The extension problem for Lie algebroids on schemes Ugo Bruzzo SISSA (International School for Advanced Studies), Trieste Universidade Federal da Para´ ıba, Jo˜ ao Pessoa, Brazil S˜ ao Paulo, November 14th, 2019 2nd Workshop of the S˜ ao Paulo Journal of Mathematical Sciences: J.-L. Koszul in S˜ ao Paulo, His Work and Legacy Ugo Bruzzo Extensions of Lie algebroids 1/26
Lie algebroids X : a differentiable manifold, or complex manifold, or a noetherian separated scheme over an algebraically closed field k of characteristic zero. Lie algebroid: a vector bundle/coherent sheaf C with a morphism of O X -modules a : C → Θ X and a k -linear Lie bracket on the sections of C satisfying [ s , ft ] = f [ s , t ] + a ( s )( f ) t for all sections s , t of C and f of O X . a is a morphism of sheaves of Lie k -algebras ker a is a bundle of Lie O X -algebras Ugo Bruzzo Extensions of Lie algebroids 2/26
Examples A sheaf of Lie algebras, with a = 0 Θ X , with a = id More generally, foliations, i.e., a is injective π Poisson structures Ω 1 − → Θ X , X Poisson-Nijenhuis bracket { ω, τ } = Lie π ( ω ) τ − Lie π ( τ ) ω − d π ( ω, τ ) Jacobi identity ⇔ [ [ π, π ] ] = 0 Ugo Bruzzo Extensions of Lie algebroids 3/26
� � � Lie algebroid morphisms f : C → C ′ a morphism of O X -modules & sheaves of Lie k -algebras f C ′ C a ′ a Θ X ⇒ ker f is a bundle of Lie algebras Ugo Bruzzo Extensions of Lie algebroids 4/26
Derived functors A an abelian category, A ∈ Ob( A ) Hom( − , A ): → Ab is a (contravariant) left exact functor, i.e., if 0 → B ′ → B → B ′′ → 0 ( ∗ ) is exact, then 0 → Hom( B ′′ , A ) → Hom( B , A ) → Hom( B ′ , A ) is exact Definition I ∈ Ob( A ) is injective if Hom( − , I ) is exact, i.e., for every exact sequence (*), 0 → Hom( B ′′ , I ) → Hom( B , I ) → Hom( B ′ , I ) → 0 is exact Ugo Bruzzo Extensions of Lie algebroids 5/26
Definition The category A has enough injectives if every object in A has an injective resolution 0 → A → I 0 → I 1 → I 2 → . . . A abelian category with enough injectives F : A → B left exact functor Derived functors R i F : A → B R i F ( A ) = H i ( F ( I • )) Example: Sheaf cohomology. X topological space, A = Sh X , B = Ab , F = Γ (global sections functor) R i Γ( F ) = H i ( X , F ) Ugo Bruzzo Extensions of Lie algebroids 6/26
The category Rep( C ) From now on, X will be a scheme (with the previous hypotheses) Given a Lie algebroid C there is a notion of enveloping algebra U ( C ) It is a sheaf of associative O X -algebras with a k -linear augmentation U ( C ) → O X Rep( C ) ≃ U ( C ) -mod ⇒ Rep( C ) has enough injectives Ugo Bruzzo Extensions of Lie algebroids 7/26
Universal enveloping algebra U ( L ) of a ( k , A )-Lie-Rinehart algebra L A k -algebra U ( L ) with an algebra monomorphism ı : A → U ( L ) and a k -module morphism : L → U ( L ), such that [ ( s ) , ( t )] − ([ s , t ])= 0 , s , t ∈ L , [ ( s ) , ı ( f )] − ı ( a ( s )( f ))= 0 , s ∈ L , f ∈ A ( ∗ ) Construction: standard enveloping algebra U ( A ⋊ L ) of the semi-direct product k -Lie algebra A ⋊ L U ( L ) = U ( A ⋊ L ) / V , V = � f ( g , s ) − ( fg , fs ) � U ( L ) is an A -module via the morphism ı due to (*) the left and right A -module structures are different morphism ε : U ( L ) → U ( L ) / I = A (the augmentation morphism) where I is the ideal generated by ( L ). Note that ε is a morphism of U ( L )-modules but not of A -modules, as ε ( fs ) = a ( s )( f ) when f ∈ A , s ∈ L . Ugo Bruzzo Extensions of Lie algebroids 8/26
Lie algebroid cohomology Given a representation ( ρ, M ) of M define M C ( U ) = { m ∈ M ( U ) | ρ ( C )( m ) = 0 } and a left exact functor I C : Rep( C ) → k -mod Γ( X , M C ) M �→ Definition (B 2016 1 ) H • ( C ; M ) ≃ R • I C ( M ) ( 1 ) J. of Algebra 483 (2017) 245–261 Ugo Bruzzo Extensions of Lie algebroids 9/26
Grothendieck’s thm about composition of derived functors A F → B G − − → C A , B , C , abelian categories A , B with enough injectives F and G left exact, F sends injectives to G -acyclics (i.e., R i G ( F ( I )) = 0 for i > 0 when I is injective) Theorem For every object A in A there is a spectral sequence abutting to R • ( G ◦ F )( A ) whose second page is E pq = R p F ( R q G ( A )) 2 Ugo Bruzzo Extensions of Lie algebroids 10/26
� � Local to global ( − ) C � Rep( C ) k X -mod Γ I C k -mod Grothendieck’s theorem on the derived functors of a composition of functors implies: Theorem (Local to global spectral sequence) There is a spectral sequence, converging to H • ( C ; M ) , whose second term is E pq = H p ( X , H q ( C ; M )) 2 Ugo Bruzzo Extensions of Lie algebroids 11/26
� � Hochschild-Serre Extension of Lie algebroids 0 → K → E → Q → 0 K is a sheaf of Lie O X -algebras ( − ) K � Rep( E ) Rep( Q ) Moreover, the sheaves H q ( K ; M ) are repre- I Q I E sentations of Q k -mod Theorem (Hochschild-Serre type spectral sequence) For every representation M of E there is a spectral sequence E converging to H • ( E ; M ) , whose second page is E pq = H p ( Q ; H q ( K ; M )) . 2 Ugo Bruzzo Extensions of Lie algebroids 12/26
The extension problem An extension π 0 → K → E − → Q → 0 (1) defines a morphims α : Q → O ut ( Z ( K )) (2) { y , x ′ } π ( x ′ ) = x α ( x )( y ) = where The extension problem is the following: Given a Lie algebroid Q , a coherent sheaf of Lie O X -algebras K , and a morphism α as in (2), does there exist an extension as in (1) which induces the given α ? We assume Q is locally free Ugo Bruzzo Extensions of Lie algebroids 13/26
� � � � � Abelian extensions If K is abelian, ( K , α ) is a representation of Q on K , and one can form the semidirect product E = K ⋊ α Q , E = K ⊕ Q as O X -modules, { ( ℓ, x ) , ( ℓ ′ , x ′ ) } = ( α ( x )( ℓ ′ ) − α ( x ′ )( ℓ ) , { x , x ′ } ) E 1 Theorem ( 2 ) If K is abelian, the extension problem � K � 0 0 Q is unobstructed; extensions are classified up to equivalence by the hypercohomology group H 2 ( Q ; K ) (1) α E 2 ( 2 ) U.B., I. Mencattini, V. Rubtsov, and P. Tortella, Nonabelian holomorphic Lie algebroid extensions, Internat. J. Math. 26 (2015) 1550040 Ugo Bruzzo Extensions of Lie algebroids 14/26
� M a representation of a Lie algebroid C . Sharp truncation of the Chevalley-Eilenberg complex σ ≥ 1 Λ • C ∗ ⊗ M defined by � C ∗ ⊗ M � Λ 2 C ∗ ⊗ M � 0 0 � · · · degree 1 We denote H i ( C ; M ) (1) := H i ( X , σ ≥ 1 Λ • C ∗ ⊗ M ) Derivation of C in M : morphism d : C → M such that d ( { x , y } ) = x ( d ( y )) − y ( d ( x )) Ugo Bruzzo Extensions of Lie algebroids 15/26
Proposition The functors H i ( C ; − ) (1) are, up to a shift, the derived functors of Der( C ; − ): Rep( C ) → k -mod M �→ Der( C , M ) i.e., R i Der( C ; − ) ≃ H i +1 ( C ; − ) (1) Ugo Bruzzo Extensions of Lie algebroids 16/26
� � Realize the hypercohomology using ˇ Cech cochains: if U is an affine cover of X , and F • a complex of sheaves on X , then H • ( X , F • ) is isomorphic to the cohomology of the total complex T of K p , q = ˇ C p ( U , F q ) π � K | U i � E | U i � Q | U i 0 0 (3) s i If U i ∈ U , Hom( Q | U i , E | U i ) → Hom( Q | U i , Q | U i ) is surjective, so that one has splittings s i , and one can define { φ ij = s i − s j } ∈ ˇ C 1 ( U , K ⊗ Q ∗ ) This is a 1-cocycle, which describes the extension only as an extension of O X -modules Ugo Bruzzo Extensions of Lie algebroids 17/26
0 → K ( U i ) → E ( U i ) → Q ( U i ) → 0 is an exact sequence of Lie-Rinehart algebras (over ( k , O X ( U i ))) which is described by a 2-cocycle ψ i in the Chevalley-Eilenberg (-Rinehart) cohomology of Q ( U i ) with coefficients in K ( U i ) ( φ, ψ ) ∈ ˇ C 1 ( U , K ⊗ Q ∗ ) ⊕ ˇ C 0 ( U , K ⊗ Λ 2 Q ∗ ) = T 2 δφ = 0 , d φ + δψ = 0 , d ψ = 0 ⇒ cohomology class in H 2 ( Q ; K ) (1) α Ugo Bruzzo Extensions of Lie algebroids 18/26
� � � � � The nonabelian case Theorem ( 2 , 3 ) If K is nonabelian, the extension problem is obstructed by a class ob ( α ) in H 3 ( Q ; Z ( K )) (1) α . If ob ( α ) = 0 , the space of equivalence classes of extensions is a torsor on H 2 ( Q ; Z ( K )) (1) α . Proof 0 Q can be written as a quotient J of a free Lie algebroid F � 0 � N � U ( F ) 0 U ( Q ) O X ( 3 ) E. Aldrovandi, U.B., V. Rubtsov, Lie algebroid cohomology and Lie algebroid extensions, J. of Algebra 505 (2018) 0 456–481 Ugo Bruzzo Extensions of Lie algebroids 19/26
N i = N i / N i +1 , J i = N i J / N i +1 J , � � for i = 0 , . . . Locally free resolution N 2 → J 1 → � N 1 → J 0 → J → 0 · · · → � � � As Hom U ( Q ) ( J , Z ( K )) ≃ Der( Q , Z ( K )), applying the functor Hom U ( Q ) ( − , Z ( K )) we obtain d 1 0 → Der( Q , Z ( K )) → Hom U ( Q ) ( � J 0 , Z ( K )) − → d 2 d 3 Hom U ( Q ) ( � K 1 , Z ( K )) → Hom U ( Q ) ( � J 1 , Z ( K )) − − → Hom U ( Q ) ( � K 2 , Z ( K )) → . . . The cohomology of this complex is isomorphic to H • +1 ( Q ; Z ( K )). Ugo Bruzzo Extensions of Lie algebroids 20/26
Recommend
More recommend
