E\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}E^{\prime}\otimes_{A^{\prime}}P such that

(5.1.1) τ′​P∘E′​φ∘φ​E=E′​φ∘φ​E∘P​τ:P​E2→E′2​P.\tau^{\prime}P\circ E^{\prime}\varphi\circ\varphi E=E^{\prime}\varphi\circ\varphi E\circ P\tau:PE^{2}\to E^{\prime 2}P.

Note that such a pair (P,φ)(P,\varphi) gives rise to a morphism of 22-representations (P⊗A−,φ):(A−diff,E⊗A−,τ)→(A′−diff,E′⊗A′−,τ′)(P\otimes_{A}-,\varphi):(A\operatorname{\!-diff}\nolimits,E\otimes_{A}-,\tau)\to(A^{\prime}\operatorname{\!-diff}\nolimits,E^{\prime}\otimes_{A^{\prime}}-,\tau^{\prime}).

We obtain a differential 22-category of 22-representations on differential algebras.

The opposite 22-representation is the data (A′,E′,τ′)(A^{\prime},E^{\prime},\tau^{\prime}) where A′=AoppA^{\prime}=A^{\operatorname{opp}\nolimits}, E′=EE^{\prime}=E and τ′=τ\tau^{\prime}=\tau. Note that (A,E,τ)(A,E,\tau) coincides with its double dual.

Assume now the 22-representation is right finite. We have two morphisms of (A,A)(A,A)-bimodules η:A→E⊗AE∨\eta:A\to E\otimes_{A}E^{\vee} and ε:E∨⊗AE→A\varepsilon:E^{\vee}\otimes_{A}E\to A (unit and counit of adjunction). We have a morphism of (A,A)(A,A)-bimodules ρ:E∨​E→E​E∨\rho:E^{\vee}E\to EE^{\vee} defined as the composition

ρ:E∨​E→∙ηE∨​E​E​E∨→E∨​τ​E∨E∨​E​E​E∨→ε∙E​E∨.\rho:E^{\vee}E\xrightarrow{\bullet\eta}E^{\vee}EEE^{\vee}\xrightarrow{E^{\vee}\tau E^{\vee}}E^{\vee}EEE^{\vee}\xrightarrow{\varepsilon\bullet}EE^{\vee}.

There is a canonical isomorphism of differential algebras End⁡(E2)opp→∼End⁡((E∨)2)\operatorname{End}\nolimits(E^{2})^{\operatorname{opp}\nolimits}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{End}\nolimits((E^{\vee})^{2}) and we still denote by τ\tau the endomorphism of (E∨)2(E^{\vee})^{2} corresponding to τ\tau.

We define the left dual 22-representation on AA with the bimodule E∨E^{\vee} and the endomorphism τ\tau.

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

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2