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The Extension Problem for Lie Algebroids on Schemes The extension problem for Lie algebroids on schemes Ugo Bruzzo SISSA (International School for Advanced Studies), Trieste Universidade Federal da Para´ıba, Jo~aoPessoa, Brazil S~aoPaulo, November 14th, 2019 2nd Workshop of the S~aoPaulo Journal of Mathematical Sciences: J.-L. Koszul in S~aoPaulo, 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 | of characteristic zero. Lie algebroid: a vector bundle/coherent sheaf C with a morphism of OX -modules a: C ! ΘX and a |-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 OX . a is a morphism of sheaves of Lie |-algebras ker a is a bundle of Lie OX -algebras Ugo Bruzzo Extensions of Lie algebroids 2/26 Examples A sheaf of Lie algebras, with a = 0 ΘX , with a = id More generally, foliations, i.e., a is injective 1 π Poisson structures ΩX −! ΘX , Poisson-Nijenhuis bracket f!; τg = Lieπ(!)τ − Lieπ(τ)! − dπ(!; τ) Jacobi identity , [[π; π]] = 0 Ugo Bruzzo Extensions of Lie algebroids 3/26 Lie algebroid morphisms 0 f : C ! C a morphism of OX -modules & sheaves of Lie k-algebras f C / C 0 0 a a ΘX ) ker f is a bundle of Lie algebras Ugo Bruzzo Extensions of Lie algebroids 4/26 Derived functors A an abelian category, A 2 Ob(A) Hom(−; A): !Ab is a (contravariant) left exact functor, i.e., if 0 ! B0 ! B ! B00 ! 0 (∗) is exact, then 0 ! Hom(B00; A) ! Hom(B; A) ! Hom(B0; A) is exact Definition I 2 Ob(A) is injective if Hom(−; I ) is exact, i.e., for every exact sequence (*), 0 ! Hom(B00; I ) ! Hom(B; I ) ! Hom(B0; 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 Ri F : A ! B Ri F (A) = Hi (F (I •)) Example: Sheaf cohomology. X topological space, A = ShX , B = Ab, F = Γ (global sections functor) Ri Γ(F ) = Hi (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 OX -algebras with a |-linear augmentation U(C ) ! OX 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 (|; A)-Lie-Rinehart algebra L A |-algebra U(L) with an algebra monomorphism {: A ! U(L) and a |-module morphism |: L ! U(L), such that [|(s); |(t)] − |([s; t])= 0 ; s; t 2 L ; [|(s);{(f )] − {(a(s)(f ))= 0 ; s 2 L; f 2 A (∗) Construction: standard enveloping algebra U(A o L) of the semi-direct product |-Lie algebra A o L U(L) = U(A o L)=V ; V = hf (g; s) − (fg; fs)i U(L) is an A-module via the morphism { due to (*) the left and right A-module structures are different morphism ": U(L) ! U(L)=I = A (the augmentation morphism) where I is the ideal generated by |(L). Note that " is a morphism of U(L)-modules but not of A-modules, as "(fs) = a(s)(f ) when f 2 A, s 2 L. Ugo Bruzzo Extensions of Lie algebroids 8/26 Lie algebroid cohomology Given a representation (ρ, M ) of M define M C (U) = fm 2 M (U) j ρ(C )(m) = 0g and a left exact functor I C : Rep(C ) ! |-mod M 7! Γ(X ; M C ) Definition (B 20161) • • 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., Ri 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 pq p q E2 = R F (R G(A)) Ugo Bruzzo Extensions of Lie algebroids 10/26 Local to global (−)C Rep(C ) / |X -mod Γ C I & |-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 pq p q E2 = H (X ; H (C ; M )) 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 OX -algebras (−)K Rep(E ) / Rep(Q) Moreover, the sheaves q I Q H (K ; M ) are repre- E I % sentations of Q |-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 pq p q E2 = H (Q; H (K ; M )): 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 ! Out(Z(K )) (2) α(x)(y) = fy; x0g where π(x0) = x The extension problem is the following: Given a Lie algebroid Q, a coherent sheaf of Lie OX -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 oα Q; E = K ⊕ Q as OX -modules, f(`; x); (`0; x0)g = (α(x)(`0) − α(x0)(`); fx; x0g) E1 Theorem (2) B If K is abelian, the extension problem 0 / K Q / 0 is unobstructed; extensions are classified C up to equivalence by the 2 (1) hypercohomology group H (Q; K )α E2 (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≥1Λ•C ∗ ⊗ M defined by 0 / 0 / C ∗ ⊗ M / Λ2C ∗ ⊗ M /··· O degree 1 i (1) i ≥1 • ∗ We denote H (C ; M ) := H (X ; σ Λ C ⊗ M ) Derivation of C in M : morphism d : C ! M such that d(fx; yg) = x(d(y)) − y(d(x)) Ugo Bruzzo Extensions of Lie algebroids 15/26 Proposition i (1) The functors H (C ; −) are, up to a shift, the derived functors of Der(C ; −): Rep(C ) ! |-mod M 7! Der(C ; M ) i.e., i i+1 (1) R Der(C ; −) ' H (C ; −) Ugo Bruzzo Extensions of Lie algebroids 16/26 Realize the hypercohomology using Cechˇ 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 = Cˇp(U; F q) π 0 / / / / 0 (3) KjUi EjUi k QjUi si If Ui 2 U, Hom(QjUi ; EjUi ) ! Hom(QjUi ; QjUi ) is surjective, so that one has splittings si , and one can define 1 ∗ fφij = si − sj g 2 Cˇ (U; K ⊗ Q ) This is a 1-cocycle, which describes the extension only as an extension of OX -modules Ugo Bruzzo Extensions of Lie algebroids 17/26 0 ! K (Ui ) ! E (Ui ) ! Q(Ui ) ! 0 is an exact sequence of Lie-Rinehart algebras (over (|; OX (Ui ))) which is described by a 2-cocycle i in the Chevalley-Eilenberg (-Rinehart) cohomology of Q(Ui ) with coefficients in K (Ui ) (φ, ) 2 Cˇ1(U; K ⊗ Q∗) ⊕ Cˇ0(U; K ⊗ Λ2Q∗) = T 2 δφ = 0; dφ + δ = 0; d = 0 2 (1) ) cohomology class in H (Q; K )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 3 (1) ob(α) in H (Q; Z(K ))α . If ob(α) = 0, the space of equivalence classes of extensions is a 2 (1) torsor on H (Q; Z(K ))α . Proof 0 Q can be written as a quotient J of a free Lie algebroid F 0 / N / U(F ) / U(Q) / 0 3 ( ) E. Aldrovandi, U.B., V. Rubtsov, Lie OX algebroid cohomology and Lie algebroid extensions, J. of Algebra 505 (2018) 0 456{481 Ugo Bruzzo Extensions of Lie algebroids 19/26 Nfi = N i =N i+1; Jfi = N i J =N i+1J ; for i = 0;::: Locally free resolution ···! Nf2 ! Jf1 ! Nf1 ! Jf0 ! J ! 0 As HomU(Q)(J ; Z(K )) ' Der(Q; Z(K )), applying the functor HomU(Q)(−; Z(K )) we obtain 0 d1 0 ! Der(Q; Z(K )) ! HomU(Q)(Jf ; Z(K )) −! 1 d2 1 d3 HomU(Q)(Kf ; Z(K )) −! HomU(Q)(Jf ; Z(K )) −! 2 HomU(Q)(Kf ; Z(K )) ! ::: •+1 The cohomology of this complex is isomorphic to H (Q; Z(K )).
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