aa r X i v : . [ m a t h . QA ] D ec ON SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS
MARCO MANETTI
Abstract.
We give some formality criteria for a differential graded Lie algebra to beformal. For instance, we show that a DG-Lie algebra L is formal if and only if the naturalspectral sequence computing the Chevalley-Eilenberg cohomology H ∗ CE ( L, L ) degeneratesat E . Introduction
The notion of formality of a differential graded commutative algebra has been quite fa-miliar in mathematics since the works by Deligne, Griffiths, Morgan, Sullivan [6], where it isproved that the de Rham algebra of a compact K¨ahler manifold X is formal and thereforeits homotopy class, controlling the real homotopy type of X , is uniquely determined by thecohomology algebra H ∗ ( X, R ).Similarly, the notion of formality of a differential graded Lie algebra has received a greatattention after the papers of Goldman, Millson [9] and Kontsevich [19]. In [9] the authorsrealize that the same approach of [6] can be used to prove the formality of the differentialgraded Lie algebra of differential forms with values in certain flat bundles of Lie algebras; asa consequence of this fact they proved that the moduli space of certain representations of thefundamental group of a compact K¨ahler manifold has at most quadratic singularities.In the paper [19] Kontsevich proved that, when A is the algebra of smooth functionson a differentiable manifold, then the natural DG-Lie algebra structure on the Hochschildcohomology complex of A with coefficients in A is formal, and then proving that every finitedimensional Poisson manifolds admits a canonical deformation quantization.Recall that a DG-Lie algebra L is formal if there exists a pair of quasi-isomorphisms ofDG-Lie algebras L ←−−− M −−−→ H with H having trivial differential. A DG-Lie algebra L is called homotopy abelian if thereexists a pair of quasi-isomorphisms of DG-Lie algebras L ←−−− M −−−→ H with H having trivial bracket and trivial differential. Thus, a DG-Lie algebra L is homotopyabelian if and only if it is formal and the cohomology graded Lie algebra H ∗ ( L ) is abelian.In view of the general principle that, in characteristic 0 every deformation problem iscontrolled by a DG-lie algebras, with quasi-isomorphic DG-Lie algebras giving the same de-formation problem [9], both the notions of formality and homotopy abelianity play a centralrole in deformation theory. Fortunately, in literature there exist several general and very usefulcriteria for homotopy abelianity. For example, it is known (see e.g. [14] and references therein)that for a morphism of DG-Lie algebras f : L → M we have: Date : December 5, 2013.2010
Mathematics Subject Classification.
Key words and phrases.
Differential graded Lie algebras, spectral sequences, L ∞ -algebras. (1) if L is homotopy abelian and f : H ∗ ( L ) → H ∗ ( M ) is surjective, then also M is homo-topy abelian;(2) if M is homotopy abelian and f : H ∗ ( L ) → H ∗ ( M ) is injective, then also L is homo-topy abelian.The above results, which are completely symmetric in their proofs, appear quite differentin their applications, being the latter used in almost all the algebraic proofs of generalizedBogomolov-Tian-Todorov theorems [15, 18], as well in deformation theory of holomorphicPoisson manifolds and coisotropic submanifolds [3, 7].The initial motivation for this paper was to seek for an analog of the above item (2) whenthe notion of homotopy abelianity is replaced with the notion of formality; it is easily verifiedthat if M is formal, then the injectivity of f : H ∗ ( L ) → H ∗ ( M ) is not sufficient to ensure theformality of L .In our proposed extension of item (2) for formality (Theorem 3.4), the cohomology gradedLie algebras H ∗ ( L ) and H ∗ ( M ) are replaced with suitable Chevalley-Eilenberg cohomologygroups. In particular we shall prove that if f : L → M is a morphism of differential gradedLie algebras, with M formal and f : H CE ( H ∗ ( L ) , H ∗ ( L )) → H CE ( H ∗ ( L ) , H ∗ ( M )) injective,then also L is formal.In doing this we have been deeply inspired by the papers [17, 23], where it is explainedwhat is the “right” obstruction to formality of a DG-algebra and by [2], where it is proved thathomotopy abelianity is equivalent to the degeneration at E of the natural spectral sequencecomputing the Chevalley-Eilenberg cohomology.The paper is organized as follows: in Section 2 we introduce the Chevalley-Eilenberg com-plex of a differential graded Lie algebra as the natural generalization of the classical Chevalley-Eilenberg complex of a Lie algebra [5]; here the choice of signs of the differential is purelyteleological and made by taking into account the sign convention used in the definition ofd´ecalage maps given in Section 7. Such a complex admits a natural filtration giving a coho-mology spectral sequence.Since almost all the proofs of this paper require a good knowledge of L ∞ [1]-algebras and L ∞ -morphisms, for the benefit of the readers which are not familiar with these notions, inSection 3 we state the main results of the paper about formality of DG-Lie algebras. Theseresults will be proved in Section 7 as particular cases of some more general results concerningthe formality of L ∞ [1]-algebras. Among the applications of these results we give a proof of thefact that for every graded vector space V the graded Lie algebra Hom ∗ K ( V, V ) is intrinsicallyformal.In Section 4 we give a short review of the definition of L ∞ [1]-algebras, L ∞ -morphismsand Nijenhuis-Richardson bracket. In Section 5 we define the Chevalley-Eilenberg spectralsequence of an L ∞ [1]-algebra and we prove that, for every r >
0, its page E r is homotopyinvariant. Finally in Section 6 we prove the formality criteria for L ∞ [1]-algebras.In the last section we show how deformation theory can be used for constructing simpleexamples of non formal differential graded Lie algebras.2. The Chevalley-Eilenberg spectral sequence
Throughout this paper every vector space, tensor product, Lie algebra etc. is consideredover a fixed field K of characteristic 0. By a DG-vector space we shall mean a Z -gradedvector space equipped with a differential of degree +1; a DG-Lie algebra is a Lie object inthe category of DG-vector spaces. Given a homogeneous vector v on a graded vector space,its degree will be denoted either v or deg( v ). N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 3
Let L = ( L, d, [ − , − ]) be a differential graded Lie algebra and let M be an L -module. Thismeans that M = ( M, d ) is a DG-vector space and it is given a morphism of DG-vector spaces[ − , − ] : M ⊗ L → M such that [ m, [ x, y ]] = [[ m, x ] , y ] − ( − x y [[ m, y ] , x ] . For instance, if f : L → M is a morphism of DG-Lie algebras, then M is an L -module via theadjoint representation [ m, x ] = [ m, f ( x )].We shall denote by H ∗ ( L ) and H ∗ ( M ) the cohomology of the DG-vector spaces ( L, d ) and(
M, d ), respectively. For every integer p ≥ CE ( L, M ) p, ∗ = Hom ∗ K ( L ∧ p , M ) , equipped with the natural differential δ : CE ( L, M ) p,q → CE ( L, M ) p,q +1 , namely( δφ )( x , . . . , x p ) = d ( φ ( x , . . . , x p )) − p X i =1 ( − φ + x + ··· + x i − φ ( x , . . . , dx i , . . . , x p ) , where every element of CE ( L, M ) p, ∗ is interpreted as a p -linear graded skewsymmetric map L × · · · × L → M . As usual we intend that L ∧ = K and then CE ( L, M ) , ∗ = M .Following [16, pag. 94], the Chevalley-Eilenberg complex of L with coefficients in M is thecomplex of DG-vector spaces: CE ( L, M ) : 0 → CE ( L, M ) , ∗ δ −→ CE ( L, M ) , ∗ δ −→ CE ( L, M ) , ∗ → · · · , i.e., the complex CE ( L, M ) : 0 → M δ −→ Hom ∗ K ( L, M ) δ −→ Hom ∗ K ( L ∧ , M ) → · · · where the differential δ is defined in the following way:(1) for m ∈ M we have ( δm )( x ) = ( − m [ m, x ];(2) for φ ∈ Hom ∗ K ( L, M ) we have( δφ )( x, y ) = ( − φ +1 (cid:0) [ φ ( x ) , y ] − ( − x y [ φ ( y ) , x ] − φ ([ x, y ]) (cid:1) ;(3) for p ≥ φ ∈ Hom ∗ K ( L ∧ p − , M ) we have:( δφ )( x , . . . , x p ) == ( − φ + p − X i χ i [ φ ( x , . . . , b x i , . . . , x p ) , x i ] − X i The Chevalley-Eilenberg cohomology H ∗ CE ( L, M ) of the differential gradedLie algebra L with coefficients in the L -module M is the cohomology of the total complexTot Q ( CE ( L, M ) , δ, δ ).In other words, H ∗ CE ( L, M ) is the cohomology of the complex · · · → A i δ + δ −−→ A i +1 → · · · ,where A n = Y p + q = n Hom q K ( L ∧ p , M ) . The Chevalley-Eilenberg complex carries the natural, decreasing, exhaustive and completefiltration F p CE ( L, M ) = Hom ∗ K M i ≥ p ^ i L, M , p ≥ . We shall denote by ( E ( L, M ) p,qr , d r ) the associated (Chevalley-Eilenberg) cohomology spectralsequence. Example 2.2. If L, M have trivial differentials, then H ∗ ( L ) = L , H ∗ ( M ) = M and therefore E ( L, M ) p,q = E ( L, M ) p,q . Moreover the spectral sequence degenerates at E (i.e., d r = 0 forevery r ≥ 2) and H iCE ( L, M ) = Y p ≥ E ( L, M ) p,i − p , where E ( L, M ) p,i − p = ker(Hom i − p K ( L ∧ p , M ) δ −→ Hom i − p K ( L ∧ p +1 , M )) δ Hom i − p K ( L ∧ p − , M ) . In general, since E ( L, M ) p, ∗ = F p CE ( L, M ) F p +1 CE ( L, M ) = Hom ∗ K ( L ∧ p , M )and the field K is assumed to be of characteristic 0, we have H ∗ ( L ∧ p ) = H ∗ ( L ) ∧ p (see e.g.[30, pag. 280]) and then E ( L, M ) p,q = H q (Hom ∗ K ( L ∧ p , M ) , δ ) = Hom q K ( H ∗ ( L ) ∧ p , H ∗ ( M ))= E ( H ∗ ( L ) , H ∗ ( M )) p,q . (2.1)The differential d : E ( L, M ) p,q → E ( L, M ) p +1 ,q depends only by the graded Lie algebra H ∗ ( L ) and its module H ∗ ( M ), giving E ( L, M ) p, ∗ = E ( H ∗ ( L ) , H ∗ ( M )) p, ∗ = H p ( CE ( H ∗ ( L ) , H ∗ ( M )) , δ ) N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 5 and therefore E ( L, M ) , ∗ = E ( H ∗ ( L ) , H ∗ ( M )) , ∗ = { derivations H ∗ ( L ) → H ∗ ( M ) }{ inner derivations } . Statement of the main results For the clarity of exposition, we list here the main results proved in this paper; the proofsrely on the theory of L ∞ [1]-algebras and will be postponed in next sections. Definition 3.1 (Euler class) . The Euler class of a morphism of differential graded Lie algebras f : L → M is the element e f ∈ E ( L, M ) , = E ( H ∗ ( L ) , H ∗ ( M )) , corresponding to the Eulerderivation e f : H ∗ ( L ) → H ∗ ( M ) , e f ( x ) = deg( x ) f ( x ) . The Euler class of a DG-Lie algebra L is defined as the Euler class of the identity on L . Lemma 3.2. Let e ∈ E ( L, L ) , be the Euler class of a differential graded Lie algebra L . If d ( e ) = · · · = d r − ( e ) = 0 for some r > , then d k = 0 for every ≤ k < r ; in particular E ( L, L ) p,qr = E ( L, L ) p,q . Every morphism of differential graded Lie algebras f : L → M induces by composition twonatural morphisms of double complexes CE ( L, L ) f ∗ / / CE ( L, M ) CE ( M, M ) f ∗ o o and then also two morphisms of spectral sequences(3.1) E ( L, L ) p,qr f ∗ / / E ( L, M ) p,qr E ( M, M ) p,qrf ∗ o o . preserving Euler classes. If f is a quasi-isomorphism, then by (2.1) the maps in (3.1) areisomorphisms for r ≥ 1. Therefore, the truncation at r ≥ of the spectral sequence E ( L, L ) p,qr and the Euler class are homotopy invariants of L . Theorem 3.3 (Formality criterion) . Let ( E ( L, L ) p,qr , d r ) be the Chevalley-Eilenberg spectralsequence of a differential graded Lie algebra L . Then the following conditions are equivalent: (1) L is formal; (2) the spectral sequence E ( L, L ) p,qr degenerates at E ; (3) denoting by e ∈ E ( L, L ) , = Der K ( H ∗ ( L ) , H ∗ ( L )) { [ x, − ] | x ∈ H ( L ) } , e ( x ) = deg( x ) · x, the Euler class of L , we have d r ( e ) = 0 ∈ E ( L, L ) r +1 , − rr for every r ≥ ; According to Lemma 3.2 the above item (3) makes sense. By the above considerations aboutthe homotopy invariance of the Chevalley-Eilenberg spectral sequence and Euler classes, theonly non trivial implication is (3 ⇒ Theorem 3.4 (Formality transfer) . Let f : L → M be a morphism of differential graded Liealgebras. Assume that (1) M is formal; (2) for every p ≥ the map f : E ( H ∗ ( L ) , H ∗ ( L )) p, − p → E ( H ∗ ( L ) , H ∗ ( M )) p, − p is injective. MARCO MANETTI Then also L is formal. As we have already pointed out, the above Item (2) holds whenever the natural map f : H CE ( H ∗ ( L ) , H ∗ ( L )) → H CE ( H ∗ ( L ) , H ∗ ( M ))is injective. Corollary 3.5. Let L, M be a differential graded Lie algebra. Then L × M is formal if andonly if both L and M are formal.Proof. Immediate consequence of Theorem 3.4, since L (resp.: M ) is a direct summand of the L -module (resp.: M -module) L × M . (cid:3) The next corollary in the Lie analog of a remarkable result by Sullivan, Halperin andStasheff [10, Cor. 6.9]. Corollary 3.6. Let L be a differential graded Lie algebra and let A be a unitary differentialgraded commutative K -algebra. If H ∗ ( A ) = 0 and L ⊗ A is formal, then also L is formal.Proof. Let’s denote by d : A i → A i +1 the differential of A , then d (1) = 0 and the assumption H ∗ ( A ) = 0 implies that the cohomology class of 1 is non trivial in H ( A ): in fact, if 1 = da for some a ∈ A − , then for every b ∈ A such that d ( b ) = 0 we have d ( ab ) = b . Thus themorphism K → A , α α 1, is injective in cohomology and therefore there exists a direct sumdecomposition A = K ⊕ B with d ( B ) ⊆ B .Now the proof follows from Theorem 3.4, since L is a direct summand of the L -module L ⊗ A = L ⊕ ( L ⊗ B ). (cid:3) Definition 3.7 ([12]) . A graded Lie algebra H is intrinsically formal if every differentialgraded Lie algebra L such that H ∗ ( L ) ∼ = H is formal.Putting M = 0 in Theorem 3.4 we recover the well known fact [12, 17] that a graded Liealgebra L with E ( L, L ) p, − p = 0 for every p ≥ Corollary 3.8. For every graded Lie algebra M and every h ∈ M the graded Lie subalgebra H = { x ∈ M | [ h, x ] = deg( x ) x } is intrinsically formal.Proof. Notice first that h ∈ H and then the Euler derivation e = [ h, − ] : H → H is an innerderivation. Let L be a differential graded Lie algebra with H ∗ ( L ) = H , then E ( L, L ) , = E ( H, H ) , = Der K ( H, H ) { [ x, − ] | x ∈ H } , and therefore the Euler class is trivial in E ( L, L ) , . (cid:3) Example 3.9. For every graded vector space V , the graded Lie algebras Hom ∗ K ( V, V ),Hom ≥ K ( V, V ) and Hom ≤ K ( V, V ) are intrinsically formal. In fact, denoting by h ∈ Hom K ( V, V ) , h ( v ) = deg( v ) v, we have [ h, f ] = e ( f ) = deg( f ) f, for every f ∈ Hom ∗ K ( V, V ) . N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 7 Example 3.10. For every graded commutative algebra A , the graded Lie algebra Der ∗ K ( A, A )is intrinsically formal. In fact, denoting by h ∈ Der K ( A, A ) , h ( v ) = deg( v ) v, we have [ h, f ] = e ( f ) = deg( f ) f, for every f ∈ Der ∗ K ( A, A ) . The same conclusion applies to every graded Lie subalgebra of Hom ∗ K ( A, A ) containing thederivation h , e.g. the algebra of differential operators.4. Review of L ∞ [1] -algebras and Nijenhuis-Richardson bracket Given a graded vector space V , the symmetric coalgebra generated by V is the gradedvector space S c ( V ) = ⊕ n ≥ V ⊙ n equipped with the coproduct∆(1) = 1 ⊗ , ∆( v ) = 1 ⊗ v + v ⊗ , and more generally ∆ : S c ( V ) → S c ( V ) ⊗ S c ( V ) , ∆( v ⊙ · · · ⊙ v n ) = n X k =0 X σ ∈ S ( n,n − k ) ǫ ( σ )( v σ (1) ⊙ · · · ⊙ v σ ( k ) ) ⊗ ( v σ ( k +1) ⊙ · · · ⊙ v σ ( n ) )where S ( k, n − k ) is the set of permutations of { , . . . , n } such that σ (1) < · · · < σ ( k ) , σ ( k + 1) < · · · < σ ( n ) , and ǫ ( σ ) is the Koszul sign. For every positive integer m the subspace L ≤ n ≤ m V ⊙ n is agraded subcoalgebra of S c ( V ).Throughout all this paper we shall use in force the following notation: whenever i, j ≥ f : S c ( V ) → S c ( W ) is a linear map, we shall denote by f ij : V ⊙ j → W ⊙ i the compositemap V ⊙ j inclusion −−−−−→ S c ( V ) f −−→ S c ( W ) projection −−−−−−→ W ⊙ i . The composition of f with the projection S c ( W ) → W is called the corestriction of f ; equiv-alently the corestriction of f is the linear map P j ≥ f j .The projection S c ( V ) → V ⊙ = K is a counity, while the inclusion K = V ⊙ → S c ( V )is an augmentation. With a little abuse of language, by a morphism f : S c ( V ) → S c ( W ) of symmetric coalgebras we shall mean a morphism of graded augmented coalgebras , i.e., amorphism of graded coalgebras such that f (1) = 1. In practice the assumption f (1) = 1is equivalent to the fact that f in non trivial: it is an easy exercise to show that, given amorphism of graded coalgebras f : S c ( V ) → S c ( W ), then either f = 0 or f (1) = 1.The following propositions are well known (see e.g. [21, 24]) and, in any case, easy to prove. Proposition 4.1. Every morphism of symmetric coalgebras f : S c ( V ) → S c ( W ) is uniquelydetermined by its corestriction, i.e., f is uniquely determined by the components f j , j > .Moreover, for every n > and v , . . . , v n ∈ V we have f ( v ⊙ · · · ⊙ v n ) − f ( v ) ⊙ · · · ⊙ f ( v n ) ∈ M
Proposition 4.2. The corestriction map gives an isomorphism of graded vector spaces Coder ∗ ( S c ( V ) , S c ( V )) → Hom ∗ K ( S c ( V ) , V ) = Y k ≥ Hom ∗ K ( V ⊙ k , V ) whose inverse map Hom ∗ K ( V ⊙ k , V ) ∋ q b q ∈ Coder ∗ ( S c ( V ) , S c ( V )) is described explicitly by the formulas b q ( v ⊙ · · · ⊙ v n ) = X σ ∈ S ( k,n − k ) ǫ ( σ ) q ( v σ (1) ⊙ · · · ⊙ v σ ( k ) ) ⊙ v σ ( k +1) ⊙ · · · ⊙ v σ ( n ) . For k = 0 the formula of Proposition 4.2 should be interpreted in the following sense: if q ∈ Hom ∗ K ( V ⊙ , V ) ∼ = Hom ∗ K ( K , V ), then b q (1) = q (1) , b q ( v ⊙ · · · ⊙ v n ) = q (1) ⊙ v ⊙ · · · ⊙ v n . Therefore, for q ∈ Hom ∗ K ( V ⊙ k , V ) we have b q ( V ⊙ n ) ⊆ V ⊙ n − k +1 .The graded commutator on Coder ∗ ( S c ( V ) , S c ( V )) induces, via the corestriction isomor-phism, a bracket[ − , − ] NR : Hom ∗ K ( S c ( V ) , V ) × Hom ∗ K ( S c ( V ) , V ) → Hom ∗ K ( S c ( V ) , V ) , known as Nijenhuis-Richardson bracket. In the notation of Proposition 4.2 we have[ − , − ] NR : Hom ∗ K ( V ⊙ n , V ) × Hom ∗ K ( V ⊙ m , V ) → Hom ∗ K ( V ⊙ n + m − , V ) , [ f, g ] NR = f b g − ( − f g g b f . Definition 4.3. An L ∞ [1]-algebra is a graded vector space V equipped with a coderivationof degree +1, Q ∈ Coder ( S c ( V ) , S c ( V )) such that Q (1) = 0 and QQ = [ Q, Q ] = 0. An L ∞ -morphism f : ( V, Q ) ( W, R ) of L ∞ [1]-algebras is a morphism of symmetric coalgebras f : S c ( V ) → S c ( W ) such that f Q = Rf .Thus, see e.g. [19], there exists a canonical bijection between the set of L ∞ -algebra struc-tures of a graded vector space V and the he set of L ∞ [1]-algebra structures of a graded vectorspace V [1]. In particular every result about L ∞ -algebras holds, mutatis mutandis, also for L ∞ [1]-algebras.Clearly we can define an L ∞ [1]-algebra also in terms of the Nijenhuis-Richardson bracket:more precisely an L ∞ [1]-algebra is an ∞ -uple ( V, q , q , . . . ), where q n ∈ Hom K ( V ⊙ n , V ) aresuch that for every n > n − X k =1 [ q k , q n − k ] NR = 0 ;the relation between the above two definitions is given by( V, Q ) ( V, Q , Q , Q , . . . ) , ( V, q , q , . . . ) V, X i> b q i ! . Notice that, if ( V, q , q , . . . ) is an L ∞ [1]-algebra we have q q = 0 and then ( V, q ) isa complex of vector spaces; we shall denote by H ∗ ( V ) its cohomology, called the tangentcohomology of V . Since the equation [ q , q ] NR = 0 may be written as q ( q ( x, y )) + q ( q ( x ) , y ) + ( − x q ( x, q ( y )) = 0we have that q factors to a graded commutative (quadratic) bracket on tangent cohomology: q : H ∗ ( V ) × H ∗ ( V ) → H ∗ ( V ) . N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 9 It is well known and in any case easy to prove that, for every L ∞ -morphism f : ( V, q , q , . . . ) ( W, r , r , . . . ) its linear component f : ( V, q ) → ( W, r )is a morphism of complexes whose restriction to tangent cohomology f : H ∗ ( V ) → H ∗ ( W )commutes with the quadratic brackets q , r . Example 4.4 (D´ecalage) . The d´ecalage functor, from the category of differential graded Liealgebras to the category of L ∞ [1]-algebra is defined as( L, d, [ − , − ]) ( V, q , q , , , . . . )where:(1) V is a graded vector space equipped with a linear map s : V → L , of degree +1,inducing an isomorphism s : V i ≃ −−→ L i +1 for every i ;(2) the maps q , q are defined by the formulas: sq ( v ) = − d ( sv ) , sq ( u, v ) = − ( − u [ su, sv ] , u, v ∈ V . Conversely, every L ∞ [1]-algebra ( V, q , q , . . . ) such that q i = 0 for every i ≥ Definition 4.5. An L ∞ -morphism f : ( V, q , q , . . . ) ( W, r , r , . . . ) is called a weak equiv-alence if induces an isomorphism between tangent cohomology groups f : H ∗ ( V ) ≃ −→ H ∗ ( W ). Definition 4.6. An L ∞ [1]-algebra ( V, q , q , . . . ) is said to be minimal if q = 0.In other words, an L ∞ [1]-algebra V is minimal if and only if H ∗ ( V ) = V . Theorem 4.7 (Minimal model theorem) . For every L ∞ [1] -algebra ( V, q , q , . . . ) there exista minimal L ∞ [1] -algebra ( W, , r , . . . ) and two weak equivalences f : ( V, q , q , . . . ) ( W, , r , . . . ) , g : ( W, , r , . . . ) ( V, q , q , . . . ) such that f g is the identity on W .Proof. See e.g. either Lemma 4.9 of [19], or Theorem 3.0.9 of [20]. (cid:3) The L ∞ [1]-algebra ( W, , r , . . . ) as in the above theorem is unique up to isomorphisms andit is called the minimal model of ( V, q , q , . . . ). It is worth to mention that, as a consequenceof Theorem 4.7, if f : ( V, q , q , . . . ) ( W, r , r , . . . ) is a weak equivalence, then there existsa weak equivalence g : ( W, r , r , . . . ) ( V, q , q , . . . ) such that g : H ∗ ( W ) → H ∗ ( V ) is theinverse of f : H ∗ ( V ) → H ∗ ( W ). Definition 4.8. An L ∞ [1]-algebra ( V, q , q , . . . ) is said to be formal if it is weak equivalentto a (pure quadratic) L ∞ [1]-algebra ( W, , r , , , . . . ). It is called homotopy abelian if it isweak equivalent to a trivial L ∞ [1]-algebra ( W, , , , , . . . )Therefore, an L ∞ [1]-algebra ( V, q , q , . . . ) is formal if and only if its minimal model isisomorphic to ( H ∗ ( V ) , , r , , , . . . ), where r is the restriction of the quadratic component q to the tangent cohomology H ∗ ( V ).It is well known, see e.g. [11], that two differential graded Lie algebras are quasi-isomorphicif and only if they have isomorphic L ∞ [1] minimal models. In particular a differential gradedLie algebra is formal (resp.: homotopy abelian) if and only if the associated L ∞ [1]-algebra isformal (resp.: homotopy abelian). Homotopy invariance of Chevalley-Eilenberg spectral sequence Let’s recall, following [8], the detailed construction of the spectral sequence associated toa differential filtered complex.Let M be an abelian group, equipped with a homomorphism d : M → M such that d = 0and a decreasing filtration F p M , p ∈ Z , such that d ( F p M ) ⊂ F p M . The associated spectralsequence ( E pr , d r ), r ≥ 0, is defined as Z pr = { x ∈ F p M | dx ∈ F p + r M } , E pr = Z pr Z p +1 r − + dZ p − r +1 r − , and the maps d r : E pr → E p + rr are induced by d in the obvious way. We have d r = 0 and there exist natural isomorphisms E pr +1 ≃ ker( d r : E pr → E p + rr ) d r ( E p − rr ) . If M = ⊕ M n is graded, d ( M n ) ⊂ M n +1 and every F p M = ⊕ n F p M n is a graded subgroup,then every group E pr inherits a natural graduation, namely: Z p,qr = { x ∈ F p M p + q | dx ∈ F p + r M p + q +1 } , E p,qr = Z p,qr Z p +1 ,q − r − + dZ p − r +1 ,q + r − r − ,E pr = M q E p,qr , d r : E p,qr → E p + r,q − r +1 r , E p,qr +1 ≃ ker( d r : E p,qr → E p + r,q − r +1 r ) d r ( E p − r,q + r − r ) . It is convenient to introduce a refinement of the usual notion of degeneration of a spectralsequence [6]. Definition 5.1. We shall say that a cohomology spectral sequence ( E p,qr , d r ) degenerates at E a,bk if the map d r : E a,br → E a + r,b − r +1 r vanishes for every r ≥ k . A spectral sequence ( E p,qr , d r )degenerates at E k if d r = 0 for every r ≥ k .For every morphism of graded coalgebras f : C → D we shall denote by Coder ∗ ( C, D ; f )the graded vector space of coderivations α : C → D , with the structure of D -comodule on C induced by the morphism f . When f is the identity we shall simply denote Coder ∗ ( C, C ) =Coder ∗ ( C, C ; Id C ). Definition 5.2. The Chevalley-Eilenberg complex of an L ∞ -morphism f : ( V, Q ) ( W, R )of L ∞ [1]-algebras is the filtered differential complex CE ( V, W ; f ) = Coder ∗ ( S c ( V ) , S c ( W ); f )= { α ∈ Hom ∗ K ( S c ( V ) , S c ( W )) | ∆ α = ( α ⊗ f + f ⊗ α )∆ } , where:(1) the filtration is defined as F p CE ( V, W ; f ) = { α ∈ Coder ∗ ( S c ( V ) , S c ( W ); f ) | α ( V ⊙ i ) = 0 , ∀ i < p } ;(2) the differential d is defined by the formula dα = Rα − ( − α αQ . N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 11 As in Proposition 4.2, the corestriction map gives an isomorphism of graded vector spacesCoder ∗ ( S c ( V ) , S c ( W ); f ) → Hom ∗ K ( S c ( V ) , W ) = Y k ≥ Hom ∗ K ( V ⊙ k , W ) , although the inverse map α b α is now described explicitly by a more complicated formula,cf. [24]. However for our applications we only need the description of b α in some particular andeasy cases, namely:(1) for w ∈ W the associated coderivation b w ∈ Coder ∗ ( S c ( V ) , S c ( W ); f ) satisfies theequalities b w (1) = w, b w ( v ) = w ⊗ f ( v ) , v ∈ V . (2) for α ∈ Hom ∗ K ( V, W ) the corresponding coderivation satisfies b α (1) = 0, b α ( v ) = α ( v )and b α ( v ⊙ v ) = α ( v ) ⊙ f ( v ) + ( − v v α ( v ) ⊙ f ( v ) , v , v ∈ V . The cohomology spectral sequence of the filtered differential complex CE ( V, W ; f ) definedabove will be denoted by ( E ( V, W ; f ) p,qr , d r ). Lemma 5.3. Let ( E ( V, W ; f ) r , d r ) be the Chevalley-Eilenberg spectral sequence of an L ∞ -morphism f : ( V, , q , q , . . . ) ( W, , r , r , . . . ) of minimal L ∞ [1] -algebras. Assume thatfor some integer k ≥ we have q = · · · = q k = 0 and r = · · · = r k = 0 . Then d r = 0 forevery ≤ r < k and therefore E ( V, W ; f ) = E ( V, W ; f ) = · · · = E ( V, W ; f ) k . Proof. By induction it is sufficient to prove that d k − = 0. We shall write Q = Q + Q ′ forthe codifferential of S c ( V ), where Q = b q , Q ′ = X i>k b q i , Q ( V ⊙ i ) ⊂ V ⊙ i − , Q ′ ( V ⊙ i ) ⊂ ⊕ j ≤ i − k V ⊙ j . Similarly we write R = R + R ′ , where R = b r , R ′ = X i>k b r i , R ( W ⊙ i ) ⊂ W ⊙ i − , R ′ ( W ⊙ i ) ⊂ ⊕ j ≤ i − k W ⊙ j . An element v ∈ E ( V, W ; f ) pk − is represented by a linear map α ∈ Hom ∗ K ( V ⊙ p , W ) suchthat R b α − ( − α b αQ ∈ F p + k − Coder ∗ ( S c ( V ) , S c ( W ); f ). Since k ≥ R b α − ( − α b αQ = 0, R b α − ( − α b αQ = R ′ b α − ( − α b αQ ′ ∈ F p + k Coder ∗ ( S c ( V ) , S c ( W ); f )and this implies that d k − ( v ) = 0. (cid:3) Lemma 5.4. Let f : ( V, Q ) ( W, R ) and g : ( W, R ) ( U, S ) be L ∞ -morphisms of L ∞ [1] -algebras. Then the composition maps give two morphisms of filtered differential complexes g ∗ : Coder ∗ ( S c ( V ) , S c ( W ); f ) → Coder ∗ ( S c ( V ) , S c ( U ); gf ) ,f ∗ : Coder ∗ ( S c ( W ) , S c ( U ); g ) → Coder ∗ ( S c ( V ) , S c ( U ); gf ) . Proof. Since f ( V ⊙ n ) ⊂ ⊕ i ≤ n W ⊙ i it is obvious that f ∗ and g ∗ preserve the filtrations. Now g ( dα ) = gRα − ( − α αQ = Sgα − ( − α gαQ = d ( gα )( dα ) f = Sαf − ( − α αRf = Sαf − ( − α αf Q = d ( αf ) . (cid:3) Proposition 5.5. In the situation of Lemma 5.4 the composition maps induce two morphismsof spectral sequences E ( W, U ; g ) p,qr f ∗ / / E ( V, U ; gf ) p,qr E ( V, W ; f ) p,qrg ∗ o o :(1) if f is a weak equivalence, then f ∗ : E ( W, U ; g ) p,qr → E ( V, U ; gf ) p,qr is an isomorphismfor every r ≥ ; (2) if g is a weak equivalence, then g ∗ : E ( V, W ; f ) p,qr → E ( V, U ; gf ) p,qr is an isomorphismfor every r ≥ .Proof. For every integer p , the corestriction map gives an isomorphism F p CE ( V, W ; f ) ≃ Y n ≥ p Hom ∗ K ( V ⊙ n , W )and therefore E ( V, W ; f ) p,q = Hom p + q K ( V ⊙ p , W ) . Given an element α ∈ Hom ∗ K ( V ⊙ p , W ) we have α = b α | V ⊙ p , b α ( V ⊙ i ) = 0 for every i < p andthen b αQ ( v ⊙ · · · ⊙ v n ) = α n X i =1 ( − v + ··· + v i − v ⊙ · · · ⊙ Q ( v i ) ⊙ · · · ⊙ v n ! R b α ( v ⊙ · · · ⊙ v n ) = R α ( v ⊙ · · · ⊙ v n ) ,d α = R α − ( − α α n X i =1 Id ⊙ i − ⊙ Q ⊙ Id ⊙ n − i ! . In other terms d is the standard differential in Hom ∗ K ( V ⊙ n , W ); by K¨unneth formula (cf. [30,pag. 280]) E ( V, W ; f ) p,q = Hom p + q K ( H ∗ ( V ) ⊙ p , H ∗ ( W )) . The conclusion of the proof is now clear. (cid:3) Definition 5.6. The Euler derivation of an L ∞ -morphism f : V W is the element e f ∈ E ( V, W ; f ) , − = Hom K ( H ∗ ( V ) , H ∗ ( W ))defined as e f ( v ) = ( v + 1) f ( v ) , v ∈ H ∗ ( V ) . We are now ready to prove the main result of this section. Theorem 5.7. Let W be the minimal model of an L ∞ [1] -algebra V . Then there exists amorphism of spectral sequences E ( V, V ) p,qr → E ( W, W ) p,qr which is an isomorphism for every r ≥ and preserves the Euler derivations.Proof. By minimal model theorem there exist two weak equivalences g : V W, f : W V such that gf is the identity on W . It now sufficient to consider the morphisms E ( V, V ) p,qr f ∗ −→ E ( W, V ; f ) p,qr g ∗ −→ E ( W, W ; gf ) p,qr = E ( W, W ) p,qr and apply Proposition 5.5. (cid:3) Lemma 5.8. Let e f ∈ E ( V, W ; f ) , − be the Euler derivation of an L ∞ -morphism. Then d ( e f ) = 0 . N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 13 Proof. By Proposition 5.5 it is not restrictive to assume both V and W minimal L ∞ [1]-algebras, say V = ( V, , q , . . . ) and W = ( W, , r , . . . ). Let’s give an explicit description ofthe two differentials E ( V, W ; f ) , d / / E ( V, W ; f ) , − d / / E ( V, W ; f ) , − W Hom K ( V, W ) Hom K ( V ∧ , W )Given w ∈ W the associated coderivation b w ∈ Coder ∗ ( S c ( V ) , S c ( W ); f ) satisfies the equali-ties b w (1) = w, b w ( v ) = w ⊗ f ( v ) , v ∈ V and then, ( d w )( v ) = r ( w, f ( v )) ∈ W . Similarly for α ∈ Hom K ( V, W ) the corresponding coderivation satisfies b α ( v ) = α ( v ), b α ( v ⊙ v ) = α ( v ) ⊙ f ( v ) + ( − v v α ( v ) ⊙ f ( v ) = α ( v ) ⊙ f ( v ) + f ( v ) ⊙ α ( v ) , and therefore( d α )( v ⊙ v ) = r ( α ( v ) , f ( v )) + r ( f ( v ) , α ( v )) − α ( q ( v , v )) . In particular( d e f )( v ⊙ v ) == r (( v + 1) f ( v ) , f ( v )) + r ( f ( v ) , ( v + 1) f ( v )) − ( v + v + 2) f ( q ( v , v ))= 0 . (cid:3) Definition 5.9. The Euler class of an L ∞ -morphism f : V W is the element e f ∈ E ( V, W ; f ) , − defined as the class of the Euler derivation modulus d ( W ). The Euler class of an L ∞ -algebrais the Euler class of the identity.It is plain from the above results that the Euler class of an L ∞ [1]-algebra is invariant underweak equivalence. 6. Formality criteria for L ∞ [1] -algebras Lemma 6.1. Let k be a positive integer and let ( V, q , q , . . . ) be an L ∞ [1] -algebra such that q i = 0 for every i = k . Then: (1) the spectral sequence E ( V, V ) p,qr degenerates at E k ; (2) if the spectral sequence E ( V, V ) p,qr degenerates at E , − k − then also q k = 0 .Proof. Via the identification CE ( V, V ) ≃ Hom ∗ K ( S c ( V ) , V ) the differential of the complexbecomes d = [ q k , − ] NR and then d (Hom ∗ K ( V ⊙ n , V )) ⊂ Hom ∗ K ( V ⊙ n + k − , V ). Therefore if α = P α i , with α i ∈ Hom ∗ K ( V ⊙ i , V ) and dα ∈ F n + k CE ( V, V ) then dα i = 0 for i ≤ n ;therefore, for every n ∈ Z and every r ≥ k we have dZ nr ⊂ d ( F n +1 CE ( V, V )) ∩ F n + r CE ( V, V ) = dZ n +1 r − and this implies d r = 0. As regards the second item, we assume k > 1, being the case k = 1 completely trivial.Considering the identity map Id V ∈ Hom K ( V, V ) as an element of F CE ( V, V ) we have: q k = 1 k − q k , Id V ] NR ∈ F k CE ( V, V ) , Id V ∈ Z , − k − . If the spectral sequence degenerates at E , − k − , then the class of q k is trivial in E k, − kk − ; inparticular q k ∈ d ( F CE ( V, V )) + F k +1 CE ( V, V )and this implies q k = 0. (cid:3) Lemma 6.2. Let ( V, , q , , . . . , , q i , . . . ) be a minimal L ∞ [1] -algebra such that q j = 0 forevery < j < i and some < i . Hence, by Lemma 5.3 we have E ( V, V ) p,q = E ( V, V ) p,qi − .Denoting by e ∈ E ( V, V ) , − its Euler class we have: (1) [ q , q i ] NR = 0 ; (2) d r ( e ) = 0 ∈ E ( V, V ) r +1 , − rr for every ≤ r < i − ; (3) if d i − ( e ) = 0 ∈ E ( V, V ) i, − ii − = E ( V, V ) i, − i , then there exists α ∈ Hom K ( V ⊙ i − , V ) such that q i = [ q , α ] NR .Proof. The first part is clear since P j [ q j , q i +2 − j ] NR = 0. The Euler class e is induced by thelinear map e ∈ Hom K ( V, V ) , e ( v ) = ( v + 1) v , and for every β ∈ Hom h K ( V ⊙ j , V ) we have[ β, e ] NR = ( j − h − β . Therefore, setting q = P q j , we have [ q , e ] NR = 0,[ q, e ] NR = ( i − q i + ( i − q i +1 + · · · ∈ F i CE ( V, V ) ∩ d ( F CE ( V, V )) . In particular [ q, e ] NR ∈ Z j +1 , − jj for every j < i and this implies that d r ( e ) = 0 ∈ E ( V, V ) r +1 , − rr for 1 ≤ r < i − 1. If d i − ( e ) = 0 ∈ E i, − ii − , then( i − q i + ( i − q i +1 + · · · ∈ Z i +1 , − ii − + dZ , − i − and then there exists a sequence α j ∈ Hom K ( V ⊙ j , V ), j ≥ 2, such that[ q, X α j ] NR − ( i − q i ∈ F i +1 CE ( V, V )this implies that [ q , α j ] NR = 0 for j < i − q , α i − ] NR = ( i − q i . In particular α = α i − i − (cid:3) Theorem 6.3. For a minimal L ∞ [1] -algebra ( V, , q , q , . . . ) with Euler class e ∈ E ( V, V ) , − ,the following conditions are equivalent: (1) there exists an L ∞ -isomorphism f : ( V, , q , , , . . . ) ( V, , q , q , . . . ) ; (2) the spectral sequence E ( V, V ) p,qr degenerates at E ; (3) d r ( e ) = 0 ∈ E ( V, V ) r +1 , − rr for every r ≥ .Proof. By Lemma 6.1 and the homotopy invariance of the Chevalley-Eilenberg spectral se-quence, we only need to prove (3 ⇒ q i = 0 for every i > N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 15 otherwise let i ≥ q i = 0; by Lemma 6.2 there exists an op-erator α ∈ Hom K ( V ⊙ i − , V ) such that [ q , α ] NR = q i . Denoting by b α ∈ Coder ( S c ( V ) , S c ( V ))the corresponding pronilpotent coderivation, by Q = P j b q j and by R = e − b α Qe b α = e [ b α, − ] ( Q ) = Q + [ b α, Q ] + 12 [ b α, [ b α, Q ]] + · · · we have that e b α : ( V, R ) ( V, Q ) is an L ∞ -morphism. Denoting by r and q = P q i thecorestrictions of R and Q , respectively, we have: r = e [ α, − ] NR ( q ) = q +[ α, q ] NR + · · · ≡ q + q i − [ q , α ] NR ≡ q mod Y j>i Hom K ( V ⊙ j , V ) . Therefore r = q + r i +1 + r i +2 + · · · and then we have an L ∞ -isomorphism e b α : ( V, q , , . . . , , r i +1 , . . . ) ( V, q , , . . . , , q i , q i +1 , . . . ) . Since e b α is the identity on V ⊙ j for every j ≤ i − 2, we can repeat the the procedure infinitelymany times and take f as the infinite composition product of the above exponentials. (cid:3) Corollary 6.4. For a minimal L ∞ [1] -algebra ( V, , q , q , . . . ) the following conditions areequivalent: (1) q i = 0 for every i ; (2) the spectral sequence E ( V, V ) p,qr degenerates at E ; (3) the spectral sequence E ( V, V ) p,qr degenerates at E , − .Proof. Immediate consequence of Theorem 6.3 and Lemma 6.1. (cid:3) Remark . In [17] it is proved (in the framework of A ∞ -algebras) that property (1) ofTheorem 6.3 is equivalent to the vanishing of a certain cohomology class, called “Kaledinclass” in [23]. For a minimal L ∞ [1]-algebras ( V, , q , . . . ), the Kaledin class may be definedin the following way: let t be a central formal indeterminate of degree 0; by extending theNijenhuis-Richardson bracket to CE ( V, V )[[ t ]] in the obvious way, by homogeneity we have[ q ( t ) , q ( t )] NR = 0 , where q ( t ) = q + tq + t q + · · · and then d ( t ) = [ q ( t ) , − ] NR is a differential on the K [[ t ]]-module CE ( V, V )[[ t ]].Taking the formal derivative on the variable t we have ∂ t q ( t ) = q + 2 tq + · · · , [ q ( t ) , ∂ t q ( t )] NR = 0and the cohomology class [ ∂ t q ( t )] ∈ H ( CE ( V, V )[[ t ]] , d ( t ))is called the Kaledin class of the minimal L ∞ [1]-algebra ( V, , q , . . . ); it is supported at t = 0,since t ∂ t q ( t ) = tq + 2 t q + · · · = [ q ( t ) , e ] NR where e is Euler class. In particular, the Kaledin class vanishes whenever the multiplicationmap H ( CE ( V, V )[[ t ]]) t −→ H ( CE ( V, V )[[ t ]]) is injective. Notice that there exists a shortexact sequence of complexes0 → ( CE ( V, V )[[ t ]] , d ( t )) t −→ ( CE ( V, V )[[ t ]] , d ( t )) t −−−→ ( CE ( V, V ) , [ q , − ] NR ) → E ( V, V ) p,qr degenerates at E a, − a for every a ≥ 1, then the map H ( CE ( V, V )[[ t ]] , d ( t )) t −−−→ H ( CE ( V, V ) , [ q , − ] NR ) is surjective.The following two corollaries follow immediately from the above results together with The-orem 5.7. Corollary 6.6. For an L ∞ [1] -algebra V with Euler class e ∈ E ( V, V ) , − , the followingconditions are equivalent: (1) V is formal; (2) the spectral sequence E ( V, V ) p,qr degenerates at E ; (3) d r ( e ) = 0 ∈ E ( V, V ) r +1 , − rr for every r ≥ . Corollary 6.7. For an L ∞ [1] -algebra V the following conditions are equivalent: (1) V is homotopy abelian; (2) the spectral sequence E ( V, V ) p,qr degenerates at E ; (3) the spectral sequence E ( V, V ) p,qr degenerates at E , − . The equivalence (1 ⇔ 2) of Corollary 6.7, together some nice applications, has been recentlyproved by R. Bandiera [2] in a different way.A well known result, which is very useful in deformation theory (see e.g. [13, 14, 15, 18]) isthat if f : V W is an L ∞ -morphism, W is homotopy abelian and f : H ∗ ( V ) → H ∗ ( W )is injective, then also V is homotopy abelian. If W is formal, then the injectivity in tangentcohomology is not sufficient to ensure the formality of V . However, we have the followingresult, proved as a consequence of Theorem 6.3. Theorem 6.8 (Formality transfer) . Let f : V W be an L ∞ -morphism of L ∞ [1] -algebrassuch that: (1) W is formal; (2) the map f ∗ : E ( V, V ) p, − p → E ( V, W ; f ) p, − p is injective for every p ≥ .Then also V is formal.Proof. It is not restrictive to assume V minimal and W purely quadratic, say f : ( V, , q , q , . . . ) ( W, , r , , , . . . ) . If q i = 0 for some i > 2, let k ≥ q k = 0; thus q i = 0 forevery 2 < i < k . According to Lemma 5.3 we have E ( V, V ) p, − p = E ( V, V ) p, − pk − , E ( V, W ; f ) p, − p = E ( V, W ; f ) p, − pk − and then also f ∗ : E ( V, V ) p, − pk − → E ( V, W ; f ) p, − pk − is injective for every p ≥ 3. We have twomorphisms of spectral sequences E ( V, V ) p,qr f ∗ / / E ( V, W ; f ) p,qr E ( W, W ) p,qrf ∗ o o Denoting by e V , e W and e f the Euler classes of V, W and f respectively we have f ∗ ( e V ) = e f = f ∗ ( e W ) and then for every 2 ≤ i < k we have f ∗ ( d i e V ) = d i ( f ∗ e V ) = d i ( f ∗ e W ) = f ∗ ( d i e W ) = 0 ∈ E ( V, W ; f ) i +1 , − ii and then d i e V = 0 ∈ E ( V, V ) i +1 , − ii for every 2 ≤ i < k . The same argument used in the proofof Theorem 6.3 implies that, up to composition with an L ∞ -isomorphism of V which is theidentity on V ⊙ i , i < k − 1, we can assume q k = 0. Repeating this step, possibly infinitelymany times for a sequence of increasing values of k , we prove the formality of V . (cid:3) N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 17 Formality criteria for differential graded Lie algebras By using the d´ecalage isomorphism we can rewrite every formality result for L ∞ [1]-algebrasin the framework of differential graded Lie algebras. If V, L are graded vector spaces, thenevery bijective linear map s : V → L of degree +1 extends naturally to a sequence of linearisomorphisms s : Hom n K ( V ⊙ k , V ) ≃ −−→ Hom n − k +1 K ( L ∧ k , L ) , n, k ∈ Z , k ≥ , defined by the formulas(7.1) ( sφ )( sv , . . . , sv k ) = ( − P i ( k − i ) v i s φ ( v , . . . , v k ) , v , . . . , v n ∈ V . Assume now that L is a differential graded Lie algebra and ( V, q , q , , . . . ) its d´ecalage,defined as in Example 4.4. A straightforward computation shows that the bijective linear mapof degree +1 s : CE ( V, V ) → CE ( L, L ) , defined in (7.1) commutes, in the graded sense, with the differentials: sδ + [ q , s ( − )] NR = sδ + [ q , s ( − )] NR = 0 . In particular, s gives a bijective morphism of spectral sequences of degree +1. s : E ( V, V ) p,qr → E ( L, L ) p,q +1 r preserving the Euler classes.After this, it is now clear that Lemma 3.2 follows from Lemma 5.3, Theorem 3.3 is a directconsequence of Corollary 6.6, while Theorem 3.4 follows from Theorem 6.8.One of the possible limitations in the application of Theorem 3.3 is that in general, thefiltration F p CE ( L, L ) is not bounded and the spectral sequence E ( L, L ) p,qr is not regular;thus it may be useful to restate our results in terms of the spectral sequences of the quotientcomplexes CE ( L, L ) /F p CE ( L, L ). Lemma 7.1. Let F p M , p ∈ Z , be a decreasing filtration of a differential graded abelian group M . Denote by E p,qr the associated spectral sequence and, for every integer b , by E ( l ) p,qr thespectral sequence of the quotient filtered complex M/F l M . Denoting by π : M → M/F l M theprojection, we have: (1) if p < l , then π : E pr → E ( l ) pr is injective; (2) if p + r ≤ l , then π : E pr → E ( l ) pr is surjective; (3) for a fixed pair a, b of integers, the spectral sequences E p,qr degenerates at E a,bk if andonly if E ( l ) p,qr degenerates at E ( l ) a,bk for every l .Proof. The first two properties are clear for r = 0; by induction we may assume r > r − 1. We have a commutative diagram E p − r +1 r − d / / π ′ (cid:15) (cid:15) E pr − d / / π (cid:15) (cid:15) E p + r − r − π ′′ (cid:15) (cid:15) E ( l ) p − r +1 r − d / / E ( l ) pr − d / / E ( l ) p + r − r − If p + r ≤ l , by induction the maps π ′ , π are isomorphisms, π ′′ is injective and then also theinduced map E pr → E ( l ) pr is an isomorphism. If p < l and p + r > l , then E ( l ) p + r − r − = 0, π isinjective and π ′ is an isomorphism; thus the induced map E pr → E ( l ) pr is injective. As regards (3), the if part follows immediately from (2). Conversely, if E p,qr degenerates at E a,bk , then for every l and every r ≥ k we have a commutative diagram: E a,br π ′ (cid:15) (cid:15) / / E a + r,b − r +1 r π (cid:15) (cid:15) E ( l ) a,br d r / / E ( l ) a + r,b − r +1 r If a + r ≥ b then E ( l ) a + r,b − r +1 r = 0, while if a + r < b , then π ′ is surjective; in both cases d r = 0. (cid:3) Thus, putting together Theorem 3.3 and Lemma 7.1 we obtain the following proposition. Proposition 7.2. Let L be a differential graded Lie algebras. For every positive integer l let τ The the role of differential graded Lie algebras in deformation theory was clear since themid sixties when Nijenhuis and Richardson [27, 28] observed that many deformation problemsare controlled by the Maurer-Cartan equation dx + 12 [ x, x ] = 0 , x ∈ L , for a suitable differential graded Lie algebra L . This point of view was extended by Deligne interms of the philosophy that “in characteristic 0 every (infinitesimal) deformation problem iscontrolled by a differential graded Lie algebra, with quasi-isomorphic DG-Lie algebras givingthe same deformation theory”, [9].A more precise statement of the above philosophy can be stated in the framework offunctors of Artin rings. Following [19], let K be a field of characteristic 0, let Art be thecategory of local Artin K -algebras with residue field K and let F : Art → Set be the functorof infinitesimal deformations of some “good” algebro-geometric structure. Then there existsa differential graded Lie algebras ( L, d, [ − , − ]) such that F ≃ Def L , whereDef L ( A ) = { x ∈ L ⊗ m A | dx + [ x, x ] = 0 } gauge action of exp( L ⊗ m A ) , m A is the maximal ideal of A and the gauge action is defined by the formula e a ∗ x := x + ∞ X n =0 [ a, − ] n ( n + 1)! ([ a, x ] − da ) , a ∈ L ⊗ m A , x ∈ L ⊗ m A . N SOME FORMALITY CRITERIA FOR DG-LIE ALGEBRAS 19 The basic theorem of deformation theory asserts that if f : L → M is a quasi-isomorphism ofdifferential graded Lie algebras, then the induced natural transformation f : Def L → Def M isan isomorphism (see [25] and reference therein). Proposition 8.1. If a differential graded Lie algebra L is formal, then the two maps Def L ( K [ t ] / ( t )) → Def L ( K [ t ] / ( t ))Def L ( K [[ t ]]) := lim ← n Def L ( K [ t ] / ( t n )) → Def L ( K [ t ] / ( t )) have the same image.Proof. Since Def L is invariant under quasi-isomorphisms we may assume that L has trivialdifferential and therefore its Maurer-Cartan equation becomes [ x, x ] = 0, x ∈ L . Therefore tx ∈ Def L ( K [ t ] / ( t )) lifts to Def L ( K [ t ] / ( t )) if and only if there exists x ∈ L such that t [ x , x ] ≡ [ tx + t x , tx + t x ] ≡ t ) ⇐⇒ [ x , x ] = 0and [ x , x ] = 0 implies that tx ∈ Def H ( K [ t ] / ( t n )) for every n ≥ (cid:3) Notice that the formality of L does not imply that Def L ( K [[ t ]]) → Def L ( K [ t ] / ( t )) issurjective. The reader can easily verify that for a generic graded vector space V and L =Hom ∗ K ( V, V ) the map Def L ( K [ t ] / ( t n +1 )) → Def L ( K [ t ] / ( t n )) is not surjective for every n ≥ L admits a local moduli space M and L is formal, then M is defined by quadratic equations;more precisely M is isomorphic to the germ at 0 of the quadratic cone defined by the equation[ x, x ] = 0, x ∈ H ( L ); for a more detailed discussion and applications we refer to [9, 26].Probably, the simplest example of local moduli space which is not defined by quadraticequations is given by the Hilbert scheme representing embedded deformations of the closedpoint inside the affine scheme Spec( K [ x ] / ( x )). By standard deformation theory, see e.g. [14],the construction of the differential graded Lie algebra L controlling this deformation problemis described by the following three steps:(1) replace the K -algebra K [ x ] / ( x ) with a Koszul-Tate resolution, for instance with theDG-algebra R = ( K [ x, y ] , d ) , deg( x ) = 0 , deg( y ) = − , d ( y ) = x , where the closed point is the subscheme defined by the differential ideal I = ( x, y ) ⊂ R .(2) consider the differential graded Lie algebra M = Der ∗ ( R, R ) and its subalgebra N = { α ∈ M | α ( I ) ⊂ I } . (3) take L as the homotopy fiber of the inclusion N ⊂ M ; as a concrete description of L we can take L = { a ( t ) ∈ M [ t, dt ] | a (0) = 0 , a (1) ∈ N } . The non formality of L can also be checked algebraically, without relying on deformationtheory. In fact M is the free K [ x ]-module generated by y ddx , ddx , y ddy , ddy , while N is the K [ x ]-submodule generated by y ddx , x ddx , y ddy , x ddy . There exists a direct sum decomposition M = N ⊕ A where A = A ⊕ A is the graded vectorspace generated by u := ddx and ddy . Since A is an abelian graded Lie subalgebra of M and d = x ddy ∈ N we can apply Voronov’s construction of higher derived brackets of an innerderivation [1, 31]. Denoting by P : M → A the projection with kernel N , the maps of degree+1: q n : A ⊙ n → A, q n ( a , . . . , a n ) = P [[ · · · [[ d, a ] , a ] , . . . ] , a n ] , n ≥ , give an L ∞ [1] structure on A which, according to [1, Thm. 1.3], is weak equivalent to thed´ecalage of L . Since[ d, u ] = (cid:20) x ddy , ddx (cid:21) = − x ddy , [[ d, u ] , u ] = (cid:20) − x ddy , ddx (cid:21) = 6 x ddy , [[[ d, u ] , u ] , u ] = (cid:20) x ddy , ddx (cid:21) = 6 ddy , we have q = q = 0 and q = 0. Thus the L ∞ [1]-algebra ( A, q , q , q , . . . ) is not formal. Acknowledgment. I’m indebted with E. Arbarello for useful discussions and for pointingmy attention on the paper [17]. References [1] R. 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