CONTINUOUS AND TWISTED L MORPHISMS

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1 CONTINUOUS AND TWISTED L MORPHISMS AMNON YEKUTIELI Abstract. The purpose of this paper is to develop a suitable notion of continuous L morphism between DG Lie algebras, and to study twists of such morphisms. 0. Introduction Let K be a field containing R. Consider two DG Lie algebras associated to the polynomial algebra K[t] := K[t,..., t n ]. The first is the algebra of poly derivations T poly K[t]), and the second is the algebra of poly differential operators D poly K[t]). A very important result of M. Kontsevich [Ko], known as the Formality Theorem, gives an explicit formula for an L quasi-isomorphism Here is the main result of our paper. U : T poly K[t]) D poly K[t]). Theorem 0.. Assume R K. Let A = i 0 Ai be a super-commutative associative unital complete DG algebra in Dir Inv Mod K. Consider the induced continuous A-multilinear L morphism U A : A T poly K[[t]]) A D poly K[[t]]). Suppose ω A Tpoly 0 K[[t]]) is a solution of the Maurer-Cartan equation in A T poly K[[t]]). Define ω := U A )ω) A Dpoly 0 K[[t]]). Then ω is a solution of the Maurer-Cartan equation in A D poly K[[t]]), and there is continuous A-multilinear L quasi-isomorphism whose Taylor coefficients are U A,ω : A T poly K[[t]]) ) ω A D poly K[[t]]) ) ω j U A,ω )α) := k 0 for α j A T poly K[[t]]) ). j+k)! j+k U A )ω k α) Below is an outline of the paper, in which we mention the various terms appearing in the theorem. In Section we develop the theory of dir-inv modules. A dir-inv structure on a K-module M is a generalization of an adic topology. The category of dir-inv modules and continuous homomorphisms is denoted by Dir Inv Mod K. The concept of dir-inv module, and related complete tensor product, are quite flexible, and Date: 6 September Key words and phrases. L morphisms, deformation quantization. Mathematics Subject Classification Primary: 53D55; Secondary: 3D0, 3N0, 3J0.

2 2 AMNON YEKUTIELI are particulary well-suited for infinitely generated modules. Among other things we introduce the notion of DG Lie algebra in Dir Inv Mod K. Section 2 concentrates on poly differential operators. The results here are mostly generalizations of material from [EGA IV]. In Section 3 we review the coalgebra approach to L morphisms. The notions of continuous, A-multilinear and twisted L morphism are defined. The main result of this section is Theorem In Section 4 we recall the Kontsevich Formality Theorem. By combining it with Theorem 3.27 we deduce Theorem 0. repeated as Theorem 4.5). In Theorem 0. the DG Lie algebras A T poly K[[t]]) and A D poly K[[t]]) are the A-multilinear extensions of T poly K[[t]]) and D poly K[[t]]) respectively, and A T poly K[[t]]) ) ω and A D poly K[[t]]) ) are their twists. The L ω morphism U A is the continuous A-multilinear extension of U, and U A,ω is its twist. Theorem 0. is used in [Ye2], in which we study deformation quantization of algebraic varieties. Acknowledgments. The author wishes to thank Vladimir Hinich, Bernhard Keller and James Stasheff for their assistance.. Dir-Inv Modules We begin the paper with a generalization of the notion of adic topology. In this section K is a commutative base ring, and C is a commutative K-algebra. The category Mod C is abelian and has direct and inverse limits. Unless specified otherwise, all limits are taken in Mod C. Definition.. ) Let M Mod C. An inv module structure on M is an inverse system {F i M} i N of C-submodules of M. The pair M, {F i M} i N ) is called an inv C-module. 2) Let M, {F i M} i N ) and N, {F i N} i N ) be two inv C-modules. A function φ : M N C-linear or not) is said to be continuous if for every i N there exists i N such that φf i M) F i N. 3) Define Inv Mod C to be the category whose objects are the inv C-modules, and whose morphisms are the continuous C-linear homomorphisms. We do not assume that the canonical homomorphism M lim i M/F i M is surjective nor injective. There is a full embedding Mod C Inv Mod C, M M, {..., 0, 0}). If M, {F i M} i N ) and N, {F i N} i N ) are two inv C-modules then M N is an inv module, with inverse system of submodules F i M N) := F i M F i N. Thus Inv Mod C is a C-linear additive category. Let M, {F i M} i N ) be an inv C-module, let M, M be two C-modules, and suppose φ : M M and ψ : M M are C-linear homomorphisms. We get induced inv module structures on M and M by defining F i M := φ F i M) and F i M := ψf i M). Recall that a directed set is a partially ordered set J with the property that for any j, j 2 J there exists j 3 J such that j, j 2 j 3. Definition.2. ) Let M Mod C. A dir-inv module structure on M is a direct system {F j M} j J of C-submodules of M, indexed by a nonempty directed set J, together with an inv module structure on each F j M, such that for every j j 2 the inclusion F j M F j2 M is continuous. The pair M, {F j M} j J ) is called a dir-inv C-module.

3 CONTINUOUS AND TWISTED L MORPHISMS 3 2) Let M, {F j M}) j J and N, {F k N} k K ) be two dir-inv C-modules. A function φ : M N C-linear or not) is said to be continuous if for every j J there exists k K such that φf j M) F k N, and φ : F j M F k N is a continuous function between these two inv C-modules. 3) Define Dir Inv Mod C to be the category whose objects are the dir-inv C- modules, and whose morphisms are the continuous C-linear homomorphisms. There is no requirement that the canonical homomorphism lim j F j M M will be surjective. An inv C-module M is endowed with the dir-inv module structure {F j M} j J, where J := {0} and F 0 M := M. Thus we get a full embedding Inv Mod C Dir Inv Mod C. Given two dir-inv C-modules M, {F j M}) j J and N, {F k N} k K ), we make M N into a dir-inv module as follows. The directed set is J K, with the component-wise partial order, and the direct system of inv modules is F j,k) M N) := F j M F k N. The condition J in part ) of the definition ensures that the zero module 0 Mod C is an initial object in Dir Inv Mod C. So Dir Inv Mod C is a C-linear additive category. Let M, {F j M} j J ) be a dir-inv C-module, let M, M be two C-modules, and suppose φ : M M and ψ : M M are C-linear homomorphisms. We get induced dir-inv module structures {F j M } j J and {F j M } j J on M and M as follows. Define F j M ) := φ F j M) and F j M := ψf j M), which have induced inv module structures via the homomorphisms φ : F j M F j M and ψ : F j M F j M. Definition.3. ) An inv C-module M, {F i M} i N ) is called discrete if F i M = 0 for i 0. 2) An inv C-module M, {F i M} i N ) is called complete if the canonical homomorphism M lim i M/F i M is bijective. 3) A dir-inv C-module M is called complete resp. discrete) if it isomorphic, in Dir Inv Mod C, to a dir-inv module N, {F j N} j J ), where all the inv modules F j N are complete resp. discrete) as defined above, and the canonical homomorphism lim j F j N N is bijective. 4) A dir-inv C-module M is called trivial if it isomorphic, in Dir Inv Mod C, to an object of Mod C, via the embedding Mod C Dir Inv Mod C. Note that M is a trivial dir-inv module iff it is isomorphic, in Dir Inv Mod C, to a discrete inv module. There do exist discrete dir-inv modules that are not trivial dir-inv modules; see Example.0. It is easy to see that if M is a discrete dir-inv module then it is also complete. The base ring K is endowed with the inv structure {..., 0, 0}, so it is a trivial dirinv K-module. But the K-algebra C could have more interesting dir-inv structures cf. Example.8). If f : C C is a homomorphism of K-algebras, then there is a functor f : Dir Inv Mod C Dir Inv Mod C. In particular any dir-inv C-module is a dir-inv K-module. Definition.4. ) Given an inv C-module M, {F i M} i N ) its completion is the inv C-module M, {F i M}i N ), defined as follows: M := lim i M/F i M and F i M := Ker M M/F i M). Thus we obtain an additive endofunctor M M of Inv Mod C.

4 4 AMNON YEKUTIELI 2) Given a dir-inv C-module M, {F j M} j J ) its completion is the dir-inv C- module M, {F j M}j J ) defined as follows. For any j J let F j M be the completion of the inv C-module F j M, as defined above. Then let M := lim j F j M and F j M := Im F j M M). Thus we obtain an additive endofunctor M M of Dir Inv Mod C. An inv C-module M is complete iff the functorial homomorphism M M is an isomorphism; and of course M is complete. For a dir-inv C-module M there is in general no functorial homomorphism between M and M, and we do not know if M is complete. Nonetheless: Proposition.5. Suppose M Dir Inv Mod C is complete. Then there is an isomorphism M = M in Dir Inv Mod C. This isomorphism is functorial. Proof. For any dir-inv module M, {F j M} j J ) let s define M := lim j F j M. So M, {F j M} j J ) is a dir-inv module, and there are functorial morphisms M M and M M. If M is complete then both these morphisms are isomorphisms. Suppose {M k } k K is a collection of dir-inv modules, indexed by a set K. There is an induced dir-inv module structure on M := k K M k, constructed as follows. For any k let us denote by {F j M k } j Jk the dir-inv structure of M k ; so that each F j M k is an inv module. For each finite subset K 0 K let J K0 := k K 0 J k, made into a directed set by component-wise partial order. Define J := K 0 J K0, where K 0 runs over the finite subsets of K. For two finite subsets K 0 K, and two elements j 0 = {j 0,k } k K0 J K0 and j = {j,k } k K J K we declare that j 0 j if j 0,k j,k for all k K 0. This makes J into a directed set. Now for any j = {j k } k K0 J K0 J let F j M := k K 0 F jk M k, which is an inv module. The dir-inv structure on M is {F j M} j J. Proposition.6. Let {M k } k K be a collection of dir-inv C-modules, and let M := k K M k, endowed with the induced dir-inv structure. ) M is a coproduct of {M k } k K in the category Dir Inv Mod C. 2) There is a functorial isomorphism M = k K M k. Proof. ) is obvious. For 2) we note that both M and k K M k are direct limits for the direct system { M j } j J. Suppose {M k } k N is a collection of inv C-modules. For each k let {F i M k } i N be the inv structure of M k. Then M := k N M k is an inv module, with inv structure F i M := k>i M ) ) k k i Fi M k. Next let {Mk } k N be a collection of dir-inv C-modules, and for each k let {F j M k } j Jk be the dir-inv structure of M k. Then there is an induced dir-inv structure on M := k N M k. Define a directed set J := k N J k, with component-wise partial order. For any j = {j k } k N J define F j M := k N F j k M k, which is an inv C-module as explained above. The dir-inv structure on M is {F j M} j J. Proposition.7. Let {M k } k N be a collection of dir-inv C-modules, and let M := k N M k, endowed with the induced dir-inv structure. Then M is a product of {M k } k N in Dir Inv Mod C.

5 CONTINUOUS AND TWISTED L MORPHISMS 5 Proof. All we need to consider is continuity. First assume that all the M k are inv C-modules. Let s denote by π k : M M k the projection. For each k, i N and i maxi, k) we have π k F i M) = F i M k. This shows that the π k are continuous. Suppose L is an inv C-module and φ k : L M k are morphisms in Inv Mod C. For any i N there exists i N such that φ k F i L) F i M k for all k i. Therefore the homomorphism φ : L M with components φ k is continuous. Now let M k be dir-inv C-modules, with dir-inv structures {F j M k } j Jk. For any j = {j k } J one has π k F j M) = F jk M k, and as shown above π k : F j M F jk M k is continuous. Given a dir-inv module L and and morphisms φ k : L M k in Dir Inv Mod C, we have to prove that φ : L M is continuous. Let {F j L} j JL be the dir-inv structure of L. Take any j J L. Since φ k is continuous, there exists some j k J k such that φ k F j L) F jk M k. But then φf j L) F j M for j := {j k } k N, and by the previous paragraph φ : F j L F j M is continuous. The following examples should help to clarify the notion of dir-inv module. Example.8. Let c be an ideal in C. Then each finitely generated C-module M has an inv structure {F i M} i N, where we define the submodules F i M := c i+ M. This is called the c-adic inv structure. Any C-module M has a dir-inv structure {F j M} j J, which is the collection of finitely generated C-submodules of M, directed by inclusion, and each F j M is given the c-adic inv structure. We get a fully faithful functor Mod C Dir Inv Mod C. This dir-inv module structure on M is called the c-adic dir-inv structure. In case C is noetherian and c-adically complete, then the finitely generated modules are complete as inv C-modules, and hence all modules are complete as dir-inv modules. Example.9. Suppose M, {F i M} i N ) is an inv C-module, and {i k } k N is a nondecreasing sequence in N with lim k i k =. Then {F i k M} k N is a new inv structure on M, yet the identity map M, {F i M} i N ) M, {F i k M} k N ) is an isomorphism in Inv Mod C. A similar modification can be done for dir-inv modules. Suppose M, {F j M} j J ) is a dir-inv C-module, and J J is a subset that is cofinal in J. Then {F j M} j J is a new dir-inv structure on M, yet the identity map M, {F j M} j J ) M, {F j M} j J ) is an isomorphism in Dir Inv Mod C. Example.0. Let M be the free K-module with basis {e p } p N ; so M = p N Ke p in Mod K. We put on M the inv module structure {F i M} i N with F i M := 0 for all i. Let N be the same K-module as M, but put on it the inv module structure {F i N} i N with F i N := p=i Ke p. Also let L be the K-module M, but put on it the dir-inv module structure {F j L} j N, with F j L := j p=0 Ke p the discrete inv module whose inv structure is {..., 0, 0}. Both L and M are discrete and complete as dir-inv K-modules, and N = p N Ke p. The dir-inv module M is trivial. L is not a trivial dir-inv K-module, because it is not isomorphic in Dir Inv Mod K to any inv module. The identity maps L M N are continuous. The only continuous K-linear homomorphisms M L are those with finitely generated images. Remark.. In the situation of the previous example, suppose we put on the three modules L, M, N genuine K-linear topologies, using the limiting processes and starting from the discrete topology. Namely M, N/F i N and F j L get the discrete

6 6 AMNON YEKUTIELI topologies; L = lim j F j L gets the lim topology; and N lim i N/F i N gets the lim topology as in [Ye, Section.]). Then L and M become the same discrete topological module, and N is the topological completion of N. We see that the notion of a dir-inv structure is more subtle than that of a topology, even though a similar language is used. Example.2. Suppose K is a field, and let M := K, the free module of rank. Up to isomorphism in Dir Inv Mod K, M has three distinct dir-inv module structures. We can denote them by M, M 2, M 3 in such a way that the identity maps M M 2 M 3 are continuous. The only continuous K-linear homomorphisms M i M j with i > j are the zero homomorphisms. M 2 is the trivial dir-inv structure, and it is the only interesting one the others are pathological ). Example.3. Suppose M = p Z M p is a graded C-module. The grading induces a dir-inv structure on M, with J := N, F j M := p= j M p, and F i F j M := p= j+i M p. The completion satisfies M = p 0 M p) p<0 M p) in Dir Inv Mod C, where each M p has the trivial dir-inv module structure. It makes sense to talk about convergence of sequences in a dir-inv module. Suppose M, {F i M} i N ) is an inv C-module and {m i } i N is a sequence in M. We say that lim i m i = 0 if for every i 0 there is some i such that {m i } i i F i0 M. If M, {F j M} j J ) is a dir-inv module and {m i } i N is a sequence in M, then we say that lim i m i = 0 if there exist some j and i such that {m i } i i F j M, and lim i m i = 0 in the inv module F j M. Having defined lim i m i = 0, it is clear how to define lim i m i = m and i=0 m i = m. Also the notion of Cauchy sequence is clear. Proposition.4. Assume M is a complete dir-inv C-module. Then any Cauchy sequence in M has a unique limit. Proof. Consider a Cauchy sequence {m i } i N in M. Convergence is an invariant of isomorphisms in Dir Inv Mod C. By Definition.3 we may assume that in the dir-inv structure {F j M} j J of M each inv module F j M is complete. By passing to the sequence {m i m i } i N for suitable i, we can also assume the sequence is contained in one of the inv modules F j M. Thus we reduce to the case of convergence in a complete inv module, which is standard. Let M, {F i M} i N ) and N, {F i N} i N ) be two inv C-modules. We make M C N into an inv module by defining F i M C N) := Im M C F i N) F i M C N) M C N ). For two dir-inv C-modules M, {F j M} j J ) and N, {F k N} k K ), we put on M C N the dir-inv module structure {F j,k) M C N)} j,k) J K, where F j,k) M C N) := ImF j M C F k N M C N). Definition.5. Given M, N Dir Inv Mod C we define N C M to be the completion of the dir-inv C-module N C M. Example.6. Let s examine the behavior of the dir-inv modules L, M, N from Example.0 with respect to complete tensor product. There is an isomorphism L K N = p N N in Dir Inv Mod K, so according to Proposition.62) there is also an isomorphism L K N = N p N in Dir Inv Mod K. On the other hand M K N

7 CONTINUOUS AND TWISTED L MORPHISMS 7 is an inv K-module with inv structure F i M K N) = M K F i N, so M K N = p N M in Dir Inv Mod K. The series p=0 e p e p converges in M K N, but not in L K N. A graded object in Dir Inv Mod C, or a graded dir-inv C-module, is an object M Dir Inv Mod C of the form M = i Z M i, with M i Dir Inv Mod C. According to Proposition.6 we have M = i Z M i. Given two graded objects M = i Z M i and N = i Z N i in Dir Inv Mod C, the tensor product is also a graded object in Dir Inv Mod C, with M C N) i := p+q=i M p C N q. In this paper algebra is taken in the weakest possible sense: by C-algebra we mean an C-module A together with an C-bilinear function µ A : A A A. If A is associative, or a Lie algebra, then we will specify that. However, commutative algebra will mean, by default, a commutative associative unital C-algebra. Another convention is that a homomorphism between unital algebras is a unital homomorphism, and a module over a unital algebra is a unital module. Definition.7. ) An algebra in Dir Inv Mod C is an object A Dir Inv Mod C, together with a continuous C-bilinear function µ A : A A A. 2) A differential graded algebra in Dir Inv Mod C is a graded object A = i Z Ai in Dir Inv Mod C, together with continuous C-bi)linear functions µ A : A A A and d A : A A, such that A is a differential graded algebra, in the usual sense, with respect to the differential d A and the multiplication µ A. 3) Let A be an algebra in Dir Inv Mod C, with dir-inv structure {F j A} j J. We say that A is a unital algebra in Dir Inv Mod C if it has a unit element A in the usual sense), such that A j J F ja. The base ring K, with its trivial dir-inv structure, is a unital algebra in Dir Inv Mod K. In item 3) above, the condition A j J F ja is equivalent to the ring homomorphism K A being continuous. We will use the common abbreviation DG for differential graded. An algebra in Dir Inv Mod C can have further attributes, such as Lie or associative, which have their usual meanings. If A Inv Mod C then we also say it is an algebra in Inv Mod C. Example.8. In the situation of Example.8, the c-adic inv structure makes C and Ĉ into unital algebras in Inv Mod C. Recall that a graded algebra A is called super-commutative if ba = ) ij ab and c 2 = 0 for all a A i, b A j, c A k and k odd. There is no essential difference between left and right DG A-modules. Proposition.9. Let A and g be DG algebras in Dir Inv Mod C. ) The completion  is a DG algebra in Dir Inv Mod C. 2) If A is complete, then the canonical isomorphism A =  of Proposition.5 is an isomorphism of DG algebras. 3) The complete tensor product A C g is a DG algebra in Dir Inv Mod C. 4) If A is a super-commutative associative unital algebra, then so is Â.

8 8 AMNON YEKUTIELI 5) If g is a DG Lie algebra and A is a super-commutative associative unital algebra, then A C g is a DG Lie algebra. Proof. ) This is a consequence of a slightly more general fact. Consider modules M,..., M r, N Dir Inv Mod C and a continuous C-multilinear linear function φ : M M r N. We claim that there is an induced continuous C-multilinear linear function φ : M k k N. This operation is functorial w.r.t. morphisms M k M k and N N ), and monoidal i.e. it respects composition in the kth argument with a continuous multilinear function ψ : L L s M k ). First assume M,..., M r, N Inv Mod C, with inv structures {F i M } i N etc. For any i N there exists i N such that φ k Fi M k ) F i N. Therefore there s an induced continuous C-multilinear function φ : M k k N. It is easy to verify that φ φ is functorial and monoidal. Next consider the general case, i.e. M,..., M r, N Dir Inv Mod C. Let {F j M k } j Jk be the dir-inv structure of M k, and let {F j N} j JN be the dir-inv structure of N. By continuity of φ, given j,..., j r ) k J k there exists j J N such that φ k F j k M k ) F j N, and φ : k F j k M k F j N is continuous. By the previous paragraph this extends to φ : F k jk M k F j N. Passing to the direct limit in j,..., j r ) we obtain φ : M k k N. Again this operation is functorial and monoidal. 2) Let A A be as in the proof of Proposition.5. This is a subalgebra. The arguments used in the proof of part ) above show that A A and A  are algebra homomorphisms. 3) Let us write A and g for the two multiplications, and d A and d g for the differentials. Then A C g is a DG algebra with multiplication and differential a γ ) a 2 γ 2 ) := ) i 2j a A a 2 ) γ g γ 2 ) da γ ) := d A a ) γ + ) i a d g γ ) for a k A i k and γ k g j k. These operations are continuous, so A C g is a DG algebra in Dir Inv Mod C. Now use part ). 4, 5) The various identities Lie etc.) are preserved by. Definition.73) ensures that  has a unit element. Definition.20. Suppose A is a DG super-commutative associative unital algebra in Dir Inv Mod C. ) A DG A-module in Dir Inv Mod C is a graded object M Dir Inv Mod C, together with a continuous C-bilinear homomorphism A M M, which makes M into a DG A-module in the usual sense. 2) A DG A-module Lie algebra in Dir Inv Mod C is a DG Lie algebra g Dir Inv Mod C, together with a continuous C-bilinear homomorphism A g g, such that g is a DG A-module, and for all a k A i k and γ k g j k. [a γ, a 2 γ 2 ] = ) i 2j a a 2 [γ, γ 2 ]

9 CONTINUOUS AND TWISTED L MORPHISMS 9 Example.2. If A is a DG super-commutative associative unital algebra in Dir Inv Mod C, and g is a DG Lie algebra in Dir Inv Mod C, then A C g is a DG Â-module Lie algebra in Dir Inv Mod C. Let A be a DG super-commutative associative unital algebra in Dir Inv Mod C, and let M, N be two DG A-modules in Dir Inv Mod C. The tensor product M A N is a quotient of M C N, and as such it has a dir-inv structure. Moreover, M A N is a DG A-module in Dir Inv Mod C, and we define M A N to be its completion, which is a DG Â-module in Dir Inv Mod C. Proposition.22. Let A and B be DG super-commutative associative unital algebras in Dir Inv Mod C, and let A B be a continuous homomorphism of DG C-algebras. ) Suppose M is a DG A-module in Dir Inv Mod C. Then B A M is a DG B-module in Dir Inv Mod C. 2) Suppose g is a DG A-module Lie algebra in Dir Inv Mod C. Then B A g is a DG B-module Lie algebra in Dir Inv Mod C. Proof. Like Proposition.9. Suppose C, C are commutative algebras in Dir Inv Mod K, and f : C C is a continuous K-algebra homomorphism. There are functors f : Dir Inv Mod C Dir Inv Mod C and f : Dir Inv Mod C Dir Inv Mod Ĉ, namely f M := C C M and f M := C C M. Let M and N be two dir-inv C-modules. We define Hom cont C M, N) := Hom Dir Inv Mod C M, N), i.e. the C-module of continuous C-linear homomorphisms. In general this module has no obvious structure. However, if M is an inv C-module with inv structure {F i M} i N, and N is a discrete inv C-module, then In this case we consider each Hom cont C M, N) = lim i Hom C M/F i M, N). F i Hom cont C M, N) := Hom C M/F i M, N) as a discrete inv module, and this endows Hom cont C M, N) with a dir-inv structure. Example.23. In the situation of Example.0 one has as dir-inv C-modules. Hom cont C N, M) = L C M Example.24. This example is taken from [Ye]. Assume K is noetherian and C is a finitely generated commutative K-algebra. For q N define B q C) = B q C) := C q+2) = C K K C. Define B q C) = B q C) to be the adic completion of B q C) with respect to the ideal KerB q C) C). There is a K-algebra homomorphism B 0 C) B q C), corresponding to the two extreme tensor factors, and in this way we view B q C) as a complete inv B 0 C)-module. There is a continuous coboundary operator that makes BC) := q N B q C) into a complex of B 0 C)-modules, and there is a quasi-isomorphism BC) C. We call BC) the complete un-normalized bar complex of C.

10 0 AMNON YEKUTIELI Next define ĈqC) = Ĉ q C) := C B q B0 C) C). This is a complete inv C- module. The complex ĈC) is called the complete Hochschild chain complex of C. Finally let C q cd C) := Homcont C Ĉ q C), C). The complex C cd C) := q N Cq cd C) is called the continuous Hochschild cochain complex of C. 2. Poly Differential Operators In this section K is a commutative base ring, and C is a commutative K-algebra. The symbol means K. We discuss some basic properties of poly differential operators, expanding results from [Ye2]. Definition 2.. Let M,..., M p, N be C-modules. A K-multilinear function φ : M M p N is called a poly differential operator over C relative to K) if there exists some d N such that for any m,..., m p ) M i and any i {,..., p} the function M i N, m φm,..., m i, m, m i+,..., m p ) is a differential operator of order d, in the sense of [EGA IV, Section 6.8]. In this case we say that φ has order d in each argument. We shall denote the set of poly differential operators M i N over C relative to K, of order d in all arguments, by And we define F d Diff poly C; M,..., M p ; N). Diff poly C; M,..., M p ; N) := d 0 F d Diff poly C; M,..., M p ; N), the union being inside the set of all K-multilinear functions M i N. By default we only consider poly differential operators relative to K. For a natural number p the p-th un-normalized B p C) was defined in Example.24. Let I p C) be the kernel of the ring homomorphism B p C) C. Define C p C) := C B0 C) B p C), the p-th Hochschild chain module of C relative to K). For any d N define and Let B p,d C) := B p C) / I p C) d+, C p,d C) := C B0C) B p,d C) C p,d C; M,..., M p ) := C p,d C) Bp 2 C) M M p ). φ uni : be the K-multilinear function p M i C p,d C; M,..., M p ) i= φ uni m,..., m p ) := m m p ). Observe that for p = we get C,d C) = P d C), the module of principal parts of order d see [EGA IV]). In the same way that P d C) parameterizes differential operators, C p,d C) parameterizes poly differential operators: Lemma 2.2. The assignment ψ ψ φ uni is a bijection Hom C Cp,d C; M,..., M p ), N ) Fd Diff poly C; M,..., M p ; N).

11 CONTINUOUS AND TWISTED L MORPHISMS Proof. The same arguments used in [EGA IV, Section 6.8] also apply here. Cf. [Ye, Section.4]. In case M = = M p = N = C we see that Diff poly C; C,..., C; C) }{{} = lim Hom C Cp,d C), C ) d 2.3) p = Hom cont C Ĉp C), C ) = C p cd C), with notation of Example.24. Proposition 2.4. Suppose C is a finitely generated K-algebra, with ideal c C. Let M,..., M p, N be C-modules, and let φ : M i N be a multi differential operator over C relative to K. Then φ is continuous for the c-adic dir-inv structures on M,..., M p, N. Proof. Suppose φ has order d in each of its arguments, and let ψ : C p,d C; M,..., M p ) N be the corresponding C-linear homomorphism. As in [Ye, Proposition.4.3], since C is a finitely generated K-algebra, it follows that B p,d C) is a finitely generated module over B 0 C); and hence C p,d C) is a finitely generated C-module. Let s denote by {F j M i } j Ji and {F k N} k K the c-adic dir-inv structures on M i and N. For any j,..., j p the B p 2 C)-module F j M F jp M p is finitely generated, and hence the C-module C p,d C; F j M,..., F jp M p ) is finitely generated. Therefore ψ C p,d C; F j M,..., F jp M p ) ) = F k N for some k K. It remains to prove that φ : p i= F j i M i F k N is continuous for the c-adic inv structures. But just like [Ye, Proposition.4.6], for any i and l one has 2.5) φf j M,..., c i+d F jl M l,..., F jp M p ) c i F k N. Suppose C is a commutative C-algebra with ideal c C. One says that C is c -adically formally étale over C if the following condition holds. Let D be a commutative C-algebra with nilpotent ideal d, and let f : C D/d be a C- algebra homomorphism such that fc i ) = 0 for i 0. Then f lifts uniquely to a C-algebra homomorphism f : C D. The important instances are when C C is étale and then c = 0); and when C is the c-adic completion of C for some ideal c A and c = C c). In both these instances C is c-adically complete; and if C is noetherian, then C C is also flat. Lemma 2.6. Let C be a c -adically formally étale C-algebra. Define C j := C /c j+. Consider C and C p,d C) as inv C-modules, with the c -adic and discrete inv structures respectively. Then the canonical homomorphism is bijective. C C C p,d C) lim j C p,d C j)

12 2 AMNON YEKUTIELI Proof. Define ideals and c p := Ker C p C ) C p C 0) ) J := Ker C j C C p,d C) C j). By the transitivity and the base change properties of formally étale homomorphisms, the ring homomorphism C p C) = C C C C = Cp C ) is c p-adically formally étale. Consider the commutative diagram of ring homomorphisms with solid arrows) C C C p C) C j C C p,d C) C p,d C j ) g f C p C = ) C j. f The ideal J satisfies J d+ = 0, and the ideal Ker C p,d C j ) j) C is nilpotent too. Due to the unique lifting property the dashed arrows exist and are unique, making the whole diagram commutative. Moreover g : C p C ) C p C j ) has to be the canonical surjection, and f is surjective. A little calculation shows that fi p C ) d+ ) = 0, and hence f induces a homomorphism f : C p,d C ) C j C C p,d C). Let C j c p,d := Ker C p,d C ) C p,d C 0) ). Another calculation shows that fc p,d j+)d+) ) = 0. The conclusion is that there are surjections C p,d C jd+j+d) f C j C C p,d C) e C p,d C j), such that e f is the canonical surjection. Passing to the inverse limit we deduce that C C C p,d C) lim j C p,d C j) is bijective. Proposition 2.7. Assume C is a noetherian finitely generated K-algebra, and C is a noetherian, c -adically complete, flat, c -adically formally étale C-algebra. Let M,..., M p, N be C-modules, and define M i := C C M i and N := C C N. ) Suppose φ : p i= M i N is a poly differential operator over C. Then φ extends uniquely to a poly differential operator φ : p i= M i N over C. If φ has order d then so does φ. 2) The homomorphism c φ c φ, is bijective. C C F d Diff poly C; M,..., M p ; N) F d Diff poly C ; M,..., M p; N ), e

13 CONTINUOUS AND TWISTED L MORPHISMS 3 Proof. By Proposition 2.4, applied to C with the 0-adic inv structure, we may assume that the C-modules M,..., M p, N are finitely generated. Fix d N. Define C j := C /c j+ and N j := C j C N. So C = lim j C j and N = lim j N j. By Lemma 2.2 and Proposition 2.4 we have 2.8) F d Diff poly C ; M,..., M p; N ) = Hom C Cp,d C ; M,..., M p), N ) = lim Hom C Cp,d C ; M j,..., M p), N j ). Now for any k j + d one has Hom C Cp,d C ; M,..., M p), N j ) = HomC Cp,d C k; M,..., M p), N j ). This is because of formula 2.5). Thus, using Lemma 2.6, we obtain Hom C Cp,d C ; M,..., M p), N j) = Hom C lim C p,d C k k; M,..., M p), N j ) = Hom C C C C p,d C; M,..., M p ), N j) Combining this with 2.8) we get = Hom C Cp,d C; M,..., M p ), N j). F d Diff poly C ; M,..., M p; N ) = lim Hom C Cp,d C; M,..., M p ), N ) j j = Hom C Cp,d C; M,..., M p ), N ). But C C is flat, C is noetherian, and C p,d C; M,..., M p ) is a finitely generated C-module. Therefore Hom C Cp,d C; M,..., M p ), N ) The conclusion is that 2.9) = C C Hom C Cp,d C; M,..., M p ), N ). F m D p+ poly C ; M,..., M p; N ) = C C F m D p+ poly C; M,..., M p ; N). Given φ : M i N of order d, let φ := φ under the isomorphism 2.9). Backtracking we see that φ is the unique poly differential operator extending φ. 3. L Morphisms and their Twists In this section we expand some results on L algebras and morphisms from [Ko] Section 4. Much of the material presented here is based on discussions with Vladimir Hinich. There is some overlap with Section 2.2 of [Fu], with Section 6. of [Le], and possibly with other accounts. Let K be a field of characteristic 0. Given a graded K-module g = i Z gi and a natural number j let T j g := g g. The direct sum Tg := }{{} j N Tj g is the j tensor algebra. Let us denote the multiplication in Tg by. This is just another way of writing, but it will be convenient to do so.)

14 4 AMNON YEKUTIELI The permutation group S j acts on T j g as follows. For any sequence of integers d = d,..., d j ) there is a group homomorphism sgn d : S j {±} such that on a transposition σ = p, p + ) the value is sgn d σ) = ) dpdp+. The action of a permutation σ S j on T j g is then σγ γ j ) := sgn d σ)γ σ) γ σj) for γ g d,..., γ j g dj. Define S j g to be the set of S j -invariants inside T j g, and Sg := j 0 S j g. The K-module Tg is also a coalgebra, with coproduct : Tg Tg Tg given by the formula γ γ j ) := j γ γ p ) γ p+ γ j ). p=0 The submodule Sg Tg is a sub-coalgebra but not a subalgebra!). The super-symmetric algebra Sg = j 0 Sj g is defined to be the quotient of Tg by the ideal generated by the elements γ γ 2 ) d d 2 γ 2 γ, for all γ g d and γ 2 g d 2. In other words, S j g is the set of coinvariants of T j g under the action of the group S j. The product in the algebra Sg is denoted by. The canonical projection is π : Tg Sg is an algebra homomorphism: πγ γ 2 ) = γ γ 2. In fact Sg is a commutative cocommutative Hopf algebra. The comultiplication : Sg Sg Sg is the unique K-algebra homomorphism such that γ) = γ + γ for all γ g. The antipode is γ γ. The projection π : Tg Sg is not a coalgebra homomorphism. However: Lemma 3.. Let τ : Sg Tg be the K-module homomorphism defined by τγ γ j ) := σ S j sgn d,...,d j )σ)γ σ) γ σj) for γ g d,..., γ j g dj. Then τ : Sg Sg is a coalgebra isomorphism, where Sg has the comultiplication and Sg has the comultiplication. Proof. Define π : Tg Sg to be the K-module homomorphism πγ γ j ) := j! πγ γ j ) = j! γ γ j for γ,..., γ j g. So π τ is the identity map of Sg, and π : Sg Sg is bijective. It suffices to prove that π π) τ τ) = π π) τ. Take any γ g d,..., γ j g d j and write d := d,..., d j ). Then ) π π) τ γ γ j ) = j p=0 σ S j p! j p)! sgn dσ)γ σ) γ σp) ) γ σp+) γ σj) ).

15 CONTINUOUS AND TWISTED L MORPHISMS 5 On the other hand π π) τ τ) ) γ γ j ) = γ γ j ) = γ + γ ) γ j + γ j ) j sgn d σ)γ σ) γ σp) ) γ σp+) γ σj) ), p=0 σ S p,j p where S p,j p is the set of p, j p)-shuffles inside the group S j. Since the algebra Sg is super-commutative the two sums are equal. The grading on g induces a grading on Sg, which we call the degree. Thus for γ i g d i the degree of γ γ j S j g is d + + d j unless γ γ j = 0). We consider Sg as a graded algebra for this grading. Actually there is another grading on Sg, by order, where we define the order of γ γ j to be j again, unless this element is zero). But this grading will have a different role. By definition the j-th super-exterior power of g is j 3.2) g := S j g[])[ j], where g[] is the shifted graded module whose degree i component is g[] i = g i+. When g is concentrated in degree 0 then these are the usual constructions of symmetric and exterior algebras, respectively. We denote by ln : Sg S g = g the projection. So lnγ) is the -st order term of γ Sg. The expression ln might stand for linear or logarithm.) Definition 3.3. Let g and g be two graded K-modules, and let Ψ : Sg Sg be a K-linear homomorphism. For any j the j-th Taylor coefficient of Ψ is defined to be j Ψ := ln Ψ : S j g g. Lemma 3.4. Suppose we are given a sequence of K-linear homomorphisms ψ j : S j g g, j, each of degree 0. Then there is a unique coalgebra homomorphism Ψ : Sg Sg, homogeneous of degree 0 and satisfying Ψ) =, whose Taylor coefficients are j Ψ = ψ j. Proof. Let ln : Sg S g = g be the projection for this coalgebra. Consider the exact sequence of coalgebras 3.5) 0 K Sg S g 0. According to [Ko, Section 4.] see also [Fu, Lemma 2..5]) the sequence {ψ j } j uniquely determines a coalgebra homomorphism Ψ : S g S g such that ln Ψ Sj g = ψ j τ Sj g for all j. Here τ : Sg Sg is the coalgebra isomorphism of Lemma 3.. Using 3.5) we can lift Ψ uniquely to a coalgebra homomorphism Ψ : Sg Sg by setting Ψ) :=. Now define the coalgebra homomorphism Ψ : Sg Sg to be Ψ := τ Ψ τ. A K-linear map Q : Sg Sg is a coderivation if where := Sg, the identity map. Q = Q + Q),

16 6 AMNON YEKUTIELI Lemma 3.6. Given a sequence of K-linear homomorphisms ψ j : S j g g, j, each of degree, there is a unique coderivation Q of degree, such that Q) = 0 and j Q = ψ j. Furthermore, one has Qγ) j Sj g for every γ Sg. Proof. According to [Ko, Section 4.3] see also [Fu, Lemma 2..2]) the sequence {ψ j } j uniquely determines a coderivation Q : S g S g such that ln Q Sj g = ψ j τ Sj g for all j. Using 3.5) this can be lifted uniquely to a coderivation Q : Sg Sg by setting Q) := 0. Now define the coderivation Q : Sg Sg to be Q := τ Q τ. We will be mostly interested in the coalgebras Sg[]) and Sg []). Observe that if Ψ : Sg[]) Sg []) is a homogeneous K-linear homomorphism of degree i, then, using formula 3.2), each Taylor coefficient j Ψ may be viewed as a homogeneous K-linear homomorphism j Ψ : j g g of degree i + j. Definition 3.7. Let g be a graded K-module. An L algebra structure on g is a coderivation Q : Sg[]) Sg[]) of degree, satisfying Q) = 0 and Q Q = 0. We call the pair g, Q) an L algebra. The notion of L algebra generalizes that of DG Lie algebra in the following sense: Proposition 3.8 [Ko, Section 4.3]). Let Q : Sg[]) Sg[]) be a coderivation of degree with Q) = 0. Then the following conditions are equivalent. i) j Q = 0 for all j 3, and Q Q = 0. ii) j Q = 0 for all j 3, and g is a DG Lie algebra with respect to the differential d := Q and the bracket [, ] := 2 Q. In view of this, we shall say that g, Q) is a DG Lie algebra if the equivalent conditions of the proposition hold. An easy calculation shows that given an L algebra g, Q), the function Q : g g is a differential, and 2 Q induces a graded Lie bracket on Hg, Q). We shall denote this graded Lie algebra by Hg, Q). Definition 3.9. Let g, Q) and g, Q ) be L algebras. An L morphism Ψ : g, Q) g, Q ) is a coalgebra homomorphism Ψ : Sg[]) Sg []) of degree 0, satisfying Ψ) = and Ψ Q = Q Ψ. Proposition 3.0 [Ko, Section 4.3]). Let g, Q) and g, Q ) be DG Lie algebras, and let Ψ : Sg[]) Sg []) be a coalgebra homomorphism of degree 0 such that Ψ) =. Then Ψ is an L morphism i.e. Ψ Q = Q Ψ) iff the Taylor coefficients ψ i := i Ψ : i g g satisfy the following identity: d ψ i γ γ i ) ) 2 k,l k+l=i k! l! σ S i ± i ) ±ψ i γ dγ k ) γ i = k= [ ψk γ σ) γ σk) ), ψ l γ σk+) γ σi) ) ] + ) ±ψ i [γk, γ l ] γ γ k γ l γ i. k<l

17 CONTINUOUS AND TWISTED L MORPHISMS 7 Here γ k g are homogeneous elements, S i is the permutation group of {,..., i}, and the signs depend only on the indices, the permutations and the degrees of the elements γ k. See [Ke, Section 6] or [CFT, Theorem 3.] for the explicit signs.) The proposition shows that when g, Q) and g, Q ) are DG Lie algebras and j Ψ = 0 for all j 2, then Ψ : g g is a homomorphism of DG Lie algebras; and conversely. It also implies that for any L morphism Ψ : g, Q) g, Q ), the map HΨ) : Hg, Q) Hg, Q ) is a homomorphism of graded Lie algebras. Given DG Lie algebras g and g we consider them as L algebras g, Q) and g, Q ), as explained in Proposition 3.8. If Ψ : g, Q) g, Q ) is an L morphism, then we shall say by slight abuse of notation) that Ψ : g g is an L morphism. From here until Theorem 3.2 inclusive) C is a commutative K-algebra, and g, g are graded C-modules. Suppose g, Q) is an L algebra structure on g such that the Taylor coefficients j Q : j g g are all C-multilinear. Then we say g, Q) is a C-multilinear L algebra. Similarly one defines the notion of C-multilinear L morphism Ψ : g, Q) g, Q ). With C and g as above let S C g be the super-symmetric associative unital free algebra over C. Namely S C g is the quotient of the tensor algebra T C g = C g g C g) by the ideal generated by the super-commutativity relations. The algebra S C g is a Hopf algebra over C, with comultiplication C : S C g S C g C S C g. The formulas are just as in the case C = K. It will be useful to note that C preserves the grading by order, namely C S i Cg) j+k=i Sj C g C S k Cg. Lemma 3.. ) Let g be a graded C-module. There is a canonical bijection Q Q C between the set of C-multilinear L algebra structures Q on g, and the set of coderivations Q C : S C g[]) S C g[]) over C of degree, such that Q C ) = 0 and Q C Q C = 0. 2) Let g, Q) and g, Q ) be two C-multilinear L algebras. The set of C- multilinear L morphisms Ψ : g, Q) g, Q ) is canonically bijective to the set of coalgebra homomorphisms Ψ C : S C g[]) S C g []) over C of degree 0, such that Ψ C ) = and Ψ C Q C = Q C Ψ C. Proof. The data for a coderivation Q C : S C g[]) S C g[]) over C is its sequence of C-linear Taylor coefficients j Q C : j Cg g. But giving such a homomorphism j Q C is the same as giving a C-multilinear homomorphism j Q : j g g, so there is a corresponding C-multilinear coderivation Q : Sg[]) Sg[]). One checks that Q Q = 0 iff Q C Q C = 0. Similarly for coalgebra homomorphisms. An element γ S C g[]) is called primitive if C γ) = γ + γ. Lemma 3.2. The set of primitive elements of S C g[]) is precisely S C g[]) = g[]. Proof. By definition of the comultiplication any γ g[] is primitive. For the converse, let us denote by µ the multiplication in S C g[]). One checks that µ C )γ) = 2 i γ for γ S i C g[]). If γ is primitive then µ C)γ) = 2γ, so indeed γ S C g[]).

18 8 AMNON YEKUTIELI Now let s assume that C is a local ring, with nilpotent maximal ideal m. Suppose we are given two C-multilinear L algebras g, Q) and g, Q ), and a C-multilinear L morphism Ψ : g, Q) g, Q ). Because the coderivation Q is C-multilinear, the C-submodule mg g becomes a C-multilinear L algebra mg, Q). Likewise for mg, and Ψ : mg, Q) mg, Q ) is a C-multilinear L morphism. The fact that m is nilpotent is essential for the next definition. Definition 3.3. The Maurer-Cartan equation in mg, Q) is for ω mg) = mg[]) 0. i= i! i Q)ω i ) = 0 An element e S C g[]) is called group-like if C e) = e e. For ω mg we define expω) := i 0 i! ωi S C g[]). Lemma 3.4. The function exp is a bijection from mg[] to the set of invertible group-like elements of S C g[]), with inverse ln. Proof. Let ω mg[] and e := expω). The element e is invertible, with inverse exp ω). Using the fact that C ω) = ω + ω it easily follows that C e) = e e. And trivially lne) = ω. For the opposite direction, let e be invertible and group-like. Write it as e = i γ i, with γ i S i C g[]). Since e is invertible one must have γ 0 C m, and γ i ms i C g[]) for all i. The equation Ce) = e e implies that C γ i ) = γ j γ k for all i. Hence j+k=i 3.5) 2 i γ i = µ C γ i )) = j+k=i γ j γ k. For i = 0 we get γ 0 = γ0, 2 and since γ 0 is invertible, it follows that γ 0 =. Let ω := γ = lne) ms C g[]) = mg[]. Using induction and equation 3.5) we see that γ i = i! ωi for all i. Thus e = expω). Lemma 3.6. Let ω mg[]) 0 = mg and e := expω). Then ω is a solution of the MC equation iff Qe) = 0. Proof. Since e is group-like and invertible by Lemma 3.4) we have and C Qe)) = Qe) e + e Qe) C e Qe)) = C e) C Qe)) = e Qe) + e Qe). So the element e Qe) is primitive, and by Lemma 3.2 we get e Qe) g[]. On the other hand hence Qe) has no 0-order term, and Q) = 0. Thus in the st

19 CONTINUOUS AND TWISTED L MORPHISMS 9 order term we get 3.7) e Qe) = ln e Qe) ) = ln ω + 2 ω2 ± )Qe) ) = ln Qe) ) = i! ln Qω i ) ) = Since e is invertible we are done. i=0 i= i! i Q)ω i ). Lemma 3.8. Given an element ω mg[], define ω mg [], e := expω) and e := expω ). Then e = Ψe). := i= i! i Ψ)ω i ) Proof. From Lemma 3.4 we see that C e) = e e, and therefore also C Ψe)) = Ψe) Ψe) S C g []). Since Ψ is C-linear and Ψ) = we get Ψe) + msg []). Thus Ψe) is group-like and invertible. According to Lemma 3.4 it suffices to prove that lne ) = lnψe)). Now lne ) = ω by definition. Since Ψ) = and ln) = 0 it follows that lnψe)) = lnψ i=0 i! ωi) ) = i=0 i! lnψωi )) = i= i! i Ψ)ω i ) = ω. Proposition 3.9. Suppose ω mg is a solution of the MC equation in mg, Q). Define ω := i= i! i Ψ)ω i ) mg. Then ω is a solution of the MC equation in mg, Q ). Proof. Let e := expω) and e := expω ). By Lemma 3.6 we get Qe) = 0. Hence Q Ψe)) = ΨQe)) = 0. According to Lemma 3.8 we have Ψe) = e, so Q e ) = 0. Again by Lemma 3.6 we deduce that ω solves the MC equation. Definition Let ω mg. ) The coderivation Q ω of S C g[]) over C, with Q ω ) := 0 and with Taylor coefficients i Q ω )γ) := j! i+j Q)ω j γ) j 0 for i and γ S i C g[]), is called the twist of Q by ω. 2) The coalgebra homomorphism Ψ ω : S C g[]) S C g []) over C, with Ψ ω ) := and Taylor coefficients i Ψ ω )γ) := j 0 j! i+j Ψ)ω j γ) for i and γ S i C g[]), is called the twist of Ψ by ω. Theorem 3.2. Let C be a commutative local K-algebra with nilpotent maximal ideal m. Let g, Q) and g, Q ) be C-multilinear L algebras and Ψ : g, Q)

20 20 AMNON YEKUTIELI g, Q ) a C-multilinear L morphism. Suppose ω mg a solution of the MC equation in mg, Q). Define ω := j! j Ψ)ω j ) mg. i= Then g, Q ω ) and g, Q ω ) are L algebras, and is an L morphism. Ψ ω : g, Q ω ) g, Q ω ) Proof. Let e := expω). Define Φ e : S C g[]) S C g[]) to be Φ e γ) := eγ. Since e is group-like and invertible it follows that Φ e is a coalgebra automorphism. Therefore Q ω := Φ e Q Φ e is a degree coderivation of S C g[]), satisfying Q ω Q ω = 0 and Q ω ) = e Qe) = 0; cf. Lemma 3.6. So g, Q ω ) is an L algebra. Likewise we have a coalgebra automorphism Φ e and a coderivation Q ω := Φ e Q Φ e of S C g []), where e := expω ). The degree 0 coalgebra homomorphism Ψ ω := Φ e Ψ Φ e satisfies Ψ ω Q ω = Q ω Ψ ω, and also Ψ ω ) = e Ψe) = e e =, by Lemma 3.8. Hence we have an L morphism Ψ ω : g, Q ω ) g, Q ω ). Let us calculate the Taylor coefficients of Q ω. For γ S i C g[]) one has i Qω )γ) = ln Q ω γ)) = lne Qeγ)). But just as in 3.7), since Qeγ) has no zero order term, we obtain And 3.22) lne Qeγ)) = lnqeγ)). lnqeγ)) = ln ) ) Q j 0 j! ωj γ = j 0 = j 0 j! lnqωj γ)) j! i+j Q)ω j γ) = i Q ω )γ). Therefore Q ω = Q ω. Similarly we see that Q ω = Q ω and Ψ ω = Ψ ω. Remark The formulation of Theorem 3.2, as well as the idea for the proof, were suggested by Vladimir Hinich. An analogous result, for A algebras, is in [Le, Section 6.]. If g, Q) is a DG Lie algebra then the sum occurring in Definition 3.20) is finite, so the coderivation Q ω can be defined without a nilpotence assumption on the coefficients. Lemma Let g, Q) be a DG Lie algebra, and let ω g be a solution of the MC equation. Then the L algebra g, Q ω ) is also a DG Lie algebra. In fact, for γ i g one has Q ω )γ ) = Q)γ ) + 2 Q)ωγ ) = dγ ) + [ω, γ ] = d + adω))γ ), 2 Q ω )γ γ 2 ) = 2 Q)γ γ 2 ) = [γ, γ 2 ], and j Q ω = 0 for j 3.

21 CONTINUOUS AND TWISTED L MORPHISMS 2 Proof. Like equation 3.22), with C := K and e :=. In the situation of the lemma, the twisted DG Lie algebra g, Q ω ) will usually be denoted by g ω. Let A be a DG super-commutative associative unital DG algebra in Dir Inv Mod K. The notion of DG A-module Lie algebra in Dir Inv Mod K was introduced in Definition.20. Definition Let A be a DG super-commutative associative unital DG algebra in Dir Inv Mod K, let g and g be DG A-module Lie algebras in Dir Inv Mod K, and let Ψ : g g be an L morphism. ) If each Taylor coefficient j Ψ : j g g is continuous then we say that Ψ is a continuous L morphism. 2) Assume each Taylor coefficient j Ψ : j g g is A-multilinear, i.e. j Ψ)a γ,..., a j γ j ) = ±a a j j Ψ)γ,..., γ j ) for all homogeneous elements a k A and γ k g, with sign according to the Koszul rule, then we say that Ψ is an A-multilinear L morphism. Proposition Let A and B be DG super-commutative associative unital DG algebras in Dir Inv Mod K, and let g and g be DG A-module Lie algebras in Dir Inv Mod K. Suppose A B is a continuous DG algebra homomorphism, and Ψ : g g is a continuous A-multilinear L morphism. Let j Ψ B : j B A g) B A g be the unique continuous B-multilinear homomorphism extending j Ψ. Then the degree 0 coalgebra homomorphism Ψ B : SB A g[]) SB A g []), with Ψ B) := and with Taylor coefficients j Ψ B, is an L morphism Ψ B : B A g B A g. Proof. First consider the continuous B-multilinear homomorphisms j Ψ B : j B A g) B A g extending j Ψ. It is a straightforward calculation to verify that the L morphism identities of Proposition 3.0 hold for the sequence of operators { j Ψ B } j. The completion process respects these identities cf. proof of Proposition.9). Theorem Let g and g be DG Lie algebras in Dir Inv Mod K, and let Ψ : g g be a continuous L morphism. Let A = i N Ai be a complete associative unital super-commutative DG algebra in Dir Inv Mod K. By Proposition 3.26 there is an induced continuous A-multilinear L morphism Ψ A : A g A g. Let ω A g 0 be a solution of the MC equation in A g. Assume d g = 0, j Ψ A )ω j ) = 0 for all j 2, and also that g is bounded below. Define ω := Ψ A )ω) A g 0. Then: ) The element ω is a solution of the MC equation in A g. 2) Given c S j A g[] ) there exists a natural number k 0 such that j+k Ψ A )ω k c) = 0 for all k > k 0. 3) The degree 0 coalgebra homomorphism Ψ A,ω : S A g[] ) S A g [] ),

22 22 AMNON YEKUTIELI with Ψ A,ω ) := and Taylor coefficients j Ψ A,ω )c) := k 0 j+k)! j+k Ψ A ) ω k c ) for c S j A g[] ), is a continuous A-multilinear L morphism Ψ A,ω : A g ) ω A g ) ω. Proof. We shall use a deformation argument. Consider the base field K as a discrete inv K-module. The polynomial algebra K[ ] is endowed with the dir-inv K-module structure such that the homomorphism i N K K[ ], whose i-th component is multiplication by i, is an isomorphism in Dir Inv Mod K. Note that K[ ] is a discrete dir-inv module, but it is not trivial. We view K[ ] as a DG algebra concentrated in degree 0 with zero differential). For any i N let A[ ] i := K[ ] A i, and let A[ ] := i N A[ ]i, which is a DG algebra in Dir Inv Mod K, with differential d A[ ] := d A. We will need a twisted version of A[ ], which we denote by A[ ]. Let A[ ] i := i A[ ] i, and define A[ ] := i N A[ ] i, which his a graded subalgebra of A[ ]. The differential is d A[ ] := d A[ ]. The dir-inv structure is such that the homomorphism i,j N Ai A[ ], whose i, j)-th component is multiplication by i+j, is an isomorphism in Dir Inv Mod K. The specialization is a continuous DG algebra homomorphism A[ ] A. There is an induced continuous A[ ] -multilinear L morphism Ψ A[ ] : A[ ] g A[ ] g. We proceed in several steps. Step. Say r 0 bounds g from below, i.e. g r = 0 for all r < r 0. Take some j. For any l {,..., j} choose p l, q l Z, γ l g p l and a l A[ ] q l. Also choose γ 0 g 0 and a 0 A[ ]. Let p := j l= p l and q := j l= q l. Because j+k Ψ A[ ] is induced from j+k Ψ, and this is a homogeneous map of degree j k, we have j+k Ψ A[ ] ) a 0 γ 0 ) k a γ ) a j γ j ) ) = ±a k 0a a j j+k Ψ)γ k 0 γ γ j ) A[ ] k+q g p+ j k. But g p+ j k = 0 for all k > p + j r 0. Using multilinearity and continuity we conclude that given any c S j A[ ] g[] ) there exists a natural number k 0 such that j+k Ψ A[ ] ) ω) k c ) = 0 for all k > k 0. Step 2. We are going to prove that ω is a solution of the MC equation in A[ ] g. It is given that ω is a solution of the MC equation in A g. Because d g = 0, this means that d A )ω) + 2 [ω, ω] = 0. Hence d A[ ] g ω) + 2 [ ω, ω] = 2 d A )ω) [ω, ω] = 0. So ω solves the MC equation in A[ ] g. Step 3. Now we shall prove that ω solves the MC equation in A[ ] g. This will require an infinitesimal argument. For any natural number m define K[ ] m := K[ ]/ m+ ) and A[ ] m := K[ ] m A. The latter is a DG algebra with differential d A[ ]m := d A. Let A[ ] m := m i=0 i A[ ] i m, which is a subalgebra of A[ ] m,