About the Hochschild-Kostant-Rosenberg theorem for differentiable manifolds
AAbout the Hochschild-Kostant-Rosenbergtheorem for differentiable manifolds
Luiz Henrique Pereira Pˆegas ∗ October 22, 2018
Abstract
In this notes it will be provided a set of techniques which can helpone to understand the proof of the Hochschild-Kostant-Rosenberg the-orem for differentiable manifolds. Precise definitions of multidiferen-tial operators and polyderivations on an algebra are given, allowing towork on these concepts, when the algebra is an algebra of functions ona differentiable manifold, in a coordinate free description. Also, it willbe constructed a cup product on polyderivations which correspondson (Hochschild) cohomology to wedge product on multivector fields.At the end, a proof of the above mentioned theorem will be given.
Contents ∗ Contact by e-mail: [email protected] or [email protected] a r X i v : . [ m a t h . R A ] J u l cknowledgements I wish to thanks prof. Dr. Eduardo Hoefel, my advisor on my MSc disser-tation in which this notes are based, for the uncountable hours of discussionand guidance. Without his patience, this work could not be possible.
The main purpose of this notes is to provide a set of techniques which canhelp one to understand the proof of the Hochschild-Kostant-Rosenberg fordifferentiable manifolds. On the way to do that, precise definitions of multi-diferential operators and polyderivations on an algebra are given. When thealgebra is an algebra of functions on a differentiable manifold, this allows usto work on these concepts in a coordinate free description. Also, it is con-structed a cup product on polyderivations which corresponds on (Hochschild)cohomology to wedge product on multivector fields.To fully understand this notes, some background on algebra and differen-tiable manifolds is desirable. In section 2 some basic concepts are presentedand some notation are fixed. So, the readers who have some knowledgeon differential geometry, derivations and differential operators on associativealgebras may goes directly to section 3. In section 3 the concept of mul-tiderivations on an associative, commutative, unital algebra is defined andsome results on algebras of smooth functions on a manifold are stated. In sec-tion 4 the concept of iterated derivations is defined and related to derivationsof higher orders on the algebra of smooth functions on a manifold. In sec-tion 5 the concept of polyderivations of an associative, commutative, unitalalgebra is defined and it is used to provide a coordinate free version of poly-derivantions on a manifold. Section 5 ends with a proof of the Hochschild-Kostant-Rosenberg theorem for differentiable manifolds which uses the toolsintroduced in the previous sections.Throughout this notes,
Hom C ( A, B ) will denote the morphisms from A to B in the category C and A ≈ C B will mean that A is isomorphic to B in the category C . As an example, Hom
Vec K ( A, B ) is the set of K -lineartransformations between the K -vector spaces A and B .2 Algebras, Derivations and the Tangent Bun-dle
In this section it will be provided some of the standard concepts needed tounderstand the techniques developed in the final sections of this notes. Itwas made for setting some notations and some “ways of thinking”. Fromnow on, k will denote a commutative unital ring and K will denote a field. Definition 2.1 (Derivation on an algebra) . Let ( A, µ, e ) be an associative k -algebra with unity e (briefly A , when the product µ is clear from the context).A derivation D on A is a k -linear map D : A → A such that D ( µ ) = µ ( D ⊗ id ) + µ ( id ⊗ D ) where id is the identity on A . The condition above is known as “Leibniz rule”.
Remark . The unity e on A provides an immersion e : k → A , so elementsin k can be viewed as elements in A through such immersion. If D is aderivation on A , by the Leibniz rule we have for all a ∈ kD ( a ) = aD ( e ) = aD ( µ ( e ⊗ e )) = a ( µ ( D ⊗ e ) + µ ( e ⊗ D )) = D ( a ) + D ( a )therefore D ( a ) = 0. Notation . The k -module of all derivations on A will be denoted by Der ( A ). Definition 2.2 (Inner derivations) . Let ( A, µ, e ) be an associative k -algebrawith unity e . A k -linear map f : A → A is an inner derivation on A if andonly if there exists a ∈ A such that f = µ ( a ⊗ id ) − µ ( id ⊗ a ) , where a ⊗ id denotes a ⊗ id : A → A ⊗ A such that ( a ⊗ id )( b ) = a ⊗ b .Remark . By denoting
IDer ( A ) the k -module of inner derivations on A ,we have IDer ( A ) ⊂ Der ( A ).Now we make precise the notion of high order differential operator andhigh order derivation on a commutative associative k -algebra. The definition,naturally, is recursive. 3 efinition 2.3 (Higher order differential operator) . Let ( A, µ, e ) be a com-mutative associative k -algebra with unity e . A k -linear map D : A → A is adifferential operator of order ≤ r on A , with r ∈ N \ { } if and only if forall a ∈ A the map ˜ D = D ( µ ( a ⊗ id )) − µ ( a ⊗ D ) is a differential operator of order ≤ r − . A map D : A → A is a differentialoperator of order 0 if and only if there exists a ∈ A such that D = µ ( a ⊗ id ) . Definition 2.4 (Higher order derivation) . Let ( A, µ, e ) be a commutativeassociative k -algebra with unity e . A derivation of order ≤ r on A is adifferential operator of order ≤ r D such that D ( a ) = 0 , ∀ a ∈ k . Theorem 2.1.
Let ( A, µ, e ) be a commutative associative k -algebra withunity e . Then derivations on A are exactly the derivations of order ≤ A and a differential operator of order ≤ D can be written uniquely as D = ∂ + D ( e ) with ∂ ∈ Der ( A ) .Proof. Here it is convenient to use de juxtaposition to denote the product µ .If D ∈ Der ( A ) then D is a derivation of order ≤ a ∈ A ˜ D ( b ) = D ( ab ) − aD ( b ) = D ( a ) b and if a ∈ k we have D ( a ) = 0.Conversely, if D : A → A is a derivation of order ≤ ∂ = D − D ( e ), i.e. , ∂ ( a ) = D ( a ) − aD ( e ) , ∀ a ∈ A satisfies ∂ ( a ) = D ( a ) − aD ( e ) = f a were f a is the element of A given by D ( ab ) − aD ( b ) = f a b (just do b = e ) thus D ( ab ) − aD ( b ) = ∂ ( a ) b hence ∂ ( ab ) = D ( ab ) − abD ( e ) = D ( ab ) − aD ( b ) + aD ( b ) − abD ( e ) == ∂ ( a ) b + aD ( b ) − abD ( e ) = ∂ ( a ) b + a ( D ( b ) − bD ( e )) == ∂ ( a ) b + a∂ ( b )which means ∂ ∈ Der ( A ). 4hen the algebra ( A, µ ) is graded we can define a derivation which “re-spects” such structure.
Definition 2.5 (Graded derivation) . Let ( A, µ ) be a graded k -algebra. A k -linear map D : A → A is a graded derivation on A of degree p if and onlyif for all homogeneous elements a ∈ A i and for all b ∈ A it satisfies D ( µ ( a ⊗ b )) = µ ( D ( a ) ⊗ b ) + ( − pi µ ( a ⊗ D ( b ))Now it is convenient to set some notions (and notations). Definition 2.6 (Superalgebra) . We say that a graded k -algebra ( A, µ ) is asuperalgebra (or supercommutative) if and only if µ satisfies for all a ∈ A i , b ∈ A j µ ( a ⊗ b ) = ( − ij µ ( b ⊗ a ) Definition 2.7 (Lie superalgebra) . A Lie superalgebra is a pair ( L, [ , ]) where L is a Z -graded K -vector space and [ , ] : L × L → L is a bilinear mapsuch thati) [ L i , L j ] ⊂ L i + j , ∀ i, j ∈ Z ;ii) [ a, b ] = − ( − ij [ b, a ] , ∀ a ∈ L i , ∀ b ∈ L j , (graded antisymmetry);iii) [ a, [ b, c ]] = [[ a, b ] , c ] + ( − ij [ b, [ a, c ]] , ∀ a ∈ L i , ∀ b ∈ L j , ∀ c ∈ L , (gradedJacobi). Definition 2.8 (Derivation on a Lie superalgebra) . Let ( L, [ , ]) be a Liesuperalgebra. A K -linear map D : L → L is a derivation of degree p on L ifand only if it satisfies for all a ∈ L i and for all b ∈ LD ([ a, b ]) = [ D ( a ) , b ] + ( − ip [ a, D ( b )] Remark . It is easy to see that if ( L, [ , ]) is a Lie superalgebra, then forany a ∈ L i , ad ( a ) : L → L given by ad ( a )( b ) = [ a, b ] is a degree i derivationon L . This is exactly what the Jacobi identity is about.In differential geometry, the concept of tangent vector on a manifoldat a point is often given by using equivalence classes of curves or simplystating the property of being a derivation at a point. In this notes we willuse an equivalent (for finite dimensional differentiable manifolds) and wellknown definition for tangent vectors which is slightly different from the usualdefinition, but reveals some interesting aspects. The construction given herefollows [7]. 5et M be a differentiable manifold. For each p ∈ M , define the R -vectorspace V p tangent at p as the following. Define the relation ∼ on C ∞ ( M ) by f ∼ g if and only if there exists an open neighbourhood U of p such that f | U = g | U . This is an equivalence relation (the reader is invited to provethis, it is not hard) and the equivalence classes induced are called germs offunctions at p . The algebraic operations on C ∞ ( M ) can be used to induce a R -algebra structure on C ∞ ( M ) / ∼ . We denote such structure by F p . Let I p be the ideal of F p of germs of functions that vanish at p . As I p is an idealof F p and I p is an ideal of I p , we have that I p /I p is a R -vector space. Thenwe define V p = ( I p /I p ) ∗ , i.e. , V p is the vector space dual to I p /I p . We willprove that V p is finite dimensional. From now on, we will denote by ( U, ϕ, m )a local chart on a differentiable manifold M where U is an open subset of M , ϕ : U → U is a diffeomorphism from U to an open subset U of R m orsimply ( U, ϕ ) if the dimension of M is clear. Proposition 2.1.
Let ( U, ϕ, m ) be a local chart of M around p . Denotingthe i -th canonical projection on R m by t i : R m → R and the i -th coordinatefunction on U by x i = t i ◦ ϕ then the set of equivalence classes of x i for i = 1 , . . . , m in I p /I p constitutes a basis for such space.Proof. Given f ∈ I p /I p , let f ∈ C ∞ ( M ) be a representing element. Notethat f ( p ) = 0. Without loss of generality we can suppose ϕ ( U ) convex and ϕ ( p ) = 0. The coordinate expression of f is given by f ◦ ϕ − : ϕ ( U ) → R ,which by the Taylor formula gives, for y = ϕ ( q ) , q ∈ U ( f ◦ ϕ − )( y ) = m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) t i ( y ) ++ m (cid:88) i,j =1 t i ( y ) t j ( y ) (cid:90) (1 − s ) ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) sy ds ( f ◦ ϕ − )( ϕ ( q )) = m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) t i ( ϕ ( q )) ++ m (cid:88) i,j =1 t i ( ϕ ( q )) t j ( ϕ ( q )) (cid:90) (1 − s ) ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) sy dsf ( q ) = m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( p ) x i ( q ) ++ m (cid:88) i,j =1 x i ( q ) x j ( q ) (cid:90) (1 − s ) ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) sy ds It is necessary because we want to use Taylor’s formula with integral reminder. f ∈ C ∞ ( M ) and x i ( p ) = 0, the term m (cid:88) i,j =1 x i x j (cid:90) (1 − s ) ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) sy ds represents the null class in I p /I p . From this we infer that we can write f = m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( p ) x i where x i is the class of x i in I p /I p . Hence the set { x i } , i = 1 , . . . m spans I p /I p . To show the linear independence, notice that m (cid:88) i =1 a i x i = 0 ⇒ m (cid:88) i =1 a i [ x i ] ∈ I p where [ x i ] is a representing of x i in I p . Writing in coordinates, we have: (cid:32) m (cid:88) i =1 a i x i (cid:33) ◦ ϕ − = m (cid:88) i =1 a i ( x i ◦ ϕ − ) = m (cid:88) i =1 a i t i which shows that m (cid:88) i =1 a i [ t i ] ∈ I ϕ ( p ) , because the map ϕ − : ϕ ( U ) → U inducesan algebra homomorphism ( ϕ − ) ∗ : F p → F ϕ ( p ) given by ( ϕ − ) ∗ ([ f ]) = [ f ◦ ϕ − ]. Thus, the first order terms vanish, which means that for each j =1 , . . . , m ∂∂t j (cid:32) m (cid:88) i =1 a i t i (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) = 0and therefore a i = 0, for all i = 1 , . . . , m .From this we conclude that V p is finite dimensional and dim ( V p ) = m .An element ξ p ∈ V p is called a tangent vector on M at p .For each p ∈ M we can associate to each tangent vector ξ i ∈ V p a linearmap v p : F p → R , given by v p ( f ) = (cid:40) ∃ c ∈ f | c ( x ) = c ∀ x ∈ M ξ p ([ f ]) , if f ∈ I p where [ f ] denotes the class corresponding to the germ f in I p /I p . Since allgerms can be written as f = ˜ f + f ( p ), where ˜ f ∈ I p and f ( p ) is the germ of7he constant function whose value is f ( p ), v p satisfies the following property v p ( f g ) = v p (( ˜ f + f ( p ))(˜ g + g ( p ))) == v p ( ˜ f ˜ g + f ( p )˜ g + g ( p ) ˜ f + f ( p ) g ( p )) == v p ( ˜ f ˜ g ) + v p ( f ( p )˜ g ) + v p ( g ( p ) ˜ f ) + v p ( f ( p ) g ( p )) == ξ p ( ˜ f ˜ g ) + f ( p ) ξ p (˜ g ) + g ( p ) ξ p ( ˜ f ) + 0 == f ( p ) ξ p (˜ g ) + g ( p ) ξ p ( ˜ f ) == f ( p ) v p (˜ g ) + g ( p ) v p ( ˜ f ) == g ( p ) v p ( f ) + f ( p ) v p ( g )When a linear map w : F p → R obeys such property we call w a derivationon F p at the point p .Conversely, if w is a derivation on F p at p , we can associate w to a uniquetangent vector η p such that η p ([ f ]) = w ( f ) for all f ∈ F p . To see this, notethat if c is the germ of a constant function w ( c ) = w ( c ·
1) = cw (1) = cw (1 ·
1) = cw (1) + cw (1) = 2 w ( c )and therefore w ( c ) = 0. By writing f as f = ˜ f + f ( p ), we have: w ( f ) = w ( ˜ f + f ( p )) = w ( ˜ f ) + w ( f ( p )) = w ( ˜ f )therefore the value of w is determined by its value at I p . If f ∈ I p , thereexists g, h ∈ I p such that f = gh . Hence w ( f ) = w ( gh ) = h ( p ) w ( g ) + g ( p ) w ( h ) = 0thus, w vanishes on I p . However, f = ˜ f + f ( p ) gives w ( f ) = w ( f − f ( p )) = w ( ˜ f )which shows that if ˜ f ≡ ˜ g mod I p , then w ( f ) = w ( g ) and w induces anunique linear map η p taking elements in I p /I p to real values. In other words η p ∈ V p .So we stablish an one to one correspondence between derivations on F p at p and elements in ( I p /I p ) ∗ . It is not hard to see that the set of suchderivations at a point, with the usual operations of addition and product byscalars, turns to be a R -vector space and the association that sends elementsin ( I p /I p ) ∗ to derivations on F p at p above mentioned is a R -vector spaceisomorphism. Thus, we can speak in elements in V p acting on a germ f ∈F p , once we implicitly understood the association above constructed. And8eyond. Define the action of a tangent vector v p ∈ V p on a function f ∈ C ∞ ( M ) by v p ( f ) = v p ( f )where f is the class of f in F p . So, v p ( g ) = v p ( f ) when g ∈ f . Linearity andLeibniz rule follows straightforward from this definition.From those facts, if M is a m -dimensional differentiable manifold, wecan construct the differentiable vector bundle ( E , M , π, GL ( R m )) with totalspace E = ∪ p ∈ M V p , base space M , projection π given by π ( v p ) = p andtypical fibre R m , with differentiable structure obtained from the structure on M . Indeed, let ( U, ϕ ) be a local chart on the m -dimensional differentiablemanifold M around p ∈ M . By using the same notations of proposition 2.1,given f ∈ C ∞ ( M ), take elements ∂∂x i | p ∈ V p for i = 1 , . . . , m , such that ∂∂x i (cid:12)(cid:12)(cid:12)(cid:12) p ( f ) = ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( p ) Now, given η ∈ π − ( U ), for all f ∈ C ∞ ( M ) η ( f ) = η (cid:32) m (cid:88) i =1 ∂ ( ˜ f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( π ( η )) x i (cid:33) = m (cid:88) i =1 ∂f∂x i (cid:12)(cid:12)(cid:12)(cid:12) π ( η ) η ( x i )allowing to write η = m (cid:88) i =1 η ( x i ) ∂∂x i (cid:12)(cid:12)(cid:12)(cid:12) π ( η ) (1)and we call this formula coordinate expression of η with respect to the localchart ( U, ϕ ) and the values η ( x i ) coordinates of η . Hence we can define amap ˜ ϕ : π − ( U ) → R m given by˜ ϕ ( η ) = ( η ( x ) , . . . , η ( x m ))Finally, define the map φ : π − ( U ) → R m × R m given by φ ( η ) = (( ϕ ◦ π )( η ) , ˜ ϕ ( η )) (2)Let A ( M ) be the atlas of M . For each local chart ( U α , ϕ α ) associate the map φ α : π − ( U α ) → R m , constructed as above. Declare a set V ⊂ E open on E if and only if there exists an open set V on R m and an index α such that V = φ − α ( V ). The collection of those sets is a base for the topology on E which makes E a topological manifold. Besides, if ( U α , ϕ α ) , ( U β , ϕ β ) ∈ A ( M ),then the map φ β ◦ φ − α : φ α ( π − ( U α )) → φ β ( π − ( U β )) is C ∞ . This shows thatthe collection ( π − ( U α ) , φ α ) defines a differentiable atlas on E .9ith this structures, we see that ( E , M , π, GL ( R m )) is a differentiablevector bundle with typical fibre R m , total space E , which is a 2 m -dimensionaldifferentiable manifold, base space M , projection π : E → M , which is adifferentiable surjective submersion, trivializations given by the maps φ α =( ϕ α ◦ π, ˜ ϕ α ), structure group GL ( R m ) in which compatibility conditions oftrivializations are satisfied, since map such as φ β ◦ φ − α are diffeomorphisms.To fix notations, we will write T M = E and call T M the tangent bundle of M , since its elements can be faced as tangent vectors at points in M . Fromnow on we will denote the vector space tangent at p by T p M = V p .We will use this Definition 2.9 (Vector fields) . Let M be a m -dimensional differentiablemanifold. The C ∞ sections of π of ( T M , M , π, GL ( R m )) are called vectorfields on M . We denote the R -vector space of vector fields with the usualoperations of addition and product by scalars pointwise by X ( M ) . Also, X has a C ∞ ( M ) -module structure given by pointwise product by functions. This leads to the following
Theorem 2.2.
Let M be a m -dimensional differentiable manifold and C ∞ ( M ) the R -algebra of C ∞ functions on M . Then X ( M ) ≈ Vec R Der ( C ∞ ( M ))The reader is invited to proof the above theorem using the tools (anddefinitions) given here as an exercise.Following this steps, we can now construct a differentiable vector bundleover a differentiable manifold M , whose differentiable sections correspondsto derivations of order ≤ to r on the algebra of functions C ∞ ( M ) (definition2.4).First, we will need the notion of high order derivation at a point. Let p ∈ M and F p be the R -algebra of germs of functions at p . If I p denotesthe ideal of germs of functions vanishing at p we have (as above) a natural R -vector space structure on I p /I rp , since I rp (here r ∈ Z , r ≥
1) is an ideal of I p . By denoting J rp = ( I p /I r +1 p ) ∗ , we can repeat all what we have done on V p and define derivation of order ≤ r at p . Definition 2.10.
Let F p be the germs of functions at a point p in a m -dimensional differentiable manifold M . We say that a R -linear map D p : F p → R is a differential operator of order ≤ r , r ≥ , at p if and only if forall g ∈ F p , the map d g : F p → R given by d g ( f ) = D p ( gf ) − g ( p ) D p ( f )10 s a differential operator of order ≤ r − at p , and called a differentialoperator of order 0 at p if it is a product by a germ of functions at p . D p iscalled a derivation of order ≤ r at p if, in addiction, it is identically null ongerms of constant functions. Compare with the definition 2.4. J rp is finite dimensional. To see this, let ( U, ϕ ) be a local chart around p ∈ M . Without loss of generality, we can suppose ϕ ( U ) convex and ϕ ( p ) = 0. Asbefore, denote the coordinates on U by x i = t i ◦ ϕ , where t i : R m → R is the i -th projection on R m . The claim follows by showing that the set of classesof the functions x i , i = 1 , . . . , m , x i x j , ≤ i ≤ j ≤ n, . . . , x i · . . . · x i r , i ≤ . . . ≤ i r is a basis for I p /I r +1 p . Let f ∈ I p /I r +1 p . Let f ∈ C ∞ ( M ) be arepresenting element of this class. By the Taylor formula, the coordinateexpression of f on U is written by f = m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) x i + 12 m (cid:88) i,j =1 ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) x i x j + . . . ++ 1 r ! m (cid:88) i ,...,i r +1 =1 x i . . . x i r +1 (cid:90) (1 − s ) r ∂ r +1 ( f ◦ ϕ − ) ∂t i . . . ∂t i r +1 (cid:12)(cid:12)(cid:12)(cid:12) sy ds By taking the quotient, we note that the last term on the right hand sidevanishes on I p /I r +1 p . Hence, the class of f is written f = m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) x i + 12 m (cid:88) i,j =1 ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) x i x j + . . . ++ 1 r ! m (cid:88) i ,...,i r =1 ∂ r ( f ◦ ϕ − ) ∂t i . . . ∂t i r (cid:12)(cid:12)(cid:12)(cid:12) x i . . . x i r which shows that x i , i = 1 , . . . , m , x i x j , ≤ i ≤ j ≤ n, . . . , x i · . . . · x i r , i ≤ . . . ≤ i r spans I p /I r +1 p , since f ◦ ϕ − is differentiable.To show the linear independence, note that m (cid:88) i =1 a i x i + (cid:88) ≤ i ≤ j ≤ m a ij x i x j + . . . + (cid:88) i ≤ ... ≤ i r a i ...i r x i . . . x i r = 0 ⇒⇒ m (cid:88) i =1 a i x i + (cid:88) ≤ i ≤ j ≤ m a ij x i x j + . . . + (cid:88) i ≤ ... ≤ i r a i ...i r x i . . . x i r ∈ I r +1 p ⇒⇒ m (cid:88) i =1 a i t i + (cid:88) ≤ i ≤ j ≤ m a ij t i t j + . . . + (cid:88) i ≤ ... ≤ i r a i ...i r t i . . . t i r ∈ I r +1 ϕ ( p ) ≤ r are null, which leads to ∂ r ∂t j . . . ∂t j r (cid:32) (cid:88) i ≤ ... ≤ i r a i ...i r t i . . . t i r (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) = 0... ∂ ∂t k ∂t l (cid:32) (cid:88) ≤ i ≤ j ≤ m a ij t i t j (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) = 0 ∂∂t j (cid:32) m (cid:88) i =1 a i t i (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) = 0hence a j ...j r = 0... a kl = 0 a j = 0for all possible combinations of indices. So, I p /I r +1 p is finite dimensional,therefore J rp is finite dimensional also.Let ξ p ∈ J rp . Associate to ξ p a linear map D p : F p → R given by D p ( f ) = (cid:40) ∃ c ∈ f | c ( x ) = c ∀ x ∈ M ξ p ([ f ]) , if f ∈ I p where [ f ] denotes the class of f in I p /I r +1 p . By writing germs f ∈ F p as f = ˜ f + f ( p ), with ˜ f ∈ I p , given g ∈ F p we have∆ g ( f ) = D p ( gf ) − g ( p ) D p ( f ) == D p (˜ g ˜ f ) + g ( p ) D p ( ˜ f ) + f ( p ) D p (˜ g ) + 2 f ( p ) g ( p ) D p (1) − g ( p ) D p ( ˜ f ) == D p (˜ g ˜ f ) + f ( p ) D p (˜ g ) = ξ p (˜ g ˜ f ) + f ( p ) ξ p (˜ g ) . (3)Note that, given f ∈ I p , for all f ∈ I p , we have δ r − f ( f ) = ξ p ( f f ) − f ( p ) ξ p ( f ) = ξ p ( f f )and successively we can see that, given f , . . . , f i δ r − if i ( f ) = ξ p ( f i f i − . . . f )12or all f ∈ I p . Now δ f r ( f ) = ξ p ( f r . . . f ) = 0which shows that δ f r − is a differential operator of order ≤ p ,viewed as restricted to I p . Restricting to I p , by construction, δ r − if i being adifferential operator of order ≤ r − i at p implies that δ r − i +1 f i − is a differentialoperator of order ≤ to r − i + 1 at p . Hence, we have ξ p differential operatorof order ≤ to r at p and also, if δ : I p → R is an operator such that forall f ∈ I p , δ ( f ) = ξ p ( f . . . f k f ), with f , . . . , f k ∈ I p , then δ is a differentialoperator of order ≤ r − k at p . Putting on equation 3, we see that ∆ g is adifferential operator of order ≤ r − p , leading to the conclusion that D p is a differential operator of order ≤ r at the point p .Conversely, let ω : F p → R be a derivation of order ≤ r at the point p .Then f ∈ F p gives ω ( f ) = ω ( ˜ f + f ( p )) = ω ( ˜ f )which shows that the value of ω depends on its evaluation at I p only. Wealso have that if f ∈ I r +1 p then there exists f , . . . , f r +1 ∈ I p such that f = f . . . f r +1 and therefore ω ( f ) = ω ( f . . . f r +1 ) = δ r − f r +1 ( f . . . f r ) + f r +1 ( p ) ω ( f . . . f r ) == δ r − f r +1 ( f . . . f r ) = δ r − f r ( f . . . f r − ) = . . . = δ f ( f ) == f ( p ) g = 0for some g ∈ F p , where δ r − if r − i +2 is a differential operator of order ≤ r − i , for i = 1 , . . . , r . So, if f ≡ g mod I r +1 p in I p then ω ( f ) = ω ( g ) and ω can beviewed as an element in ( I p /I r +1 p ) ∗ . It follows that there exists an one to onecorrespondence between derivations of order ≤ r at the point p and elementsin J rp .We define the action of a derivation of order ≤ r at the point p , D p , ona function f ∈ C ∞ ( M ) as given by D p ( f ) = ξ p ([ f ])where ξ p is the element related to D p by the above correspondence and [ f ] isthe equivalence class of the function f in I p /I r +1 p .We can now prove the following theorem. Theorem 2.3.
Let M be a m -dimensional differentiable manifold. Thereexists a differentiable vector bundle J r ( M ) , whose base space is M and whosespace of differentiable sections Γ( J r ( M )) is isomorphic, as R -vector space,to the space of derivations of order ≤ r on C ∞ ( M ) . roof. Let us take the differentiable vector bundle J r ( M ) = (cid:91) p ∈ M J rp , with J rp = ( I p /I r +1 p ) ∗ whose coordinate functions φ : π − ( U ) → R K , with π theprojection of J r ( M ) on M and U an open subset of M , are of the form φ ( ω p ) = ( x i ( π ( ω p )) , ω p ( x i ) , ω p ( x i x j ) , . . . , ω p ( x i . . . x i r )), where the indices areincreasing, 1 to m , for all ω p ∈ J rp . Note that K = (cid:80) rk =1 (cid:0) m + k − k (cid:1) .Let ω : M → J r ( M ) a differentiable section of J r ( M ). We associate ω tothe map D : C ∞ ( M ) → C ∞ ( M ) given by D ( f )( p ) = ω ( p )( f ) ∀ f ∈ C ∞ ( M )The map D above defined is a derivation of order ≤ r on C ∞ ( M ). To seethis, first note that if c is a constant function, then D ( c )( p ) = ω ( p )( c ) = 0 ∀ p ∈ M hence D vanishes on constants. Second, given g ∈ C ∞ ( M ), the operator ∆ g given by ∆ g ( f ) = D ( gf ) − gD ( f ) ∀ f ∈ C ∞ ( M )is such that∆ g ( f )( p ) = D ( gf )( p ) − g ( p ) D ( f )( p ) = ω ( p )( gf ) − g ( p ) ω ( p )( f ) == δ g ( p )( f )which is a differentiable section (by construction) of J r − ( M ). However,Γ( J ( M )) = X ( M ) and theorem 2.2 shows that Γ( J ( M )) ≈ Vec R Der ( C ∞ ( M )).Therefore, by induction, D is a derivation of order ≤ r on C ∞ ( M ).The assignment ω (cid:55)→ D is clearly linear. Lets show it is injective. Suppose ω associated to D identically null. We have D ( f ) = 0 , ∀ f ∈ C ∞ ( M ) D ( f )( p ) = 0 , ∀ f ∈ C ∞ ( M ) , ∀ p ∈ M ω ( p )( f ) = 0 , ∀ p ∈ M , ∀ f ∈ C ∞ ( M ) ω ( p ) = 0 , ∀ p ∈ M ω = 0Lets show it is surjective. Let D : C ∞ ( M ) → C ∞ ( M ) be a derivation oforder ≤ r on C ∞ ( M ). For each p ∈ M , define ω p : F p → R by ω p ( f ) = D ( f )( p ) ∀ f ∈ C ∞ ( M )where f on the left hand side is for the equivalence class in F p of the functionrepresented by the symbol f on the right hand side. ω p is well defined because14f f, g ∈ C ∞ ( M ) are such that f ≡ g in F p , we can take a local chart ( U, ϕ )around p such that U ⊂ W , where W is an open subset of M in which f and g coincides, ϕ ( p ) = 0 and ϕ ( U ) is convex. Now, on U , f and g are written f = f ( p ) + m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) x i + 12 m (cid:88) i,j =1 ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) x i x j + . . . ++ 1 r ! m (cid:88) i ,...,i r +1 =1 x i . . . x i r +1 (cid:90) (1 − s ) r ∂ r +1 ( f ◦ ϕ − ) ∂t i . . . ∂t i r +1 (cid:12)(cid:12)(cid:12)(cid:12) sy ds and g = g ( p ) + m (cid:88) i =1 ∂ ( g ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) x i + 12 m (cid:88) i,j =1 ∂ ( g ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) x i x j + . . . ++ 1 r ! m (cid:88) i ,...,i r +1 =1 x i . . . x i r +1 (cid:90) (1 − s ) r ∂ r +1 ( g ◦ ϕ − ) ∂t i . . . ∂t i r +1 (cid:12)(cid:12)(cid:12)(cid:12) sy ds where t i : R m → R is the i -th canonical projection on R m . Hence, on U , D ( f ) = m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) D ( x i ) + 12 m (cid:88) i,j =1 ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) D ( x i x j ) + . . . ++ 1 r ! m (cid:88) i ,...,i r +1 =1 D ( x i . . . x i r +1 ) (cid:90) (1 − s ) r ∂ r +1 ( f ◦ ϕ − ) ∂t i . . . ∂t i r +1 (cid:12)(cid:12)(cid:12)(cid:12) sy ds and D ( g ) = m (cid:88) i =1 ∂ ( g ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) D ( x i ) + 12 m (cid:88) i,j =1 ∂ ( g ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) D ( x i x j ) + . . . ++ 1 r ! m (cid:88) i ,...,i r +1 =1 D ( x i . . . x i r +1 ) (cid:90) (1 − s ) r ∂ r +1 ( g ◦ ϕ − ) ∂t i . . . ∂t i r +1 (cid:12)(cid:12)(cid:12)(cid:12) sy ds But since D is a derivation of order ≤ r , we have D ( x i . . . x i r +1 )( p ) = 0for all relevant combination of indices. As f and g are in the same germ offunctions at p , all partial derivatives up to order r of its coordinate expres-15ions coincide, leading to ω p ( f ) = D ( f )( p ) == m (cid:88) i =1 ∂ ( f ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) D ( x i )( p ) + 12 m (cid:88) i,j =1 ∂ ( f ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) D ( x i x j )( p ) + . . . ++ 1 r ! m (cid:88) i ,...,i r +1 =1 D ( x i . . . x i r +1 )( p ) (cid:90) (1 − s ) r ∂ r +1 ( f ◦ ϕ − ) ∂t i . . . ∂t i r +1 (cid:12)(cid:12)(cid:12)(cid:12) sy ds =+ m (cid:88) i =1 ∂ ( g ◦ ϕ − ) ∂t i (cid:12)(cid:12)(cid:12)(cid:12) D ( x i )( p ) + 12 m (cid:88) i,j =1 ∂ ( g ◦ ϕ − ) ∂t i ∂t j (cid:12)(cid:12)(cid:12)(cid:12) D ( x i x j )( p ) + . . . ++ 1 r ! m (cid:88) i ,...,i r +1 =1 D ( x i . . . x i r +1 )( p ) (cid:90) (1 − s ) r ∂ r +1 ( g ◦ ϕ − ) ∂t i . . . ∂t i r +1 (cid:12)(cid:12)(cid:12)(cid:12) sy ds == D ( g )( p ) = ω p ( g )As D is a derivation of order ≤ r , it follows by induction on r that ω p is aderivation of order r at p . Hence, ω p ∈ J rp , for all p ∈ M . Let ω : M → J r ( M )be the map given by ω ( p ) = ω p . For all f ∈ C ∞ ( M ), we have ω ( p )( f ) = ω p ( f ) = D ( f )( p )showing that p (cid:55)→ ω ( p )( f ) is differentiable, because D ( f ) is, and so ω is adifferentiable section of J r ( M ).Hence, Γ( J r ( M )) is isomorphic, as R -vector space, to the space of deriva-tions of order ≤ r on C ∞ ( M ).It follows from the last theorem that if M is a m -dimensional manifold,a derivation of order ≤ r , D : C ∞ ( M ) → C ∞ ( M ) is written locally as D ( f ) = r (cid:88) k =1 (cid:88) ≤ i ≤ ... ≤ i k ≤ m a i ...i k ( x , . . . , x m ) ∂ k f∂x i . . . ∂x i k , with a i ...i k differentiable [4]. Let (
A, µ, e ) be an associative commutative unital K -algebra. Denote C n ( A, A ) =
Hom
Vec K ( A ⊗ n , A ) , ∀ n ∈ Z , n ≥ C • ( A, A ) = (cid:77) n ≥ C n ( A, A )16 efinition 3.1 (Partial composition) . Let f ∈ C m +1 ( A, A ) and g ∈ C n +1 ( A, A ) .For ≤ i ≤ m + 1 , the i -th partial composition of f and g is the linear map ◦ i : C m +1 ( A, A ) ⊗ C n +1 ( A, A ) → C m + n +1 ( A, A ) given by f ◦ i g = f ( id ⊗ ( i − A ⊗ g ⊗ id ⊗ ( m − i +1) A ) where id A denotes the identity of A . Definition 3.2 (Total composition) . The total composition ◦ : C • ( A, A ) ⊗ C • ( A, A ) → C • ( A, A ) is the linear map which, for each f ∈ C m +1 ( A, A ) and g ∈ C n +1 ( A, A ) , associates f ◦ g ∈ C m + n +1 ( A, A ) given by f ◦ g = m +1 (cid:88) i =1 ( − n ( i +1) f ◦ i g. Definition 3.3 (Cup product) . The cup product is the degree 0 K -linearmap (cid:94) : C • ( A, A ) ⊗ C • ( A, A ) → C • ( A, A ) , which for each f ∈ C m +1 ( A, A ) and g ∈ C n +1 ( A, A ) , associates f (cid:94) g = ( − ( m +1)( n +1) µ ◦ ( f ⊗ g )i.e. , if a , . . . , a m , a m +1 , . . . , a m + n +1 ∈ A , we have ( f (cid:94) g )( a ⊗ . . . ⊗ a m ⊗ a m +1 ⊗ . . . ⊗ a m + n +1 ) == µ ( f ( a ⊗ . . . ⊗ a m ) ⊗ g ( a m +1 ⊗ . . . ⊗ a m + n +1 )) . Proposition 3.1. ( C • ( A, A ) , (cid:94) ) is an associative graded K -algebra.Proof. By construction, C • ( A, A ) is a graded K -vector space. By K -linearityand 0 degree of cup product, we only have to prove the associativity. Notethat µ ∈ C ( A, A ), which leads to µ = µ ◦ µ = µ ( µ ⊗ id A − id A ⊗ µ ) = 0So, if f ∈ C m +1 ( A, A ) , g ∈ C n +1 ( A, A ) , h ∈ C l +1 ( A, A ), and denoting σ = ( m + 1)( n + 1) + ( m + 1)( l + 1) + ( n + 1)( l + 1) we have( f (cid:94) g ) (cid:94) h − f (cid:94) ( g (cid:94) h ) = ( − σ µ ( f ⊗ g ⊗ h ) = 0 Definition 3.4 (Hochschild cohomology) . The Hochschild cohomology of anassociative K -algebra A with coefficients in A is the cohomology of the com-plex (cid:47) (cid:47) A δ H (cid:47) (cid:47) C ( A, A ) δ H (cid:47) (cid:47) . . . δ H (cid:47) (cid:47) C n ( A, A ) δ H (cid:47) (cid:47) . . . here the coboundary operator δ H , called the Hochschild differential, is givenby ( δ H f )( a ⊗ . . . ⊗ a n ) := µ ( a ⊗ f ( a ⊗ . . . ⊗ a n )) ++ n − (cid:88) i =0 ( − i +1 f ( a ⊗ . . . ⊗ µ ( a i ⊗ a i +1 ) ⊗ . . . ⊗ a n ) ++( − n +1 µ ( f ( a ⊗ . . . ⊗ a n − ) ⊗ a n ) for any f ∈ C n ( A, A ) , for all a i ∈ A, i = 0 , . . . , n . Proposition 3.2.
The Hochschild differential δ H is a degree 1 derivation for ( C • ( A, A ) , (cid:94) ) .Proof. For simplicity, we will denote the product µ of the algebra A byjuxtaposition. By linearity, it is enough to consider the evaluation of δ H onproducts of homogeneous terms. Let f ∈ C m +1 ( A, A ) and g ∈ C n +1 ( A, A ).For any a i ∈ A, i = 0 , . . . , m + n + 2, we have δ H ( f (cid:94) g )( a ⊗ . . . ⊗ a m + n +2 ) = a ( f (cid:94) g )( a ⊗ . . . ⊗ a m + n +2 ) ++ m + n +1 (cid:88) i =0 ( − i +1 ( f (cid:94) g )( a ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a m + n +2 ) ++ ( − m + n +3 ( f (cid:94) g )( a ⊗ . . . ⊗ a m + n +1 ) a m + n +2 == a f ( a ⊗ . . . ⊗ a m +1 ) g ( a m +2 ⊗ . . . ⊗ a m + n +2 ) ++ m (cid:88) i =0 ( − i +1 f ( a ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a m +1 ) g ( a m +2 ⊗ . . . ⊗ a m + n +2 ) ++ m + n +1 (cid:88) i = m +1 ( − i +1 f ( a ⊗ . . . ⊗ a m ) g ( a m +1 ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a m + n +2 ) ++ ( − m + n +3 f ( a ⊗ . . . ⊗ a m ) g ( a m +1 ⊗ . . . ⊗ a m + n +1 ) a m + n +2 == ( a f ( a ⊗ . . . ⊗ a m ) + m (cid:88) i =0 ( − i +1 f ( a ⊗ . . . ⊗ a i a i +1 ⊗ . . . ⊗ a m +1 ) ++ ( − m +2 f ( a ⊗ . . . ⊗ a m ) a m +1 ) g ( a m +2 ⊗ . . . ⊗ a m + n +2 ) ++ ( − m +1 f ( a ⊗ . . . ⊗ a m )( a m +1 g ( a m +2 ⊗ . . . ⊗ a m + n +2 ) ++ n (cid:88) i =0 ( − i +1 g ( a m +1 ⊗ . . . ⊗ a m + i +1 a m + i +2 ⊗ . . . ⊗ a m + n +2 ) ++ ( − n +2 g ( a m +1 ⊗ . . . ⊗ a m + n +1 ) a m + n +2 ) == (( δ H f ) (cid:94) g )( a ⊗ . . . ⊗ a m + n +2 ) + ( − m +1 ( f (cid:94) δ H g )( a ⊗ . . . ⊗ a m + n +2 )18 efinition 3.5 (Gerstenhaber bracket) . The Gerstenhaber bracket is thedegree -1 K -linear map [ , ] : C • ( A, A ) ⊗ C • ( A, A ) → C • ( A, A ) which, foreach f ∈ C m +1 ( A, A ) and g ∈ C n +1 ( A, A )[ f, g ] = f ◦ g − ( − mn g ◦ f. Proposition 3.3. ( C • ( A, A ) , [ , ]) is a Lie superalgebra with respect to thereduced (by one) degree. The reader can find a proof of this fact in [2].
Proposition 3.4.
Let ( A, µ ) be an associative K -algebra. For any f ∈ C m +1 ( A, A ) δ H ( f ) = ( − m [ µ, f ] where [ , ] is the Gerstenhaber bracket.Proof. Let f ∈ C m +1 ( A, A ). Since µ ∈ C ( A, A ), it follows that[ µ, f ] = µ ◦ f − ( − m f ◦ µ = µ ( f ⊗ id A ) + ( − m µ ( id A ⊗ f ) ++ ( − m m (cid:88) i =0 ( − i +1 f ( id ⊗ iA ⊗ µ ⊗ id ⊗ ( m − i ) A ) == ( − m δ H ( f ) Proposition 3.5.
Let A be a K -algebra, where K has characteristic differentfrom 2. Fix a product ν ∈ C ( A, A ) . Then ν is associative if and only if [ ν, ν ] = 0 .Proof. Given f ∈ C m +1 ( A, A ), we have δ H ( f ) = δ H ( δ H ( f )) == δ H (( − m [ ν, f ]) = ( − m +2 [ ν, [ ν, f ]] = [[ ν, ν ] , f ] − [ ν, [ ν, f ]] ∴ δ H ( f ) = 12 [[ ν, ν ] , f ]To finish the proof, we just have to note that12 [ ν, ν ] = ν ( ν ⊗ id A ) − ν ( id A ⊗ ν ) . Remark . If (
A, µ ) is an associative K -algebra and id A : A → A denotesthe identity map on A , then δ H ( id A ) = µ ( id A ⊗ id A ) + µ ( id A ⊗ id A ) − id A ( µ ) = µ efinition 3.6 (Multiderivations) . The space of the multiderivations on theassociative unital K -algebra ( A, µ, e ) , denoted by M Der ( A ) , is the subalgebraof ( C • ( A, A ) , (cid:94) ) generated by Der ( A ) . Note that
M Der ( A ) is a graded algebra with M Der ( A ) = (cid:77) n ≥ M Der n ( A ) , where M Der n ( A ) = M Der ( A ) ∩ C n ( A, A ). Theorem 3.1 (The
M Der ( A ) subcomplex) . Every multiderivation is a Hochschildcocycle.Proof.
Lets proceed by induction to prove that δ H is identically null on M Der ( A ). Let X ∈ Der ( A ). For all a, b ∈ Aδ H X ( a ⊗ b ) = µ ( a ⊗ X ( b )) − X ( µ ( a ⊗ b )) + µ ( X ( a ) ⊗ b ) = 0Now, suppose the result for elements in M Der n − ( A ) and consider D ∈ M Der n ( A ). The space M Der ( A ) is generated by Der ( A ), so D can bewritten as linear combinations of elements of the form X (cid:94) ˜ D , where X ∈ Der ( A ) and ˜ D ∈ M Der n − ( A ). By linearity, its enough to consider theevaluation of δ H in such elements. It follows from the fact that δ H is a degree1 derivation on ( C • ( A, A ) , (cid:94) ) that δ H ( X (cid:94) ˜ D ) = ( δ H X ) (cid:94) ˜ D − X (cid:94) δ H ˜ D = 0Hence, M Der ( A ) is a subcomplex of ( C • ( A, A ) , (cid:94) ) and δ H is identically nullon M Der ( A ).The next theorem relates contravariant tensor fields on a differentiablemanifold M with multiderivations on the algebra C ∞ ( M ). Before statingthe results it is worthy to relate such fields with multiderivations at a point,an analogous to the concept of derivation at a point (see section 2). Letsprecise the notion of multiderivation at a point.Let M be a m -dimensional differentiable manifold. The tangent spaceat every point of M , being finite dimensional, allows identify ( T p M ) ⊗ n and( I p /I p ) ∗⊗ n , where I p denotes the ideal of germs of functions which vanish at p . Consider the differentiable tensor bundle ( T M ) ⊗ n . Let p ∈ M and τ p ∈ ( T M ) ⊗ n such that τ p ∈ V ⊗ np = ( I p /I p ) ∗⊗ n . By denoting F p the R -vector20pace of germs of functions at the point p , define the linear map ϑ p : F ⊗ np → R given by ϑ p ( f ⊗ . . . ⊗ f n ) = ∃ c ∈ f i | c ( x ) = c ∀ x ∈ M , for any i, i = 1 , . . . , nτ p ([ f ] ⊗ . . . ⊗ [ f n ]) , if f i ∈ I p , ∀ i = 1 , . . . , n where [ f i ] denotes the equivalence class of the germ f i in I p /I p . Since everygerm f can be written as f = ˜ f + f ( p ), where ˜ f ∈ I p and f ( p ) is the germof the constant function whose value is f ( p ), ϑ p satisfies the following: ϑ p ( f ⊗ . . . ⊗ f i g i ⊗ . . . ⊗ f n ) == g i ( p ) ϑ p ( f ⊗ . . . ⊗ f i ⊗ . . . ⊗ f n ) + f i ( p ) ϑ p ( f ⊗ . . . ⊗ g i ⊗ . . . ⊗ f n )for every i . A linear map ω : F ⊗ np → R such that ω ( f ⊗ . . . ⊗ f i g i ⊗ . . . ⊗ f n ) == g i ( p ) ω ( f ⊗ . . . ⊗ f i ⊗ . . . ⊗ f n ) + f i ( p ) ω ( f ⊗ . . . ⊗ g i ⊗ . . . ⊗ f n )for every i = 1 , . . . , n is called a multiderivation of degree n, at the point p .On the other hand, if ω : F ⊗ np → R is a multiderivation at p , one canrelate this to an unique element η p ∈ V ⊗ np , such that η p ([ f ] ⊗ . . . ⊗ [ f n ]) = ω ( f ⊗ . . . ⊗ f n ), for all f ⊗ . . . ⊗ f n ∈ F ⊗ np . To see this, first note that if c represents a constant function, then ω ( f ⊗ . . . ⊗ c ⊗ . . . ⊗ f n ) = c ω ( f ⊗ . . . ⊗ · ⊗ . . . ⊗ f n ) == c ( ω ( f ⊗ . . . ⊗ ⊗ . . . ⊗ f n ) + ω ( f ⊗ . . . ⊗ ⊗ . . . ⊗ f n )) == 2 ω ( f ⊗ . . . ⊗ c ⊗ . . . ⊗ f n )Hence, ω ( f ⊗ . . . ⊗ c ⊗ . . . ⊗ f n ) = 0. It follows that if f ⊗ . . . ⊗ f n ∈ F ⊗ np ,writing every f i as f i = ˜ f i + f i ( p ), with ˜ f i ∈ I p , by linearity of ω , we have ω ( f ⊗ . . . ⊗ f n ) = ω ( ˜ f ⊗ . . . ⊗ ˜ f n )which shows that the value of ω is determined only by its value in I ⊗ np .Now, suppose f i ∈ I p for some i . Then there exists g i , h i ∈ I p such that f i = g i h i , resulting ω ( f ⊗ . . . ⊗ f i ⊗ . . . ⊗ f n ) = ω ( f ⊗ . . . ⊗ g i h i ⊗ . . . ⊗ f n ) == g i ( p ) ω ( f ⊗ . . . ⊗ h i ⊗ . . . ⊗ f n ) + h i ( p ) ω ( f ⊗ . . . ⊗ g i ⊗ . . . ⊗ f n ) = 0So, if ˜ f i ≡ ˜ g i mod I p , ∀ i = 1 , . . . , n , then ω ( f ⊗ . . . ⊗ f n ) = ω ( g ⊗ . . . ⊗ g n )and ω induces an unique linear map η p ∈ ( I p /I p ) ∗⊗ n . Therefore there exists21 bijection between V ⊗ np and multiderivations of degree n at p . Hence, anelement η p ∈ V ⊗ np can be thought as a multiderivation of degree n at p .Define the action of ϑ p ∈ V ⊗ np on elements in ( C ∞ ( M )) ⊗ n by ϑ p ( F ⊗ . . . ⊗ F n ) = ϑ p ( f ⊗ . . . ⊗ f n ) on decomposable elements and extended itby linearity, where f i denotes a representing element of F i ∈ C ∞ ( M ) in F p , i = 1 , . . . , n .We can now prove the following Theorem 3.2.
Let M be an m -dimensional differentiable manifold. Then Γ(( T M ) ⊗ n ) ≈ Vec R M Der n ( C ∞ ( M )) , ∀ n ≥ . Proof.
Given τ ∈ Γ(( T M ) ⊗ n ), define a linear map ¯ τ : C ∞ ( M ) ⊗ n → C ∞ ( M ),given by ¯ τ ( f ⊗ . . . ⊗ f n )( p ) = τ p ( f ⊗ . . . ⊗ f n ) , ∀ p ∈ M Note that τ p ∈ ( I p /I p ) ∗⊗ n means that τ p can be written as linear combinationof elements of the form v p ⊗ . . . ⊗ v np with v ip ∈ ( I p /I p ) ∗ , ∀ i = 1 , . . . , n , and( v p ⊗ . . . ⊗ v np )( f ⊗ . . . ⊗ f n ) = v p ( f ) . . . v np ( f n ). Since τ is a differentiablesection, we have ¯ τ ∈ M Der n ( C ∞ ( M )).The assignment τ (cid:55)→ ¯ τ is clearly linear. We claim it is a bijection.To show that it is injective, it is enough to see that¯ τ ( f ⊗ . . . ⊗ f n ) = 0 , ∀ f i ∈ C ∞ ( M ) , i = 1 , . . . , n ⇒⇒ ¯ τ ( f ⊗ . . . ⊗ f n )( p ) = 0 , ∀ p ∈ M , ∀ f i ∈ C ∞ ( M ) , i = 1 , . . . , n ⇒⇒ τ p ( f ⊗ . . . ⊗ f n ) = 0 , ∀ p ∈ M , ∀ f i ∈ C ∞ ( M ) , i = 1 , . . . , n ⇒⇒ τ p = 0 , ∀ p ∈ M ⇒⇒ τ = 0where in the last but one step it was used the following fact. Taking a localchart ( U, ϕ ) at the point p , with ϕ ( p ) = 0 and taking x i = t i ◦ ϕ , with t i : R m → R the canonical projection on the i -th component (see section 2),we can write τ p ( f ⊗ . . . ⊗ f n ) = (cid:88) ( i ,...,i n ) a i ...i n ∂∂x i ⊗ . . . ⊗ ∂∂x i n where ( i , . . . , i n ) under the summation symbol means that the sum mustbe evaluated for each i j , with j = 1 , . . . , n , i j = 1 , . . . , m . Evaluating τ p onelements of the form ( x i ⊗ . . . ⊗ x i n ) successively, we have a i ...i n = 0 for anycombination of indices i j . 22o show that it is surjective, consider D ∈ M Der n ( C ∞ ( M )). For each p ∈ M , define the linear map τ p : F ⊗ np → R given by τ p ( f ⊗ . . . ⊗ f n ) = D ( f ⊗ . . . ⊗ f n )( p ) , ∀ f ⊗ . . . ⊗ f n ∈ F p where f i on the left hand side denotes the germ of the function f i written onthe right. For now on in this proof we will denote germs and functions by thesame symbol to avoid cumbersome notation. It is clear when a symbol is fora function or for a germ from the operator on such symbol. Lets show that τ p is well defined. Let f i , g i ∈ C ∞ ( M ) with i = 1 , . . . , n , such that f i ≡ g i on F p , for each i = 1 , . . . , n . Let W i be open neighbourhoods of p ∈ M such that f i | W i = g i | W i , i = 1 , . . . , n . Let ( V, ϕ ) be a local chart such that ϕ ( p ) = 0. Taking U = V ∩ W ∩ . . . ∩ W n we have ( U, ϕ ) still a local chartaround p . If necessary, we can shrink U to make ϕ ( U ) open and convex on R m . As f i and g i coincides on U for each i , it also coincides ˜ f i = f i − f i ( p )and ˜ g i = g i − g i ( p ) on U for each i . Hence, ∂ ( ˜ f i ◦ ϕ − ) ∂t j (cid:12)(cid:12)(cid:12)(cid:12) = ∂ (˜ g i ◦ ϕ − ) ∂t j (cid:12)(cid:12)(cid:12)(cid:12) , ∀ j = 1 , . . . , m, ∀ i = 1 , . . . , n. By D ∈ M Der n ( C ∞ ( M )), we have D ( f ⊗ . . . ⊗ f n ) = D (( ˜ f + f ( p )) ⊗ . . . ⊗ ( ˜ f n + f n ( p ))) = D ( ˜ f ⊗ . . . ⊗ ˜ f n )which results in τ p ( f ⊗ . . . ⊗ f n ) = D ( f ⊗ . . . ⊗ f n )( p ) = D ( ˜ f ⊗ . . . ⊗ ˜ f n )( p ) == m (cid:88) j ,...,j n =1 ∂ ( ˜ f i ◦ ϕ − ) ∂t j (cid:12)(cid:12)(cid:12)(cid:12) . . . ∂ ( ˜ f i ◦ ϕ − ) ∂t j n (cid:12)(cid:12)(cid:12)(cid:12) D ( x j ⊗ . . . ⊗ x j n )( p ) ++ m (cid:88) j ,...,j n =1 l ,...,l n =1 (cid:90) (1 − s ) ∂ ( ˜ f ◦ ϕ − ) ∂t j ∂t l (cid:12)(cid:12)(cid:12)(cid:12) sy ds . . . (cid:90) (1 − s ) ∂ ( ˜ f n ◦ ϕ − ) ∂t j n ∂t l n (cid:12)(cid:12)(cid:12)(cid:12) sy ds ·· D ( x j x l ⊗ . . . ⊗ x j n x l n )( p ) == m (cid:88) j ,...,j n =1 ∂ (˜ g i ◦ ϕ − ) ∂t j (cid:12)(cid:12)(cid:12)(cid:12) . . . ∂ (˜ g i ◦ ϕ − ) ∂t j n (cid:12)(cid:12)(cid:12)(cid:12) D ( x j ⊗ . . . ⊗ x j n )( p ) ++ m (cid:88) j ,...,j n =1 l ,...,l n =1 (cid:90) (1 − s ) ∂ (˜ g ◦ ϕ − ) ∂t j ∂t l (cid:12)(cid:12)(cid:12)(cid:12) sy ds . . . (cid:90) (1 − s ) ∂ (˜ g n ◦ ϕ − ) ∂t j n ∂t l n (cid:12)(cid:12)(cid:12)(cid:12) sy ds ·· D ( x j x l ⊗ . . . ⊗ x j n x l n )( p ) = D (˜ g ⊗ . . . ⊗ ˜ g n )( p ) == D ( g ⊗ . . . ⊗ g n )( p ) = τ p ( g ⊗ . . . ⊗ g n )23ecause D ( x j x l ⊗ . . . ⊗ x j n x l n )( p ) = 0 for any combination of indices ( j k , l k ).Thus, τ p is well defined as a linear map on F ⊗ np to real values, for all p ∈ M .Besides that, we have τ p ( f ⊗ . . . ⊗ f i g i ⊗ . . . ⊗ f n ) = D ( f ⊗ . . . ⊗ f i g i ⊗ . . . ⊗ f n )( p ) == g i ( p ) D ( f ⊗ . . . ⊗ f i ⊗ . . . ⊗ f n )( p ) + f i ( p ) D ( f ⊗ . . . ⊗ g i ⊗ . . . ⊗ f n )( p ) == g i ( p ) τ p ( f ⊗ . . . ⊗ f i ⊗ . . . ⊗ f n ) + f i ( p ) τ p ( f ⊗ . . . ⊗ g i ⊗ . . . ⊗ f n )on each entry. Therefore τ p is a multiderivation of degree n at the point p ,for each p ∈ M . Finally, lets construct the map τ : M → ( T M ) ⊗ n given by τ ( p ) = τ p . The map τ is a differentiable section, because for each p ∈ M τ ( p )( f ⊗ . . . ⊗ f n ) = τ p ( f ⊗ . . . ⊗ f n ) = D ( f ⊗ . . . ⊗ f n )( p )and D ( f ⊗ . . . ⊗ f n ) ∈ C ∞ ( M ) for any linear combination of elements f ⊗ . . . ⊗ f n ∈ C ∞ ( M ).Thus, the stated assignment is an isomorphism of R -vector spaces betweenΓ(( T M ) ⊗ n ) and M Der n ( C ∞ ( M )).The last theorem reveals that if M is a m -dimensional differentiable man-ifold, an element in D ∈ M Der n ( C ∞ ( M )) can be written in local coordinatesas D = m (cid:88) j ,...,j n =1 D ( x j ⊗ . . . ⊗ x j n ) ∂∂x j ⊗ . . . ⊗ ∂∂x j n . Definition 4.1 (Iterated Derivation) . Let A be a commutative associativeunital K -algebra. The space of iterated derivations, denoted by SDer ( A ) , isthe subalgebra of ( C ( A, A ) , ◦ ) generated by Der ( A ) . We denote by SDer n ( A ) the set of elements D ∈ SDer ( A ) which can be written as linear combinationsof elements of the form X ◦ . . . ◦ X r , with X i ∈ Der ( A ) , ∀ i = 1 , . . . , r, r ≤ n .Remark . Note that
SDer ( A ) can not be written as direct sum of thespaces SDer n ( A ). However, if r ≤ n then we have SDer r ( A ) ⊂ SDer n ( A ).Hence, we have a filtration on the algebra SDer ( A ). Theorem 4.1. If D ∈ SDer n ( A ) , then D is a derivation of order ≤ n . A similar notion for Lie algebras can be found in [6]. roof. Denote the product on A by juxtaposition. We proceed by inductionon n . Surely, if X ∈ Der ( A ), then X is a derivation of order ≤
1. Suppose D ∈ SDer n ( A ) and the result valid for n −
1. By linearity, it is enough toconsider D as D = ˜ D ◦ X n , where ˜ D ∈ SDer n − ( A ) and X n ∈ Der ( A ).By the induction hypothesis and the fact that SDer r − ( A ) ⊂ SDer r ( A ) forall r ≥
1, it is enough to consider the high order terms of ˜ D , i.e. terms as X ◦ . . . ◦ X n − . To show that X ◦ . . . ◦ X n − is a differential operator oforder ≤ n , given a ∈ A , we must show that the operator ∆ a , given by∆ a ( b ) = ( X ◦ . . . ◦ X n )( ab ) − a ( X ◦ . . . ◦ X n )( b )for all b ∈ A , is a differential operator of order ≤ n −
1. We have∆ a ( b ) = ( X ◦ . . . ◦ X n )( ab ) − a ( X ◦ . . . ◦ X n )( b ) == ( X ◦ . . . ◦ X n )( a ) · b + n (cid:88) i =1 ( X ◦ . . . ◦ ˆ X i ◦ . . . ◦ X n )( a ) X i ( b ) ++ (cid:88) ≤ i 1, for all a ∈ A . Thus D is a differential operator of order ≤ n . By considering operators as ˜ D ◦ X , with X ∈ Der ( A ), it is clear that˜ D ( X ( α )) = 0, for all α ∈ K (properly identified as element of A ). Hence, D is a derivation of order ≤ n . Theorem 4.2. Let M be an m -dimensional differentiable manifold. If D isa derivation of order ≤ r on C ∞ ( M ) , then D ∈ SDer r ( C ∞ ( M )) .Proof. We proceed by induction. If D is a derivation of order ≤ 1, then D ∈ Der ( C ∞ ( M )) therefore D ∈ SDer ( C ∞ ( M )). Suppose the result for r − 1. Let D be a derivation of order ≤ r on C ∞ ( M ). Then, by theorem 2.3, D can be related to an element D ∈ Γ( J r ( M )). For each p ∈ M , define thelinear map Φ r,p : I p /I r +1 p → I p /I rp which associates the equivalence class ofa germ of a function f in I p /I r +1 p to its class in I p /I rp . This is well defined,25ecause I r +1 p ⊂ I rp and it is a projection because, by the Taylor’s formula, if f has a representing in I p /I rp , then it has a representing in I p /I r +1 p such that[ f ] r = Φ r,p ([ f ] r +1 ). Note that if f ∈ I rp mod I r +1 p , then Φ r,p ( f ) = 0, and bythe other hand, if Φ r,p ( f ) = 0, then f ∈ I rp mod I r +1 p . Thus, Ker (Φ r,p ) ≈ Vec R I rp /I r +1 p . We have, naturally, I p /I r +1 p ≈ Vec R I p /I rp ⊕ I rp /I r +1 p .The dual map to Φ r,p is Φ ∗ r,p : J r − p → J rp , given by(Φ ∗ r,p ( u ))( f ) = u (Φ r,p ( f ))remembering that J rp = ( I p /I r +1 p ) ∗ . Φ ∗ r,p is injective. This follows from thefact of being dual to a surjective linear map between vector spaces, becauseif u ∈ J r − p is such that Φ ∗ r,p ( u ) = 0, then (Φ ∗ r,p ( u ))( f ) = 0 for all f ∈ I p /I r +1 p and then, u (Φ r,p ( f )) = 0 for all f ∈ I p /I r +1 p . As Φ r,p is surjective, given g ∈ I p /I rp , there exists f ∈ I p /I r +1 p such that g = Φ r,p ( f ). Hence, u ( g ) = 0for all g ∈ I p /I rp therefore u = 0.The map Φ ∗ r : J r − ( M ) → J r ( M ) such that Φ ∗ r ( ξ ) = Φ ∗ r,π ( ξ ) ( ξ ) is a mor-phism of differentiable vector bundles. We have Φ ∗ r fibre preserving and linearon fibres by construction. Furthermore, if ξ ∈ J r − ( M ), locally, ξ is writtenas ξ = r − (cid:88) k =1 (cid:88) ≤ i ≤ ... ≤ i k ≤ m ξ ( x i . . . x i k ) ∂ k ∂x i . . . ∂x i k But Φ ∗ r ( ξ ) is written locally as ξ = r − (cid:88) k =1 (cid:88) ≤ i ≤ ... ≤ i k ≤ m ξ ( y i . . . y i k ) ∂ k ∂y i . . . ∂y i k because terms of order r does not belong to the range of Φ ∗ r . By the factthat local charts on J r − ( M ) and J r ( M ) are fibred charts, there exists adiffeomorphism sending the coordinate expression of ξ in terms of y i andderivatives, to the coordinate expression of ξ in terms of x i and derivatives.By the match of those expressions follows Φ ∗ r differentiable.Given p ∈ M , by the induction hypothesis and the inclusion above, it isenough to consider derivations of order ≤ r such that in a neighbourhood of p have only terms of order r . Let η a such derivation and ( U, x , . . . , x m ) alocal chart around p for which this occurs. η being a derivation of order ≤ r leads to η ( x i . . . x i r ) = ∆ i r ( x i . . . x i r − ) + x i r η ( x i . . . x i r − )with ∆ i r differential operator of order ≤ r − 1. By the choice of the localchart, we have η ( x i . . . x i r − ) = 0, because η has only terms of order r .Therefore η ( x i . . . x i r ) = ∆ i r ( x i . . . x i r − )26nd from this follows that ∆ i r is a derivation of order ≤ r − η can bewritten in terms of ∆ i r in this way η = m (cid:88) k =1 (cid:88) i ≤ ... ≤ i r − η ( x i . . . x i r − x k ) r ! ∂ r ∂x i . . . ∂x i r − ∂x k == m (cid:88) k =1 (cid:88) i ≤ ... ≤ i r − ∆ k ( x i . . . x i r − ) r ! ∂ r − ∂x i . . . ∂x i r − ∂∂x k == m (cid:88) k =1 ∆ k r ! ∂∂x k (4)As ∆ k is a derivation of order ≤ r − 1, the induction hypothesis allows towrite ∆ k = v k ◦ u k where v k is a vector field and u k is a derivation of order ≤ r − 2, both definedon U .By the equation 4, we have η = m (cid:88) k =1 ∆ k r ! ∂∂x k = m (cid:88) k =1 ( v k ◦ u k ) r ! ∂∂x k = m (cid:88) k =1 v k (cid:18) u k r ! ∂∂x k (cid:19) as the term u k r ! ∂∂x k is a composition of derivations, it is itself a derivation oforder ≤ r − U . Therefore η = m (cid:88) k =1 v k ◦ w k with v k ∈ Γ( J ( U )) and w k ∈ Γ( J r − ( U )), for each k = 1 , . . . , m . For thesake of simplicity, we denote this by η = v ◦ u .Let { U α } be a locally finite open covering of M and { ρ α } a partition ofunity subordinated to such covering. For each index α , we can find v α and u α as above, such that η = v α ◦ u α Lets construct the fields ζ ∈ X ( M ), ξ, θ ∈ Γ( J r − ( M )) by ζ = (cid:88) λ ρ λ v λ , ξ = (cid:88) ν ρ ν u ν , θ = (cid:88) β γ β u β where γ β = (cid:80) α ρ α v α ( ρ β ). Note that θ is well defined, because if ρ β hassupport on U β , so are its derivatives and then, given p ∈ M , γ β ( p ) does notvanish only for a finite number of indices β . Furthermore, given f ∈ C ∞ ( M ) ρ λ v λ ( ρ ν u ν ( f )) = ρ λ ρ ν v λ ( u ν ( f )) + ρ λ v λ ( ρ ν ) u ν ( f )27eading to ρ λ ρ ν η ( f ) = ρ λ v λ ( ρ ν u ν ( f )) − ρ λ v λ ( ρ ν ) u ν ( f )because if U λ ∩ U ν = ∅ , then either ρ λ or ρ ν vanish, and then ρ λ ρ ν v λ ◦ u ν = ρ λ ρ ν η , and if U λ ∩ U ν (cid:54) = ∅ , then u νp = u λp at each p ∈ U λ ∩ U ν , leading to ρ λ ρ ν v λ ◦ u ν = ρ λ ρ ν η .Hence, η ( f ) = (cid:88) λ,ν ρ λ ρ ν η ( f ) = (cid:88) λ,ν ρ λ v λ ( ρ ν u ν ( f )) − (cid:88) λ,ν ρ λ v λ ( ρ ν ) u ν ( f ) == (cid:88) λ ρ λ v λ (cid:32)(cid:88) ν ρ ν u ν ( f ) (cid:33) − (cid:88) ν γ ν u ν ( f ) == ( ζ ◦ ξ )( f ) − θ ( f )By the induction hypothesis, ξ, θ ∈ Γ( J r − ( M ) can be related to elementsin SDer r − ( C ∞ ( M )), leading to η ∈ SDer r ( C ∞ ( M )). As an arbitrary ele-ment D ∈ Γ( J r ( M ) is a linear combination of elements in Γ( J r − ( M ) andelements of order r , it follows that D ∈ SDer r ( C ∞ ( M )) . The Hochschild-Kostant-Rosenberg theorem is usually stated as an isomor-phism of graded algebras between Hochschild homology and universal differ-ential forms (given by the K¨ahler differentials) of a smooth algebra. A proofof this version can be found in [5]. However, we want a dual version of thisfact, by relating Hochschild cohomology of an algebra and its multilineartransformations. The process of taking duals often involves some restrictionto a nice subspace. For infinite dimensional cases, the dual of a vector spaceis too big and an analogous copy of the original space that retains or pre-serves the desired properties lies in a specific kind of subspace. In the case ofthe Hochschild-Kostant-Rosenberg theorem we must restrict the Hochschildcohomology to the subcomplex of polyderivations. Definition 5.1 (Polyderivations on an algebra) . Let A be a commutative as-sociative unital K -algebra. The space of polyderivations on the algebra A , de-noted by D poly ( A ) , is the subalgebra of ( C • ( A, A ) , (cid:94) ) generated by SDer ( A ) .We denote D npoly ( A ) = D poly ( A ) ∩ C n ( A, A ) . Also, we denote by D n,rpoly ( A ) thespace of polyderivations of degree n and order ≤ r i.e. elements in C n ( A, A ) which are polyderivations generated by SDer r ( A ) . heorem 5.1. ( D poly ( A ) , δ H ) is a filtered subcomplex of ( C • ( A, A ) , δ H ) .Proof. For the sake of simplicity, we denote the product on A by juxtaposi-tion. Take an element D ∈ D n,rpoly ( A ). Then D is a linear combination of ele-ments of the form D (cid:94) . . . (cid:94) D n , with D i ∈ SDer r ( A ), for all i = 1 , . . . , n .However, if D i ∈ SDer r ( A ), then it is linear combination of elements of theform X i ◦ . . . ◦ X ij , j ≤ r , with X ij ∈ Der ( A ), for all i = 1 , . . . , n , for all j ≤ r .Then, if a, b ∈ Aδ H ( X i ◦ . . . ◦ X ij )( a ⊗ b ) == a ( X i ◦ . . . ◦ X ij )( b ) − ( X i ◦ . . . ◦ X ij )( ab ) + ( X i ◦ . . . ◦ X ij )( a ) b == − j − (cid:88) k =1 (cid:88) I k ( X i ˆ I k )( a )( X iI k )( b ) (5)where I k denotes a set of indices, subset of { , . . . , j } , with exactly k elements l , . . . , l k such that l < . . . < l k , for k ≤ j , X i ˆ I k denotes the composite X i ◦ . . . ◦ ˆ X il s ◦ . . . ◦ X ij , in which are absent all elements X il s , l s ∈ I k , in order,and X iI k denotes the composite X il ◦ . . . ◦ X il k in that order.Hence, δ H ( X i ◦ . . . ◦ X ij ) ∈ D ,j − poly ( A ). As δ H is a degree 1 derivation on( C • ( A, A ) , (cid:94) ), it follows that δ H ( D (cid:94) . . . (cid:94) D n ) = n (cid:88) i =1 ( − i +1 D (cid:94) . . . (cid:94) δ H ( D i ) (cid:94) . . . (cid:94) D n (6)By linearity, D ∈ D n,rpoly ( A ), results δ H ( D ) ∈ D n +1 ,rpoly ( A ). It shows that( D poly ( A ) , δ H ) is subcomplex of ( C • ( A, A ) , δ H ), filtered by order of deriva-tions. Definition 5.2 (Alternator on D n,rpoly ( A )) . If A is a commutative associativeunital K -algebra, where K is a field with characteristic 0, we define for n ≥ the linear map Alt : D n,rpoly ( A ) → D n,rpoly ( A ) given, on decomposable elements,by Alt ( D (cid:94) . . . (cid:94) D n ) = 1 n ! (cid:88) σ ∈ S n ε ( σ ) D σ (1) (cid:94) . . . (cid:94) D σ ( n ) where σ denotes a permutation in S n , the set of all permutations on n ele-ments, and ε ( σ ) denotes the signal of this permutation. Proposition 5.1. Let D ∈ D n,rpoly ( C ∞ ( M )) such that D is closed for theHochschild differential. Then there exists a cochain E ∈ D n − ,r +1 poly ( C ∞ ( M )) and an alternating element η ∈ M Der n ( C ∞ ( M )) such that D = δ H ( E ) + η (7)29 he proof of the proposition is quite technical and can be found in [3].Remark . Let A be a commutative associative unital K -algebra. We de-note D ( A ) = A ⊕ D poly ( A ). Note that ( D ( A ) , δ H ) is subcomplex of theHochschild complex ( C • ( A, A ) , δ H ). Theorem 5.2 (The Hochschild-Kostant-Rosenberg theorem for differen-tiable manifolds ) . Let M be a m -dimensional differentiable manifold. Thereis a quasi-isomorphism between the complexes ( D ( C ∞ ( M )) , δ H ) and (Ω • ( M ) , d ) ,where d : Ω • ( M ) → Ω • ( M ) is the null differential on polyvector fields Ω • ( M ) =Γ(Λ T M ) .Proof. Let Alt ( M Der n ( C ∞ ( M ))) be the range of the alternator on M Der n ( C ∞ ( M )) = D n, poly ( C ∞ ( M )). Define the linear map ψ : Ω n ( M ) → Alt ( M Der n ( C ∞ ( M ))) given, on decomposable elements, by ψ ( X ∧ . . . ∧ X n ) = Alt ( X (cid:94) . . . (cid:94) X n )for n ≥ 1. Note that ψ is fibre preserving. Lets show that ψ is injective.Let η ∈ Ω n ( M ) such that ψ ( η ) = 0. At each p ∈ M , η p is written as linearcombination of elements in a base for Λ p ( T p M ), of the form X i p ∧ . . . ∧ X i n p .However, ψ ( X i p ∧ . . . ∧ X i n p ) = Alt ( X i p (cid:94) . . . (cid:94) X i n p ) = Alt ( X i p ⊗ . . . ⊗ X i n p ) == X i p ∧ . . . ∧ X i n p because at each point the cup product (cid:94) coincides with tensor product,once each X ip can be viewed as a linear functional. Thus, ψ ( η ) = 0 results η p = 0 for all p , and then η = 0. Lets show that ψ is surjective. Let N ∈ Alt ( M Der n ( C ∞ ( M ))). By linearity, it is enough to consider N in theform 1 n ! (cid:88) σ ∈ S n ε ( σ ) X σ (1) (cid:94) . . . (cid:94) X σ ( n ) . Now, take η ∈ Ω n ( M ) as X ∧ . . . ∧ X n .It follows that ψ ( X ∧ . . . ∧ X n ) = Alt ( X (cid:94) . . . (cid:94) X n ) = 1 n ! (cid:88) σ ∈ S n ε ( σ ) X σ (1) (cid:94) . . . (cid:94) X σ ( n ) By linearity, ψ ( η ) = N . Hence, we have an one-to-one association betweenalternating elements in M Der n ( C ∞ ( M )) and n -vector fields. For now on, weshall no more distinguish such elements. We call J n the family of maps takingcochains D ∈ D n,rpoly ( C ∞ ( M )) and sending to J n ( D ) = Alt ( D ), for n ≥ This proof follows the technique in [1] as identity on C ∞ ( M ). As C ∞ ( M ) is commutative, δ H vanish on C ∞ ( M ).Thus, J ◦ δ H = d ◦ J . Let D be a n -coboundary, n > 1. Then there existsa ( n − E such that D = δ H ( E ). The formulae 5 and 6 shows that δ H ( E ) is a linear combination of terms which are symmetric on two entries,hence it must be Alt ( δ H ( E )) = 0. It follows that J n ◦ δ H = d ◦ J n − , because d is identically null. Hence, each J n induces a morphism on cohomology J ∗ n : H n ( D ( C ∞ ( M ))) → H n (Ω n ( M )).By the fact that d is the null differential on (Ω n ( M ) , d ) we have H n (Ω n ( M ))isomorphic as R -vector space to Ω n ( M ), for all n ≥ J ∗ is isomorphism. Let D be a n -cocycle, n ≥ 1. Fromproposition 5.1 we have D = δ H ( E ) + η , where E is a ( n − η ∈ Ω n ( M ). It follows that if θ ∈ H n ( D poly ( C ∞ ( M ))), whose representingelement in D poly ( C ∞ ( M )) is D , then D can be written as D = δ H ( E ) + η and thus J ∗ n ( θ ) = [ J n ( D )] = [ J n ( δ H ( E ) + η )] = [ η ] = ηJ ∗ n is injective. Indeed, if θ is such that J ∗ n ( θ ) = 0, then [ J n ( D )] = 0 hence J n ( δ H ( E ) + η ) = J n ( η ) = 0, resulting η = 0 because J n ( η ) = η . Thus, D = δ H ( E ) and then θ is the null class. Now, J ∗ n is surjective. To show this, notethat Ω n ( M ) is isomorphic to Alt ( M Der n ( C ∞ ( M ))), which is contained in M Der n ( C ∞ ( M )), which is contained in D n,rpoly ( C ∞ ( M )), for all r ≥ 1. Hence,given η ∈ Ω n ( M ) we associate η ∈ Ω n ( M ) to it. However, by theorem 3.1, η isa n -cocycle. Thus, J n ( η ) = η . Also, by η alternating and by formulae 5 and 6, η can not be a coboundary, therefore the class of η in H n ( D poly ( C ∞ ( M ))) cannot be the null class. It follows that J ∗ n is an isomorphism on cohomology forall n and hence ( D ( C ∞ ( M )) , δ H ) and (Ω • ( M ) , d ) are quasi-isomorphics. References [1] Cahen, M., De Wilde, M., Gutt, S.: Local cohomology of the algebra of C ∞ functions on a connected manifold, Lett. in Math. Phys. , 4:157-167,1980.[2] Gerstenhaber, M.: The cohomology structure of an associative ring, Ann. of Math. , 78(2):267-288, 1963.[3] Gutt, S., Rawnsley, J.: Equivalence of star products on a symplec-tic manifold: an introduction to Deligne’s ˇCech cohomology classes, J.Geom. Phys. , 29:347-392, 1999. 314] Kostrikin, A. I., Shafarevich I. R. Basic Notions of Algebra , Springer-Verlag, New York, 1990.[5] Loday, J. Cyclic Homology , Springer-Verlag, Berlin, 1992.[6] Sardanashvily, G.: Differential operators on Lie and graded Lie algebras,arXiv:1004.0058v1[math-ph][7] Warner, F. W.