Noncommutative localization in smooth deformation quantization
aa r X i v : . [ m a t h . QA ] O c t Noncommutative localization in smooth deformationquantization
Martin Bordemann,
[email protected] ,Benedikt Hurle,
[email protected] ,Hamilton Menezes de Araujo,
[email protected] ,Université de Haute Alsace, Mulhouse,October 30, 2020
Abstract
In this paper we shall show the equivalence of algebraic and analytic localisation foralgebras of smooth deformation quantization for several situations. The proofs arebased on old work by Whitney, Malgrange and Tougeron on the commutative algebraof smooth function rings from the 60’s and 70’s. ontents
Introduction 11 Review of basic concepts 3 C ∞ ( X, K )
18B Proof of Proposition 1.2 20 ntroduction
Localization in commutative algebra means a universal construction where a set of chosen elementsin a given commutative ring is made invertible (they will become denominators): the outcome iscalled a ring of fractions. The classical example is the well-known passage from the integers tothe field of rational numbers. It is a very important tool in algebraic and analytical geometry. Indifferential geometry, however, localization is rather used in the analytic sense, i.e. the passage fromglobally defined smooth functions to those which are only defined on an open subset. It followsfrom the classical works by Whitney, Malgrange and Tougeron that these analytical localizationsare often isomorphic to certain algebraic localizations in the smooth (or even C k , k ∈ N ) case.Based on old work by O.Ore in the 1930’s localization can be transferred to noncommutativealgebras: it turns out that there is a completely general construction which –however– is in somesituations not very practical: on the other hand if there is an additional condition on the set ofpotential denominators, the famous right (or left) Ore condition, the construction shares almost allproperties of the commutative localization.In this work we should like to study noncommutative localization of algebras arising in defor-mation quantization: in this theory –invented by [1] in 1978– formal associative deformations ofthe algebra of all smooth complex valued functions on a Poisson manifold, so-called star-products,are studied aiming at an interpretation of the noncommutative multiplication of operators used inquantum mechanics. It is well-known that the first order commutator of such a deformation alwaysgives rise to a Poisson bracket, but it is highly non-trivial to show that every Poisson bracket arisesas a first order commutator of a deformation: this latter result is the famous Kontsevich formalityTheorem, [10].We consider star-products given by formal power series of bidifferential operators (as almost every-one): these multiplications immediately define star-products of locally defined functions by suitable‘restrictions’.We first show that this analytical localization is isomorphic to the commutative algebraic localiza-tion with respect to the set of all those formal power series of smooth functions whose zeroth orderterm does nowhere vanish on the given open set. As a by-product we have the result that thismultiplicative set satisfies the right Ore condition.In a similar way we can show that the set of all analytical germs of a star-product algebra at agiven point of the manifold is isomorphic the noncommutative localization of the complement ofthe maximal ideal of all those formal power series of functions whose term of order zero vanishes atthe point.We also sketch a more algebraic frame work to compare the commutative localization of bidifferentialoperators giving rise in a deformation of the localized algebra and the noncommutative localizationof the deformed algebra by a rather natural mutliplicative set: here the question ‘Does localizationcommutes with deformation arises’.The paper is organised as follows: in the first Section we recall some basic concepts of the commu-tative algebra of smooth function algebras and (non)commutative localization following Tougeron’sbook [23] and Lam’s very nice text-book [11].In Section 2 we show the first localization result concerning open sets: an important tool isTougeron’s fonction aplatisseur which makes a given locally defined smooth function globally de-fined by multiplication with a suitbal function being nowhere zero on the given open set. In thisproof, we had to make explicit use of the semi-norms defining the Fréchet topology of the smoothfunction space.In Section 3 we prove a similar result for germs, heavily relying on the first Theorem.In an outlook, we discuss commutative localization of multidifferential operators and prime ideals1f the function algebra which rise yet another localization problem.At the end we describe a simple example of a multiplicative set which is not Ore. Acknowledgements
The authors would like to thank Alberto Elduque, Camille Laurent-Gengoux, Daniel Panazzolo,Leonid Ryvkin, Zoran Škoda, and Friedrich Wagemann for many valuable discussions and hints.2
Review of basic concepts
Let K be a fixed commutative associative unital ring such that K = 0 = 0 K . All K -algebrasare supposed to be associative and unital. We shall include unital K -algebras R isomorphic to { } for which R = 0 = 0 R . Note that associative unital rings are always Z -algebras in a naturalway. In order to avoid clumsy notation we shall not write R , K , R or K , but simply and where the precise interpretation should be clear from the context. For the convenience of the reader we shall give the following elementary survey. For more informationsee e.g. [3].Recall the elementary commutative algebra of unital K -algebras of functions on a set: let X be aset, K be a commutative domain, and R be a given unital subalgebra of the K -algebra of all thefunctions X → K . As usual, for any subset Y ⊂ X let I ( Y ) ⊂ R be the set of functions vanishing on Y which is always an ideal of R (the vanishing ideal of Y ), and for any subset J ⊂ R let Z ( J ) ⊂ X be the subset of those points of X on which all functions in J vanish. Clearly Y ⊂ Z ( I ( Y )) and J ⊂ I ( Z ( J )) . Moreover for any two ideals I and I of R it follows that Z ( I ) ∪ Z ( I ) = Z ( I I ) (here the fact that K is a domain is used), hence the set of all subsets Z ( I ) , I ideal of R , satisfiesthe axioms of the closed sets of a topology on X called the Zariski topology on X w.r.t. R . Theseclosed subsets could be called R -algebraic sets : in the particular case K = K is a field, X = K n , and R the polynomial ring K [ x , . . . , x n ] these are the algebraic subsets , whereas for R being the algebraof analytic functions (for K = R or K = C ) these sets are called analytic subsets . Note that theZariski closure of a set Y ⊂ X is equal to Z ( I ( Y )) . On the other hand the inclusion J ⊂ I ( Z ( J )) issometimes expressible by what is called a Nullstellensatz : in algebraic geometry over algebraicallyclosed fields I ( Z ( J )) is equal to the set of all polynomials in R such that a certain power is in J .Recall that an ideal I ⊂ R is called proper iff I = R iff I . Moreover, a proper ideal is called maximal iff it is equal to any other proper ideal containing it. More generally, a proper ideal p of R is called a prime ideal iff the factor algebra R/ p is a domain iff the complementary set S p = R \ p isa multiplicative subset , i.e. ∈ S p and if s, s ′ ∈ S p then ss ′ ∈ S p . Recall that Krull’s Lemma statesthat given any multiplicative subset S ⊂ R and ideal J with J ∩ S = ∅ there is a prime ideal p ⊃ J with S ∩ p = ∅ , see e.g. [6, p.391, Prop. 7.2, Prop. 7.3]. Let X be an N -dimensional differentiable manifold (whose underlying topological space we shallalways assume to be Hausdorff and second countable). Let K denote either the field of all realnumbers, R , or the field of all complex numbers, C . For any real vector bundle E over X we shalldenote by the same symbol E its complexification. Consider the K -algebra A = C ∞ ( X, K ) of allsmooth K -valued functions f on X . Even in the case where X is an open subset of R n the algebraicproperties of A are rather different from the function algebras used in algebraic or analytic geometry.There has been much work on that in the past by Malgrange and Tougeron, see [14], [23], based onthe classical works by Whitney. We shall give a short outline of the features we shall need.The K -vector space A is given a well-known Fréchet topology which can be conveniently definedin the following terms: fix a Riemannian metric h on X , and let ∇ denote its Levi-Civita connection.For any nonnegative integer n denote by S n T ∗ X the n th symmetric power of the cotangent bundleof X : its smooth sections can be viewed as smooth functions on the tangent bundle τ X : T X → X n in the direction of the fibres. For any section α ∈ Γ ∞ ( X, S n T ∗ X ) let Dα ∈ Γ ∞ ( X, S n +1 T ∗ X ) its symmetrized covariant derivative w.r.t. ∇ whichcan be seen as a symmetric version (depending on ∇ !) of the exterior derivative. Finally for anysmooth function f : X → K let D n f ∈ Γ ∞ ( X, S n T ∗ X ) be the n fold iterated symmetrized covariantderivative of f , hence D f = f , Df = df , D n +1 f = D ( D n f ) . For any compact set K ⊂ X and anynonnegative integer m define a system of functions p K,m : A → R by p K,m ( f ) = max {| D n f ( v ) | | n ≤ m, τ X ( v ) ∈ K and h ( v, v ) ≤ } . (1.1)which will define an exhaustive system of seminorms, hence a locally convex topological vector spacewhich is known to be metric and sequentially complete, hence Fréchet. It is not hard to see thatthe choice of another Riemannian metric will give another system of seminorms which is equivalentto the first one. For flat R n equipped with the usual euclidean scalar product these seminormsare easily seen to be equivalent to the usual seminorms used in analysis where the higher partialderivatives are expressed by multi-indices. Pointwise multiplication and evaluation at a point arewell-known to be continuous w.r.t. to the Fréchet topology.For later use we shall give the usual definition of multidifferential operators D : A × · · · × A → A ofrank p as a p -linear map (over the ground field K ) such that there is a nonnegative integer l and foreach chart ( U, ( x , . . . , x N )) there are smooth functions D α ··· α p : U → K indexed by p multi-indices α , . . . , α p ∈ N N such that for each point x ∈ U , and smooth functions f , . . . , f p ∈ A there is thefollowing local expression D ( f , . . . , f p )( x ) = X | α | ,..., | α p |≤ l D α ··· α p ( x ) ∂ | α | (cid:0) f | U (cid:1) ∂x α ( x ) · · · ∂ | α p | (cid:0) f p | U (cid:1) ∂x α p ( x ) (1.2)ointwise multiplication, evaluation at a point and where as usual | α | = | ( i , . . . , i N ) | = i + . . . + i N and ∂ | α | φ∂x α is short for ∂ i ... + iN φ∂ ( x ) i ··· ∂ ( x N ) iN . Note that the value of D ( f , . . . , f p ) at x only depends onthe restriction of the functions f , . . . , f p to any open neighbourhood of x : it follows that multi-differential operators can always be localized in the analytical sense that they give rise to unique well-defined multi-differential operators D U on C ∞ ( U, K ) for any open subset U ⊂ X such that D and D U intertwine the restriction map η : A → C ∞ ( U, K ) in the obvious way. Multi-differential operators arewell-known to be continuous w.r.t. to the Fréchet topology. Furthermore, recall the usual composi-tion rule of multi-differential operators (inherited by the usual rule for multi-linear maps): given twomulti-differential operators D (of rank p ) and D ′ (of rank q ) and a positive integer ≤ i ≤ p thenthe map D ◦ i D ′ defined by ( f , . . . , f p + q − ) D (cid:0) f , . . . , f i − , D ′ ( f i , . . . , f i + p − ) , f i + p , . . . , f p + q − (cid:1) is a multidifferential operator of rank p + q − . This composition rule obviously is compatible withlocalization in the sense that (cid:0) D ◦ i D ′ (cid:1) U = D U ◦ i D ′ U .Recall that for any given point x ∈ X the binary relation ∼ x on A defined by f ∼ x g iff the twosmooth functions f and g have the same Taylor series at x w.r.t. to some chosen chart around x is an equivalence relation which does not depend on the chosen chart, that an equivalence class iscalled an infinite jet , and the class of f is called the infinite jet j ∞ x ( f ) of f , see e.g. [9, p.117 Section12]. A is well-known to have two ‘bad’ features from the point of view of commutative algebra: it hasvery many nontrivial zero-divisors and it is NOT noetherian: if for each nonnegative integer n we denote by I n the ideal of all smooth K -valued functions on R vanishing on the closed interval h − n +1 , n +1 i , then the ascending sequence I n ⊂ I n +1 never stabilizes after a finite number of steps.For a given ideal J of A its zero set Z ( J ) ⊂ X is of course a closed subset of X , and the closure J of J (w.r.t. the Fréchet topology) in A remains an ideal. The vanishing ideal of any set is closed in4he Fréchet topology, hence there is the chain of inclusions J ⊂ J ⊂ I ( Z ( J )) . Since for any point x not contained in Z ( J ) there is a function g in the ideal J not vanishing at x , a simple partition ofunity argument shows that any smooth K -valued function whose support is compact and has emptyintersection with Z ( J ) must be an element of J . It follows in particular that an ideal J containsthe ideal D ( X ) of all smooth K -valued functions having compact support iff Z ( J ) = ∅ iff J is densein A . In the particular case where X is compact this means that the only dense ideal is equal to A . Returning to general X it follows that every proper ideal of A which is closed w.r.t. the Fréchettopology has a non-empty set of common zeros. In particular, every closed proper maximal ideal of A is equal to the vanishing ideal I x = I ( { x } ) of some point x ∈ X . More generally, every primeideal p of A is either dense, hence Z ( p ) = ∅ , or has zero-set Z ( p ) = { x } (if Z ( p ) contained twodistinct points x , x there would be two smooth functions ϕ , ϕ such that ϕ ( x ) = 1 = ϕ ( x ) ,but ϕ ϕ = 0 contradicting the fact that p is prime). For the closure of an ideal there is the veryuseful Whitney’s Spectral Theorem stating that a function g belongs to the closure J of an ideal J ifffor each x ∈ X there is a function h ∈ J (which may depend on x ) whose infinite jet j ∞ x ( h ) is equalto j ∞ x ( g ) , see e.g. [23, p. 91, Cor 1.6., cas q = 1 ]. It is not hard to see that J always contains theideal of all those smooth functions vanishing in some open neighbourhood of Z ( J ) , and Whitney’sSpectral Theorem ensures that the ideal I ∞ Z ( J ) of all those functions whose infinite jets vanish on allpoints of Z ( J ) is always contained in J . Moreover, ideals having finitely many analytic generatorsare always closed, see e.g. [23, p.119, Cor. 1.6.], but there are also closed ideals having finitely manynonanalytic generators, see e.g. [23, p.104, Rem. 4.7, Exemp. 4.8.]. For more details see AppendixA. This Section recalls well-known results which we recall from the excellent text-book [11] in a cate-gorically ‘tuned’ version. See also the rather useful review [21] for more aspects.
Recall that for any domain R it is always possible to construct a field, called the field of fractions of R ,by formally inverting all nonzero elements. More generally, recall the localization of a commutative K -algebra R : let S ⊂ R be a multiplicative subset (which is characterized by containing the unitand for any two of its elements its product). Then the following binary relation ∼ on R × S definedby ( r , s ) ∼ ( r , s ) if and only if ∃ s ∈ S : r s s = r s s (1.3)is an equivalence relation, and the set of all classes (written as (symbolic) fractions rs ) forms acommutative K -algebra R S –by means of the usual addition and multiplication rules of fractions–called the quotient ring , and a ring homomorphism (the numerator morphism ) η ( R,S ) = η : R → R S given by r r which in particular defines the K -algebra structure of R S . Let U ( R ) ⊂ R denote themultiplicative group of invertible elements of R . A morphism of unital K -algebras Φ : R → R ′ iscalled S -inverting (for a multiplicative subset S ⊂ R ) if for each s ∈ S the image Φ( s ) is invertiblein R ′ , hence Φ( S ) ⊂ U ( R ′ ) . The following properties of the constructions can be observed: Proposition 1.1.
Let R be a commutative K -algebra and S ⊂ R be a multiplicative subset. Thenthe following is true:a. η ( R,S ) ( S ) ⊂ U ( R S ) , that is, the homomorphism η ( R,S ) sends elements of S to invertible ele-ments of R S . Moreover, for any commutative unital K -algebra R equipped with a multiplicative ubset S ⊂ R , the pair ( R S , η ( R,S ) ) is universal in the sense that any S -inverting morphismof unital K -algebras uniquely factorizes, i.e. the following diagram commutes: R η / / α ❆❆❆❆❆❆❆❆ R Sf (cid:15) (cid:15) R ′ (1.4) where f is a morphism of unital K -algebras determined by α , see e.g. [13, p.55, Ch.III] fordefinitions of universal objects.b. Every element of R S is written as a fraction η ( r ) η ( s ) − , for some r ∈ R and s ∈ S .c. ker( η ( R,S ) ) = { r ∈ R | rs = 0 for some s ∈ S } . We shall give a more categorical description in the next Section.
Let R be an associative unital K -algebra which is not necessarily commutative. Again, we call S ⊂ R a multiplicative subset if for all s, s ′ ∈ S we have ss ′ ∈ S and R = 1 ∈ S . As above, let U ( R ) ⊂ R denote the multiplicative subset (which is even a group) of invertible elements of R .Let K Alg be the category of all associative unital K -algebras. Moreover, let K AlgMS be thecategory of all pairs ( R, S ) of associative unital K -algebras R with a muliplicative subset S ⊂ R where the morphisms ( R, S ) → ( R ′ , S ′ ) are morphisms of unital K -algebras R → R ′ mapping S into S ′ . Since any morphism of unital K -algebras maps the group of invertible elements in thegroup of invertible elements there is an obvious functor U : K Alg → K AlgMS given on objectsby U ( R ) = (cid:0) R, U ( R ) (cid:1) .For commutative K -algebras , the above localization description in Proposition 1.1, a. , gives rise toa functor L : K AlgMS → K Alg associating to each pair ( R, S ) the quotient ring R S , and it is nothard to see that it is a left adjoint of the functor U , see e.g. [13, p.79, Ch.IV] for definitions: the unitof the adjunction gives back the canonical numerator morphism η , and the counit is an isomorphismsince localization w.r.t. the group of all invertible elements is isomorphic to the original algebra.In the general noncommutative situation such a localization functor L : K AlgMS → K Alg does also always exist, see e.g. [11, Prop.(9.2), p.289] for a proof. We present in the followingcategorical form:
Proposition 1.2.
There is an adjunction of functors K AlgMS
L−−−−−−→←−−−−−−U K Alg where L is the left adjoint to the above functor U such that each component η ( R,S ) of the unit η : I K AlgMS . −→ U L of the adjunction satisfies the universal property a. of the previous Proposition1.1 in the general noncommutative case. We refer to L as a localization functor .For a given ( R, S ) in K AlgMS we denote by R S the K -algebra L ( R, S ) given by the functor L , andby η ( R,S ) : R → R S the component of the unit of the adjunction. Then η ( R,U ( R )) : R → R U ( R ) is anisomorphism, the inverse being the component ǫ R of the counit ǫ : LU . −→ I K Alg of the adjunction.Moreover, every element of the K -algebra R S is a finite sum of products of the form ( η = η ( R,S ) ) η ( r ) (cid:0) η ( s ) (cid:1) − · · · η ( r N ) (cid:0) η ( s N ) (cid:1) − (1.5)6 which may be called ‘multifractions’) with r , . . . , r N ∈ R and s , . . . , s N ∈ S (note that r or s N may be equal to the unit of R ). The idea of the proof of [11, Prop.(9.2), p.289] is as follows: there is a natural surjective morphismof unital K -algebras ˆ ǫ R from the free K -algebra generated by the K -module R , T K R , to R whichprovides us with a natural categorical presentation of R ‘by generators and relations’: this morphismis given by the R -component of the counit ˆ ǫ of the well-known adjunction K Mod T K −−−−−−−→←−−−−−−O K Alg where O is the forgetful functor and T K the free algebra functor. Let κ ( R ) ⊂ T K R denote the kernelof ˆ ǫ R . The next step is to add to the generating K -module R the free K -module KS with basis S ,and to consider the two-sided ideal κ ( R, S ) in the free algebra T K ( R ⊕ KS ) generated by κ ( R ) andby the subsets { ( s, ⊗ (0 , s ) − T | s ∈ S } and { (0 , s ) ⊗ ( s, − T | s ∈ S } of T K (cid:0) R ⊕ KS (cid:1) wherethe multiplication ⊗ and the unit T are taken in the free algebra T K (cid:0) R ⊕ KS (cid:1) . The localizedalgebra L ( R, S ) = R S is then defined by R S = T K (cid:0) R ⊕ KS (cid:1) /κ ( R, S ) , and the ‘numerator morphism’ η ( R,S ) : R → R S is simply the canonical injection of R into T K R ⊂ T K (cid:0) R ⊕ KS (cid:1) followed by theobvious projection. It follows that for every s ∈ S its image η ( R,S ) ( s ) has an inverse by construction.The verification that this leads to a well-defined functor L which is a left adjoint to the functor U is lengthy, but straight-forward. We have put a more detailed proof in Appendix B.The preceding construction shows that the functor L provides us with an abstract universalnumerator map η ( R,S ) which is S -inverting in the sense that every η ( R,S ) ( s ) , s ∈ S , is invertible in R S and a natural isomorphism ǫ R of an algebra with its localization w.r.t. its group of units.However, the construction by generators and relations renders the localized algebra R S quite implicitand not always computable.Moreover, even for multiplicative subsets S ⊂ R not containing it may happen that the localizedalgebra R S is trivial as example (9 . of [11, p.289] shows. This can never happen in the commutativecase since the equation = is equivalent to the fact that ∈ S . This shows the lack of controlover the kernel of the ‘numerator morphism’ η ( R,S ) .Thirdly, the presentation of elements of R S in terms of sums of ‘ multifractions ’ as equation (1.5)shows is quite clumsy, and on would prefer simple right or left fractions.In order to motivate the conditions on S in the following Definition we look at the multifractionswhich span the localized K -algebra R S , see eqn (1.5): it may be desirable to transform a multi-fraction in a simple right fraction, and a partial step may consist in transforming a left fraction (cid:0) η ( s ) (cid:1) − η ( r ) (with r ∈ R and s ∈ S ) directly into a right fraction η ( r ′ ) (cid:0) η ( s ′ ) (cid:1) − (for some r ′ ∈ R and s ′ ∈ S ) which implies that every multifraction is equal to a right fraction by applying thisstep a finite number of times. This above condition implies the equation η ( rs ′ ) = η ( sr ′ ) and thusmotivates the stronger condition that for any pair ( r, s ) ∈ R × S there is a pair ( r ′ , s ′ ) ∈ R × S suchthat rs ′ = sr ′ , and this the well-known right Ore condition : Definition 1.1.
Let R be an associative unital K -algebra, and S ⊂ R be a multiplicative subset.i. A K -algebra ˇ R S equipped with a morphism of unital K -algebras ˇ η ( R,S ) = ˇ η : R → ˇ R S is saidto be a right K -algebra of fractions of ( R, S ) if the following conditions are satisfied:a. ˇ η ( R,S ) is S -inverting,b. Every element of ˇ R S is of the form ˇ η ( r ) (cid:0) ˇ η ( s ) (cid:1) − for some r ∈ R and s ∈ S ;c. ker(ˇ η ) = { r ∈ R | rs = 0 , for some s ∈ S } =: I ( R,S ) =: I . i. S is called a right denominator set if it satisfies the following two properties:a. For all r ∈ R and s ∈ S we have rS ∩ sR = ∅ ( S right permutable or right Ore set ),i.e. there are r ′ ∈ R and s ′ ∈ S such that rs ′ = sr ′ .b. For all r ∈ R and for all s ′ ∈ S : if s ′ r = 0 then there is s ∈ S such that rs = 0 ( S rightreversible ). In case R is commutative every multiplicative subset is a right denominator set. Moreover the groupof all invertible elements U ( R ) of any unital K -algebra is obviously a right denominator set.The next Theorem shows that such a right algebra of fractions exists iff S is a right denominatorset, see also [11, Thm (10.6), p.300]: Theorem 1.1.
Let R be a unital K -algebra and S ⊂ R be a multiplicative subset. Then the followingis true:1. The K -algebra R has a right K -algebra of fractions ˇ R S with respect to the multiplicative subset S if and only if S is a right denominator set.2. If this is the case each such pair ( ˇ R S , ˇ η ) is universal in the sense of diagram (1.4) and each ˇ R S is isomorphic to the canonical localized algebra R S of Proposition 1.2.3. Each ˇ R S is isomorphic to the quotient set RS − := ( R × S ) / ∼ with respect to the followingbinary relation ∼ on R × S ( r , s ) ∼ ( r , s ) ⇔ ∃ b , b ∈ R such that s b = s b ∈ S and r b = r b ∈ R (1.6) which is an equivalence relation generalizing relation (1.3). RS − carries a canonical unital K -algebra structure, i.e. addition and multiplication on equivalence classes r s − and r s − (with r , r ∈ R and s , s ∈ S ) is given by r s − + r s − = ( r c + r c ) s − , and ( r s − )( r s − ) = ( r r ′ )( s s ′ ) − (1.7) where we have written s c = s c = s ∈ S (with c ∈ S and c ∈ R ) and r s ′ = s r ′ (with s ′ ∈ S and r ′ ∈ R ) using the right Ore property. The numerator morphism η I : R → RS − isgiven by η I ( r ) = r − for all r ∈ R . For a proof, see e.g. [16, p.244, Thm. 25.3]. We shortly describe the idea of the proof : whereas in part 1. the verification of the implication“ ( i. ) = ⇒ ( ii. ) ” in Definition 1.1 is straight-forward, the converse implication “ ( i. ) ⇐ = ( ii. ) ”of Definition 1.1 is much more involved: the traditional ‘steep and thorny way’ (originally set upby Øystein Ore, [15]) consists of a concrete construction of the K -algebra RS − upon using theabove relation (1.6) –which reflects the idea of creating ‘common denominators’– and defining andverifying the canonical K -algebra structure (1.7) on the quotient set R × S/ ∼ by hand which iselementary, but extremely tedious (even the fact that the above relation (1.6) is transitive requiressome work). We refer to Lam’s book [11, p.300-302] for some of the details.There is a different more elaborate way to prove part 1. and the rest of the Theorem (see [16, p.244,Thm. 25.3] and [11, Remark (10.13), p.302, and footnote 70]): it is instructive to look first at theequivalence relations created by an arbitrary S -inverting morphism of unital K -algebras α : R → R ′ ,the classes being defined by the fibres of the map p α : R × S → R ′ given by p α ( r, s ) = α ( r ) (cid:0) α ( s ) (cid:1) − ,which is already very close to relation (1.6): thanks to the fact that the right fractions α ( r ) (cid:0) α ( s ) (cid:1) − We are indebted to A. Eduque for having pointed out this reference to us. K -subalgebra of R ′ (here the Ore axiom is needed) it creates an algebra structure on thequotient set isomorphic to the aforementioned subalgebra of R ′ whence there is no need of tediousverifications of identities of algebraic structures. The central point then is to construct a unital K -algebra R ′ and an S -inverting morphism α : R → R ′ whose kernel is minimal, hence equal to I ( R,S ) which finally shows that the above algebra RS − exists and does everything it should do. Forthis construction, the following trick is used: after ‘regularizing’ R by passing to the factor algebra R = R/I ( R,S ) (where the image multiplicative set S does no longer contain right or left divisors ofzero) one looks at the endomorphism algebra of the injective hull E of the right R -module R .Every left multiplication with elements of R can nonuniquely be extended to E , and the extensionsof left multiplications with elements of S turn out to be invertible (here the Ore axiom is needed). R ′ will then be given by the subalgebra generated by all extensions of left multiplications and theinverses of left multiplications with elements of S modulo the two-sided ideal of all R -linear maps E → E vanishing on R : this will resolve the ambiguity of extension, and R injects in R ′ , theinjection being S -inverting.Moreover, in any noncommutative domain (no nontrivial zero divisors) which is right noetherian (i.e. where every ascending chain of right ideals stabilizes) the subset of nonzero elements is alwaysa right denominator set (see [11, p.304, Cor. (10.23)] or [2, p.14, Beisp. 2.3 b)]). In particular,this applies to every universal enveloping algebra over a finite-dimensional Lie algebra (over a fieldof characteristic zero) and for the Weyl-algebra generated by K n . On the other hand, for thefree algebra R = T K V generated by a vector space V of dimension ≥ (which is well-knownto be isomorphic to the universal enveloping algebra of the free Lie algebra generated by V ) themultiplicative subset of all nonzero elements is neither a right nor a left denominator set: for twolinearly independent elements v and w in V we clearly have vR ∩ wR = { } . Hence the abovestatement about universal enveloping algebras does no longer apply to infinite-dimensional Liealgebras like the free Lie algebra generated by V . Moreover inverse images of right denominatorsubsets are in general no right denominator subsets as the example of the natural homomorphism T K V → S K V of the free to the free commutative algebra generated by V shows: as S K V is acommutative domain, the subset S = S K V \{ } is a right denominator set whereas its inverse image T K V \ { } is not. On the other hand every homomorphic image of a right (or left) denominator setclearly is again a right (or left) denominator set. However, there may be subsets of right (or left)denominator sets which are no longer right (or left) denominator sets, as we shall see later. We want to recall some basic definitions and facts about the deformation quantization of smoothmanifolds and star products, see [1], [25] for more information.Given a K vector space V we denote by V [[ λ ]] the K [[ λ ]] -module of formal power series. Anelement of v ∈ V [[ λ ]] can be written uniquely as v = P ∞ i =0 v i λ i with v i ∈ V , and for a given v ∈ V [[ λ ]] and i ∈ N we shall always write v i ∈ V for the i th component of v as a formal power series. Wealso note that for two K -vector spaces V, W we have
Hom K [[ λ ]] ( V [[ λ ]] , W [[ λ ]]) ∼ = Hom K ( V, W )[[ λ ]] .In the following considerations of differential geometry we set K = R or K = C , and for any smoothdifferentiable manifold X we write C ∞ ( X ) = C ∞ ( X, K ) . Definition 1.2 (Star product) . A (formal) star product ⋆ on a manifold X is a K [[ λ ]] -bilinearoperation C ∞ ( X )[[ λ ]] × C ∞ ( X )[[ λ ]] → C ∞ ( X )[[ λ ]] –which can always be written as a formal series f ⋆ g = P ∞ k =0 λ k C k ( f, g ) for all f, g ∈ C ∞ ( X ) – satisfying the following properties for all f, g, ∈C ∞ ( X ) (see Section 1.1.2 for definitions and notations): • P kl =0 (cid:0) C l ◦ C k − l − C l ◦ C k − l (cid:1) = 0 (associativity), C ( f, g ) = f g , • ⋆ f = f ⋆ f ,with K -bilinear operators C k : C ∞ ( X ) ⊗ K C ∞ ( X ) → C ∞ ( X ) which we always assume to be bidiffer-ential operators. Note that every star-product ∗ can be analytically localized to an associative star-product ∗ U definedon C ∞ ( U )[[ λ ]] by the localization of all the bidifferential operators C k to C kU (see Section 1.1.2 formore details).The following well-known explicit star-product ⋆ s on R with coordinates ( x, p ) will be used inthe sequel: f ⋆ s g = ∞ X k =0 λ k k ! ∂ k f∂p k ∂ k g∂x k (1.8)for any two functions f, g ∈ C ∞ ( R ) . In the physics literature λ corresponds to ( − i ~ ) . Moreover, forfunctions polynomial in the ‘momenta’ p it is obvious that the above series converge, and for λ = 1 one obtains the usual formula for the symbol calculus of multiplication of differential operators onthe real line (where partial derivatives are always brought to the right and replaced by the newvariable p ).We mention the following facts although they are not necessary for the main subject of thispaper:We define the star commutator for a, b ∈ C ∞ ( X )[[ λ ]] by [ a, b ] ⋆ = a ⋆ b − b ⋆ a . As usual, the starcommutator satisfies the Leibniz-identity, i.e. [ a, b⋆c ] ⋆ = [ a, b ] ⋆ ⋆c + b⋆ [ a, c ] ⋆ , and the Jacobi-identityand thus defines the structure of a non-commutative Poisson algebra. Also the adjoint action is aderivation of C ∞ ( X )[[ λ ]] for all a ∈ C ∞ ( X )[[ λ ]] .From this it can easily be deduced that the first order term of a star product defines a Poissonbracket as follows { f, g } = 12 ( C ( f, g ) − C ( g, f )) = 12 λ [ f, g ] | λ =0 for f, g ∈ C ∞ ( X ) . (1.9)For C ∞ ( X ) it is well-known that every Poisson bracket comes from a unique Poisson structure π which is a smooth bivector field π , i.e. a smooth section in Λ T X satisfying the identity [ π, π ] S = 0 where [ , ] S denotes the Schouten bracket, see e.g. [25, p.84-87]: the relation is { f, g } = π ( df, dg ) .The very difficult converse problem whether the Poisson bracket associated to any given Poissonstructure π arises as the first order commutator of a star-product had been solved by M. Kontsevich,see [10].We also note that two star products ⋆ , ⋆ ′ are called equivalent if there exists a formal power seriesof differential operators T = id + P ∞ k =1 λ k T k , with T (1) = 1 such that T ( f ) ⋆ T ( g ) = T ( f ⋆ ′ g ) for all f, g ∈ C ∞ ( X )[[ λ ]] . The operator T in the above definition is always invertible and indeed,given a star product ⋆ , f ⋆ ′ g := T − ( T ( f ) ⋆ T ( g )) always gives a new equivalent star product. Twoequivalent star products clearly give rise to the same Poisson bracket. Let ( X, π ) be a Poisson mannifold, let ∗ = P ∞ k =0 λ k C k be a star-product on ( X, π ) , and let Ω ⊂ X be a fixed open set. We set K = K [[ λ ]] , and consider the K -algebra (cid:0) R = C ∞ ( X )[[ λ ]] , ∗ (cid:1) . Moreover,since the star-product ∗ only involves bidifferential operators, it restricts to a star-product ∗ Ω on10ormal power-series φ ∈ R Ω := C ∞ (Ω , K )[[ λ ]] such that (cid:0) R Ω , ∗ Ω (cid:1) is also a K -algebra. It follows thatthe restriction map η Ω = η : R → R Ω : f f | Ω is a morphism of unital K -algebras. We define thefollowing subset S ⊂ R : S := { g ∈ R | ∀ x ∈ Ω : g ( x ) = 0 } (2.1)Clearly, the constant function is in S , and for any g, h ∈ S we have ( g ∗ h ) ( x ) = g ( x ) h ( x ) = 0 (for all x ∈ Ω ) whence S is a multiplicative subset of the unital K -algebra R .We can now consider the noncommutative localization of R with respect to S and compare it withthe unital K -algebra R Ω : Theorem 2.1.
Using the previously fixed notations we get for any open set Ω ⊂ X :1. ( R Ω , ∗ Ω ) equipped with the restriction morphism η consitutes a right K -algebra of fractions for ( R, S ) .2. As an immediate consequence we have that S is a right denominator set.3. This implies in particular that the algebraic localization RS − of R with respect to S is isomorphic to the concrete localization R Ω as unital K -algebras. Proof. We have to check properties ( i.a. ) , ( i.b. ) , and ( i.c. ) of Definition 1.1: • “ η is S -inverting ” (property ( i.a. ) ): indeed, this is a classical reasoning from deformation quanti-zation which we shall repeat for the convenience of the reader. Let g ∈ S and γ = η ( g ) its restrictionto Ω . Take ψ ∈ R Ω and try to solve the equation γ ∗ Ω ψ = 1 . At order k = 0 we get the condition γ ψ = 1 , but since γ ( x ) = 0 for all x ∈ Ω the function x ψ ( x ) := γ ( x ) − is well-defined andsmooth in C ∞ (Ω , K ) . Suppose by induction that the functions ψ , . . . , ψ k ∈ C ∞ (Ω , K ) have alreadybeen found in order to satisfy equation γ ∗ Ω ψ = 1 up to order k . At order k + 1 ≥ the conditionreads (cid:0) γ ∗ Ω ψ (cid:1) k +1 = k +1 X l, p, q = 0 l + p + q = k + 1 C l ( γ p , ψ q ) = γ ψ k +1 + F k +1 ( ψ , . . . , ψ k , γ , . . . , γ k +1 ) where the term starting with F k +1 denotes the difference (cid:0) γ ∗ Ω ψ (cid:1) k +1 − γ ψ k +1 which obviouslydoes not contain ψ k +1 . Again, since γ is nowhere zero on Ω the function ψ k +1 can be computedfrom this equation by multiplying with x γ ( x ) − . Hence there is a solution ψ ∈ R Ω of equation γ ∗ Ω ψ = 1 . In a completely analogous way there is a solution ψ ′ ∈ R Ω of the equation ψ ′ ∗ Ω γ = 1 .By associativity of ∗ Ω we get ψ = ψ ′ as the unique inverse of γ in the unital K -algebra R Ω . • “ Every φ ∈ R Ω is equal to η ( f ) ∗ Ω η ( g ) ∗ Ω − for some f ∈ R and g ∈ S ” (property ( i.b. ) ): themain idea is to transfer the proof of Lemme 6.1 of Jean-Claude Tougerons’s book [23, p.113] tothe non-commutative situation. Let φ = P ∞ i =0 λ i φ i ∈ R Ω . We then fix the following data whichwe get thanks to the fact that X and therefore each open set Ω is a second countable locallycompact topological space: there is a sequence of compact sets ( K n ) n ∈ N of X , a sequence of opensets ( W n ) n ∈ N , and a sequence of smooth functions ( g n ) n ∈ N : X → R such that [ n ∈ N K n = Ω , and ∀ n ∈ N : K n ⊂ W n ⊂ W n ⊂ K ◦ n +1 and g n ( x ) = x ∈ W n , x K n +1 ,y ∈ [0 ,
1] else . .
11e denote by γ j the restriction η ( g j ) of g j to Ω for each nonnegative integer j . The idea is to definethe denominator function g as a (non formal!) converging sum g = P ∞ j =0 ǫ j g j . Choose a sequence ( ǫ j ) j ∈ N of strictly positive real numbers such that ∀ j ∈ N : ǫ j p K j +1 ,j ( g j ) < j and ∀ i ≤ j ∈ N : ǫ j i X l =0 p K j +1 ,j (cid:0) C l ( φ i − l , g j ) (cid:1) < j (see eqn (1.1) for the definition of the seminorms p K,m ) which is possible since for each nonnegativeinteger j there are only finitely many seminorms and functions involved. For all nonnegative integers i, j, N we define the functions g ( N ) ∈ C ∞ ( X, K ) , and ψ ij , ψ ( i,N ) ∈ C ∞ (Ω , K ) : g ( N ) := N X j =0 ǫ j g j , ψ ij := i X l =0 C l (cid:0) φ i − l , γ j (cid:1) , ψ ( i,N ) := N X j =0 ǫ j ψ ij = i X l =0 C l (cid:0) φ i − l , γ ( N ) (cid:1) , and since supp( g j ) ⊂ K j +1 ⊂ Ω , hence supp( g ( N ) ) ⊂ K N +1 ⊂ Ω , there are unique functions f ij ∈ C ∞ ( X, K ) such that f ij ( x ) := (cid:26) ψ ij ( x ) if x ∈ Ω , x Ω . , hence η ( f ij ) = ψ ij and supp( f ij ) ⊂ K j +1 . For each nonnegative integer N we set f ( i,N ) := P Nj =0 ǫ j f ij ∈ C ∞ ( X, K ) with supp( f ( i,N ) ) ⊂ K N +1 .Clearly, η ( f ( i,N ) ) = φ ( i,N ) .We shall now prove that both sequences ( g ( N ) ) N ∈ N , and for each nonnegative integer i , ( f ( i,N ) ) N ∈ N are Cauchy sequences in the complete metric space C ∞ ( X, K ) . First, it is obvious that for any twocompact subsets K, K ′ and nonnegative integers N, N ′ we always have for all f ∈ C ∞ ( R n , K )if K ⊂ K ′ and m ≤ m ′ then p K,m ( f ) ≤ p K ′ ,m ′ ( f ) . (2.2)Fix a nonnegative integer i . Let ǫ ∈ R , ǫ > , K ⊂ X a compact subset, and m ∈ N . Then there isa nonnegative integer N such that N < ǫ, m ≤ N , and i ≤ N Then for all nonnegative integers
N, p with N ≥ N we get (since for all j ∈ N such that N + 1 ≤ j we have m ≤ N ≤ N ≤ j and i ≤ N , and supp( f i,j ) ⊂ K ◦ j +1 ⊂ K j +1 ) p K,m (cid:0) f ( i,N + p ) − f ( i,N ) (cid:1) = p K,m N + p X j = N +1 ǫ j f i,j ≤ N + p X j = N +1 ǫ j p K,m (cid:0) f i,j (cid:1) = N + p X j = N +1 ǫ j p K ∩ K j +1 ,m (cid:0) ψ ij (cid:1) ≤ N + p X j = N +1 ǫ j p K j +1 ,j i X l =0 C l ( φ i − l , g j ) ! ≤ N + p X j = N +1 ǫ j i X l =0 p K j +1 ,j ( C l ( φ i − l , g j )) < N + p X j = N +1 j = 12 N (cid:18) − p (cid:19) < N ≤ N < ǫ. It follows that for each i ∈ N the sequence ( f ( i,N ) ) N ∈ N is a Cauchy sequence in the locally convexvector space C ∞ ( X, K ) hence converges to a smooth function f i = P ∞ j =0 ǫ j f i,j . Replacing in theabove reasoning the function φ by the constant function on Ω it follows that the sequence12 g ( N ) ) N ∈ N converges to a smooth function g : X → R . Now let x ∈ Ω . Then there is a nonnegativeinteger j such that x ∈ K j . It follows from the nonnegativity and the definition of all the g j andfrom the strict positivity of ǫ j that g ( x ) = ∞ X j =0 ǫ j g j ( x ) ≥ ǫ j g j ( x ) = ǫ j > showing that g takes strictly positive values on Ω whence g ∈ S .Now let x Ω . Then for any v ∈ T x X with h ( v, v ) ≤ we have that ∀ m ∈ N : ( D m g ( N ) )( v ) = N X j =0 ǫ j ( D m g j )( v ) = 0 because each g j has compact support in K j +1 ⊂ Ω . Since g ( N ) → g for N → ∞ it follows bythe continuity of differential operators and evaluation functionals that D m g ( N ) ( v ) → D m g ( v ) , andhence ∀ x ∈ X \ Ω , ∀ m ∈ N , ∀ v ∈ T x X, h ( v, v ) ≤ D m g )( v ) = 0 , (2.3)and in a completely analogous manner ∀ x ∈ X \ Ω , ∀ m ∈ N , ∀ v ∈ T x X, h ( v, v ) ≤ D m f i )( v ) = 0 . Hence the infinite jets of all the functions g and f i , i ∈ N , vanish outside the open subset Ω . J.-C. Tougeron calls the function g fonction aplatisseur for the family ( φ i ) i ∈ N in case C l = 0 for l ≥ .Now we get (cid:0) φ ∗ U η ( g ( N ) ) (cid:1) i = i X l =0 C l (cid:0) φ i − l , η ( g ( N ) ) (cid:1) = ψ ( i,N ) = η ( f ( i,N ) ) . Since the restriction map η : C ∞ ( X, K ) → C ∞ (Ω , K ) is continuous (where the Fréchet topology on C ∞ (Ω , K ) is induced by those seminorms p K,m where K ⊂ Ω ) as are the bidifferential operators C l we can pass to the limit N → ∞ in the above equation and get φ ∗ Ω η ( g ) = ∞ X i =0 λ i (cid:0) φ ∗ Ω η ( g ) (cid:1) i = ∞ X i =0 λ i η ( f i ) =: η ( f ) . Since g ∈ S it follows that η ( g ) is invertible in R Ω by property ( i.a ) of Definition 1.1, and thepreceding equation implies φ = η ( f ) ∗ Ω η ( g ) ∗ Ω − thus proving property ( i.b ) of Definition 1.1. • The kernel of η is equal to the space of functions f ∈ R such that there is g ∈ S with f ∗ g = 0 (property ( i.c ) of Definition 1.1. Clearly if there is f ∈ R and g ∈ S such that f ∗ g = 0 then η ( f ) ∗ Ω η ( g ) = 0 , and since η ( g ) is invertible in R Ω we have η ( f ) = 0 .Conversely, if f ∈ R such that η ( f ) = 0 , then for all integers i ∈ N and for all x ∈ Ω we have f i ( x ) = 0 . Hence the infinite jet of each f i vanishes at each point x ∈ Ω since Ω is open. Take the fonction aplatisseur g ∈ S constructed in the preceding part of the proof for φ = 1 , φ i = 0 for all i ≥ . Then we get ∀ x ∈ X : ( f ∗ g ) i ( x ) = i X l =0 C l ( f i − l , g )( x ) = (cid:26) x ∈ Ω since every jet of each f i vanishes in Ω , x Ω since every jet of g vanishes outside of Ω , where we have used eqn (2.3) for the second alternative of the above statement. This proves part of the Theorem.Statements and are immediate consequences of and Theorem 1.1.13 emark: For zero Poisson structure and trivial deformation C l = 0 for all l ≥ the above resultspecializes to the classical result that algebraic and analytic localisation with respect to an opensubset Ω ⊂ X are isomorphic for the commutative K -algebra C ∞ ( X, K ) .Moreover, since for any closed set F ⊂ X Tougeron’s above construction gives us a smooth function g : X → R which is nowhere zero on the open set Ω = X \ F and zero outside Ω , hence on F , onegets the well-known result that the Zariski topology on X induced by the commutative K -algebra C ∞ ( X, K ) coincides with the usual manifold topology because each set Z ( I ) is closed by continuityof all the functions in the ideal I , and conversely every closed set F is the zero set Z ( gA ) of theideal gA (where A = C ∞ ( X, K ) ). Let ( X, π ) again be a Poisson manifold, and let ∗ = P ∞ l =0 λ l C l be a bidifferential star-product. Let K = K [[ λ ]] , and we denote the unital K -algebra (cid:0) C ∞ ( X, K )[[ λ ]] , ∗ (cid:1) by R . For any open set U ⊂ X let R U denote the unital K -algebra (cid:0) C ∞ ( U, K )[[ λ ]] , ∗ U (cid:1) , where ∗ U denotes the obvious action of thebidifferential operators in ∗ to the local functions in C ∞ ( U, K ) . We write R X = R . For any twoopen sets with U ⊃ V , denote by η UV : R U → R V be the restriction morphism where we write η U for η XU . Clearly, for U ⊃ V ⊃ W one has the categorical identities η VW ◦ η UV = η UW and η UU = id U .Denoting by X the topology of X it is readily checked that the family (cid:0) R U (cid:1) U ∈ X with the restrictionmorphisms η UV defines a sheaf of K -algebras over X , see e.g. the book [8] for definitions.Let x a fixed point in X , and let X x ⊂ X the set of all open sets containing x . We recall thedefinition of the stalk at x , R x of the sheaf (cid:0) R U (cid:1) U ∈ X whose elements are called germs at x : it isdefined as the inductive limit (or colimit, see [13]) lim U ∈ X x R U . In order to perform computationswe recall the more down-to-earth definition: let ˜ R x be the disjoint union of all the R U , i.e. the setof all pairs ( U, f ) where U is an open set containing x and f ∈ C ∞ ( U, K )[[ λ ]] . Define an addition + and a multiplication ∗ on these pairs by ( U, f )+(
V, g ) := (cid:0) U ∩ V, η UU ∩ V ( f )+ η VU ∩ V ( g ) (cid:1) and ( U, f ) ∗ ( V, g ) := (cid:0) U ∩ V, η UU ∩ V ( f ) ∗ U ∩ V η VU ∩ V ( g ) (cid:1) , and it is easily checked that the addition is associative and commutative, that the multiplication isassociative, and that there is the distributive law. Furthermore, the sum of ( U, f ) and ( V, equals (cid:0) U ∩ V, η UU ∩ V ( f ) (cid:1) which is equal to ( U, f ) ∗ ( V,
1) = ( V, ∗ ( U, f ) . Next the binary relation ∼ x defined by ( U, f ) ∼ x ( V, g ) iff ∃ W ∈ X x with W ⊂ U ∩ V : η UW ( f ) = η VW ( g ) turns out to be an equivalence relation. Denoting by R x the quotient set ˜ R x / ∼ x and by η Ux : R U → R x the restriction of the canonical projection ˜ R x → R x to R U ⊂ ˜ R x (where η Xx will be shortened by η x : R → R x ) it is easy to see that the above addition and multiplicationpasses to the quotient, that all the zero elements ( U, are equivalent as are all the unit elements ( U, , and that this defines the structure of a unital associative K -algebra denoted by (cid:0) R x , ∗ x (cid:1) onthe quotient set such that all maps η Ux : (cid:0) R U , ∗ U (cid:1) → (cid:0) R x , ∗ x (cid:1) are morphisms of unital K -algebras.Note the following equations for all open sets U ⊃ V : η Vx ◦ η UV = η Ux . (3.1)Define the following subset S = S ( x ) and I = I x of R : S = S ( x ) = { g ∈ R | g ( x ) = 0 } and I = I x = { g ∈ R | g ( x ) = 0 } . (3.2)14t is easy to see that S = R \ I , that S is a multiplicative subset of R , and that I x is a maximalideal of R (the quotient R/I is isomorphic to the quotient K/ ( λK ) ∼ = K which is a field).We now have the following analog of Theorem 2.1: Theorem 3.1.
Using the previously fixed notations we get for any point x ∈ X :1. ( R x , ∗ x ) together with the morphism η x : R → R x consitutes a right K -algebra of fractionsfor ( R, S ( x )) .2. As an immediate consequence we have that S ( x ) is a right denominator set.3. This implies in particular that the algebraic localization RS − of R with respect to S = S ( x ) is isomorphic to the concrete stalk R x as unital K -algebras. Proof. Once again, we have to check properties ( i.a. ) , ( i.b. ) , and ( i.c. ) of Definition 1.1: • “ η x is S -inverting ” (property ( i.a. ) ): indeed, let g ∈ S ( x ) . Since g ( x ) = 0 there is an openneighbourhood U of x such that g ( y ) = 0 for all y ∈ U . Hence the restriction η U ( g ) is invertiblein ( R U , ∗ U ) by Theorem 2.1. Using eqn (3.1) we see that η x ( g ) = η Ux (cid:0) η U ( g ) (cid:1) , and the r.h.s. isinvertible in R x as the image of an invertible element η U ( g ) in R U with respect to the morphismof unital K -algebras η Ux . • “ Every φ ∈ R x is equal to η x ( f ) ∗ x η x ( g ) ∗ x − for some f ∈ R and g ∈ S ( x ) ” (property ( i.b. ) ):indeed, let φ ∈ R x . By definition of R x as a quotient set there is an open neighbourhood U of x and an element ψ ∈ R U with η Ux ( U, ψ ) = φ . According to the preceding Theorem 2.1 thereare elements f, g ∈ R with g ( y ) = 0 for all y ∈ U such that η U ( f ) = ψ ∗ U η U ( g ) . In particular, g ( x ) = 0 , hence g ∈ S ( x ) . Applying η Ux to the preceding equation we get (upon using eqn (3.1)) η x ( f ) = η Ux (cid:0) η U ( f ) (cid:1) = (cid:16) η Ux ( ψ ) (cid:17) ∗ x (cid:16) η Ux (cid:0) η U ( g ) (cid:1)(cid:17) = φ ∗ x (cid:0) η x ( g ) (cid:1) proving the result since g ∈ S ( x ) and η x ( g ) is invertible in the unital K -algebra ( R x , ∗ x ) . • The kernel of η x is equal to the space of functions f ∈ R such that there is g ∈ S ( x ) with f ∗ g = 0 (property ( i.c ) ). Indeed, given f ∈ R with η x ( f ) = 0 then there is an open neighbourhood W of x such that η W ( f ) = η W (0) = 0 . By the preceding Theorem 2.1 there is an element g ∈ S W ⊂ S ( x ) (which can be chosen to be a fonction aplatisseur ) such that f ∗ g = 0 . This proves of theTheorem. and are immediate consequences of part and Theorem 1.1. Warning:
The stalk R x is taken in the sense of sheaves of K [[ λ ]] -algebras. Another interpretationwould be two consider the sheaf (cid:0) C ∞ ( U, K ) (cid:1) U ∈ X x of commutative K -algebras and the classical stalk C ∞ ( X, K ) x : in a completely analogous fashion it can be shown that it is isomorphic to the algebraiclocalization with respect to the multiplicative set of functions which do not vanish at x . Howeverthe K [[ λ ]] -module C ∞ ( X, K ) x [[ λ ]] is NOT in general isomorphic to the above R x : if f = P ∞ l =0 λ l f l is a series of smooth functions such that f l vanishes on an closed ball of radius ǫ l > around x where ǫ l → (for l → ∞ ) and is non-zero outside, then the germ of each f l vanishes, but there isno common open neighbourhood of x such that f restricted to that neighbourhood vanishes. In this Section we shall describe a more algebraic frame work to generalize the two precedingSections. 15et A be a commutative associative unital K -algebra. We recall briefly the well-known algebraicdefinition of an (algebraic) multidifferential operator where we follow the book [25, p.566-578], seealso the arcicle [12]:We shall write unadorned tensor products ⊗ short for ⊗ K . A multidifferential operator D of rank p on A is a K -linear map D : A ⊗ p → A satisfying certain properties:We denote by L a : A → A , L a ( b ) = ab for a, b ∈ A the left multiplication, and similarly for eachinteger ≤ i ≤ p the map L ia : A ⊗ p → A ⊗ p , L ia ( a ⊗ · · · ⊗ a i ⊗ a p ) = a ⊗ · · · ⊗ ( aa i ) ⊗ · · · ⊗ a p .Further we denote by k = ( k , . . . , k p ) ∈ Z p a multi-index and by e i ∈ Z p the multi-index which is in the i -th position and zero otherwise. We shall use the partial ordering ≺ on Z p defined by k ≺ l iff for all ≤ i ≤ p we have k i ≤ l i . Definition 4.1.
We define the left A -module of p -multidifferential operators DiffOp k ( A, . . . , A ; A ) on A of order k ∈ N p inductively by DiffOp k ( A, . . . , A ; A ) = { } if there exists k i < and DiffOp k ( A, . . . , A ; A ) = (cid:8) D ∈ Hom K ( A ⊗ n , A ) | ∀ a ∈ A ∀ ≤ i ≤ p : L a ◦ D − D ◦ L ia ∈ DiffOp k − e i ( A, . . . , A ; A ) (cid:9) (4.1) for k ∈ N p . Furthermore, we set DiffOp ( p ) ( A ; A ) = S k ∈ Z p DiffOp k ( A, . . . , A ; A ) . Since clearly k ≺ l implies that DiffOp k ( A, . . . , A ; A ) ⊂ DiffOp l ( A, . . . , A ; A ) there is the well-known result that each A -module of p -multidifferential operators is exhaustively filtered by theabelian group Z p .Furthermore, for A = C ∞ ( X ) for a manifold X this algebraic definition is well-known to coincidewith the analytic definition, see e.g. [25, p. 575, Satz A.5.2.] which means that in local charts a(algebraically defined) differential operator looks as in eqn (1.2).Returning to general A there is a well-known procedure to localize multidifferential operators: Proposition 4.1.
Let S ⊂ A be a multiplicative subset, let A S be the ordinary commutativelocalization of A w.r.t. S , and let η ( A,S ) = η : A → A S be the numerator morphism. Let D ∈ DiffOp ( p ) ( A ; A ) a multidifferential operator of rank p .Then there exists a unique multidifferential operator D S ∈ DiffOp ( p ) ( A S ; A S ) of rank p such that η ◦ D = D S ◦ η ⊗ p .Furthermore, given another multidifferential operator D ′ ∈ DiffOp( A ; A ) we have ( D ◦ i D ′ ) S = D S ◦ i D ′ S for each integer ≤ i ≤ p .Proof. This follows from the similar statement for differential operators, see e.g. [24, Prop.3.3]. Thesecond part follows from the uniqueness of the localization.Observe now that the Definition 1.2 of star-products can be generalized to any commutativeassociative unital K -algebra A whence the significance ‘bidifferential’ for the K -bilinear maps C k : A × A → A is now given by the algebraic Definition 4.1. Proposition 4.2.
Let A be a commutative unital K -algebra and a differential star product ⋆ = P ∞ i =0 λ i C i on R := A [[ λ ]] where the C i are bidifferential operators on A . For any multiplicativesubset S ⊂ A there exists a unique star product ⋆ S on A S [[ λ ]] such that the numerator map η canonically extended as a K [[ λ ]] -linear map (also denoted η ) A [[ λ ]] → A S [[ λ ]] is a morphism ofunital K [[ λ ]] -algebras.Proof. This follows from the previous Proposition by considering the localization of the bidifferentialoperators C i . It remains associative since the localization is compatible with composition.16ith the above structures A, S , ⋆ consider the subset S = S + λR ⊂ R = A [[ λ ]] . (4.2)then we have the Proposition 4.3.
The subset S = S + λR is a multiplicative subset of the algebra ( R, ⋆ ) , and itsimage under η consists of invertible elements of the K [[ λ ]] -algebra (cid:0) A S [[ λ ]] , ⋆ S (cid:1) .It follows that there is a canonical morphism Φ : (cid:16)(cid:0) A [[ λ ]] (cid:1) S ⋆ S (cid:17) → (cid:0) A S [[ λ ]] , ⋆ S (cid:1) . (4.3)Indeed, since the deformation terms of ⋆ come in higher orders of λ it is clear that S is multiplicative.Since η ( S ) is invertible in A S this also holds for the ‘deformation’ S . The existence of Φ is clearfrom the universal property of the localized algebra, see Proposition 1.2.Here we come to the general problem Localization commutes with deformation ? as to under which circumstances the above morphism Φ (4.3) is an isomorphism.For localization with respect to open sets (see Section 2 ( S = S Ω ) the morphism Φ is an isomorphismwhereas it is not injective for the germs ( S = A \ I x ) in Section 3) as the warning at the end ofthe Section indicates. The following example provides a non-Ore subset which is a subset of an Ore subset: consider A = C ∞ ( R , R ) with the standard star-product ⋆ given by formula (1.8). Let R = A [[ λ ]] , and let Ω ⊂ R be the open set of all ( x, p ) ∈ R where p = 0 . We have the Proposition 5.1.
The subset S = { , p, p , p , . . . } ⊂ R is a multiplicative subset of ( R, ⋆ ) which iscontained in the Ore subset S Ω (see Section 2) but which is neither right nor left Ore.Proof. Since p m ⋆p n = p m + n it si clear that S is a multiplicative subset of R which clearly is a subsetof S Ω . Set r = ( x, p ) e x and s = ( x, p ) p . Let r ′ ∈ R be an arbitrary series r ′ = P ∞ k =0 λ k r ′ k and s ′ = ( x, p ) p n for some fixed nonnegative integer n . Suppose the right Ore condition holds,i.e. r ⋆ s ′ = s ⋆ r ′ . We get e x p n = pr ′ ( x, p ) + λ ∂r ′ ∂x ( x, p ) = pr ′ ( x, p ) + ∞ X k =1 λ k (cid:18) pr ′ k ( x, p ) + ∂r ′ k − ∂x ( x, p ) (cid:19) . This shows that n = 0 is impossible since in order λ the left hand side does not vanish for p = 0 whereas the right hand side does. In case n ≥ we must have r ′ ( x, p ) = e x p n − because the equation e x p n = pr ′ ( x, p ) implies e x p n − = r ′ ( x, p ) , and using Hadamard’s Lemma n − times we get that there is a smooth function f with r ′ ( x, p ) = f ( x, p ) p n − implying e x = f ( x, p ) .Next, in order λ we must have r ′ ( x, p ) = − e x p n −
17o satisfy the equation pr ′ ( x, p ) = − ( ∂r ′ /∂x )( x, p ) by an analogous reasoning. By induction we get r ′ n − ( x, p ) = ( − n − e x , and then at order λ n we get a contradiction: we must have pr ′ n ( x, p ) = ( − n e x and the left hand side vanishes for p = 0 whereas the right hand side does not. Hence S is not rightOre.In order to check the left Ore condition choose again r ( x, p ) = e x and s ( x, p ) = p and if there are r ′ ∈ R and s ′ ( p ) = p n such that r ′ ⋆ s = s ′ ⋆ r we get r ′ ( x, p ) p = p n ∗ e x = ( p + λ ) n e x . It follows that the left hand side always vanishes for p = 0 and the right hand side never, contra-diction, whence S is not left Ore. A Properties of Prime Ideals in C ∞ ( X, K ) It is instructive to first recall prime ideals of the commutative K -algebra A = C ∞ ( X, K ) where X is a differentiable Hausdorff and second countable manifold and K is equal to R or C . Let x ∈ X be a given point. We denote by I x ⊂ A the ideal of functions in A vanishing at x , by I ∞ x ⊂ A theideal of functions g in A whose infinite jet j ∞ x ( g ) vanishes, and J x ⊂ A the ideal of functions in A whose germ at x (see the preceding Section 3) vanishes. Clearly we have the inclusion J x ⊂ I ∞ x ⊂ I x . (A.1)As usual we write B for the closure of a subset B ⊂ A with respect to the Fréchet topology. Wecollect some facts on prime ideals of A which can be deduced from Krull’s Lemma (see e.g. [6,p.391, Prop. 7.2 and Prop. 7.3]) and Whitney’s Spectral Theorem (see e.g. Tougeron’s book [23,p.91, Cor. 1.6]: Proposition A.1.
With the abovementioned notations let p ⊂ A be a prime ideal.1. Z ( p ) = ∅ if and only if p is dense, p = A .2. For each function f ∈ A whose support is non compact there is a dense prime ideal p ′ ⊂ A not containing f .3. Let x ∈ X . Then x ∈ Z ( p ) if and only if Z ( p ) = { x } if and only if J x ⊂ p = A .4. For each x ∈ X : J x = I ∞ x , and the latter is a closed proper prime ideal of A which is equalto the intersection T n ∈ N I x n . Moreover I ∞ x I ∞ x = I ∞ x .5. Let p a proper prime ideal of A . Then p is closed if and only if there is a unique x ∈ X suchthat Z ( p ) = { x } and I ∞ x ⊂ p ⊂ I x .Moreover, for each given x ∈ X : the set of all closed proper prime ideals p ⊂ A with Z ( p ) = { x } is in bijection with the set of all the proper prime ideals of the formal powerseries algebra K [[ x , . . . , x n ]] via the map p p /I ∞ x . . Let x ∈ X . For each function g ∈ I ∞ x with g J x there is a (necessarily non closed) primeideal p ′′ of A such that J x ⊂ p ′′ ⊂ I x , but g p ′′ , hence I ∞ x p ′′ .Proof. is true for any ideal of A : if the set of common zeros is empty, then for each point x ∈ X there is an element g x of the ideal not vanishing on x , and by continuity nowhere vanishing on someopen neighbourhood of x . A simple partition of unity argument shows that the ideal D ( X, K ) of allsmooth functions having compact support is contained in the ideal, and D ( X, K ) is well-known tobe dense in A . If the set of common zeros is not empty then the ideal is contained in the vanishingideal of that set which is a closed proper ideal of A . Apply Krull’s Lemma to the ideal D ( X, K ) of A and to the multiplicative set generated by g . The first implication “ ⇐ ” being trivial, suppose that Z ( p ) contains another point y . But thenthere would be two smooth functions f , f with disjoint compact supports such that f ( x ) =1 = f ( y ) and hence f ( y ) = 0 = f ( x ) . By assumption f p and f p , but f f = 0 ∈ p ,contradiction to the fact that p was a prime ideal! This proves the first “ ⇒ ”. Moreover suppose Z ( p ) = { x } and suppose that there is g ∈ J x p . But then there would be an open neighbourhood U of x on which g vanishes identically, hence its product gh with a smooth function h having supportinside U with h ( x ) = 1 would vanish, hence be in p although g, h p . Contradiction to the factthat p was a prime ideal! This proves the inclusion J x ⊂ p , and since J x ⊂ I x by assumption, p is a proper ideal. Conversely, if J x ⊂ p , suppose that there is γ ∈ p such that γ ( x ) = 0 .Then the nonnegative function | γ | : x
7→ | γ ( x ) | would be in p and strictly positive at x . Therewould be two open neighbourhoods U ⊂ U ⊂ V of x such that | γ | was strictly positive on V .There would also be a nonnegative function ρ in A which vanishes on U and is strictly positive on X \ U . Clearly ρ ∈ J x ⊂ p . Then the sum ζ = γ + ρ whold have only strictly positive values byconstruction and would be contained in p . Therefore ζ would be invertible, and hence ∈ p hence p = A contradicting the fact that p was proper by hypothesis. This proves the second “ ⇒ ”. Let first be g ∈ I ∞ x and y ∈ X . If y = x , then there is an open neighbourhoods U, V of x with U ⊂ U ⊂ V and y V . Take a smooth nonnegative function χ on X which takes the value on U and zero outside of V . Then the function ˜ g = g (1 − χ ) belongs to J x , and j ∞ y ( g ) = j ∞ y (˜ g ) .For y = x we have j ∞ y ( g ) = j ∞ x ( g ) = 0 = j ∞ x (0) . Hence the infinite jet of g at any point y of X coincides with the infinite jet at y of a function in J x , hence by Whitney’s spectral theorem thefunction g belongs to the closure of J x . Otherwise I ∞ x is clearly a closed ideal of A since all the jetsof finite order, j rx are continuous maps, whence the first equality is shown. By E.Borel’s classicalLemma (see e.g. [17]) it is known that the quotient A/I ∞ x is isomorphic to the formal power seriesalgebra in n variables which is well-known to be domain whence I ∞ x is a closed proper prime ideal.Finally, since obviously for all nonnegative integers l ≥ k we have j kx (cid:0) I l +1 x (cid:1) = { } it is clear thatthe intersection is contained in I ∞ x . The converse inclusion follows from [23, p.93, Lemme 2.4] forthe particular case where the closed set equals { x } : it shows the equation I ∞ x I ∞ x = I ∞ x which byiteration proves the converse inclusion since I ∞ x ⊂ I x . Suppose first that p is closed and proper: then its zero set Z ( p ) is non empty by , and accordingto Z ( p ) is equal to a singleton { x } , and J x ⊂ p ⊂ I x . Since p is closed it contains the closureof J x which is I ∞ x according to whence the inclusion I ∞ x ⊂ p ⊂ I x . Conversely suppose thatthe latter inclusions hold, and let g ∈ p . By Whitney’s Spectral Theorem there is h ∈ p such that j ∞ x ( g ) = j ∞ x ( h ) . It follows that g = h + ζ with ζ ∈ Ker( j ∞ x ) = I ∞ x ⊂ p , hence g ∈ p , and p is closed.The last statement follows from the fact that the morphism φ : A → A/I ∞ x ∼ = K [[ x , . . . , x n ]] issurjective and that A/ p ∼ = K [[ x , . . . , x n ]] / ( φ ( p )) by Noether’s isomorphism theorem. Hence if p isa proper prime ideal, then φ ( p ) is a proper prime ideal, and if q ⊂ K [[ x , . . . , x n ]] is a proper primeideal then by elementary commutative algebra its inverse image p = φ − ( q ) is a proper prime idealof A containing I ∞ x whence p is a closed proper prime ideal by . Apply Krull’s Lemma to the ideal J x and the multiplicative subset generated by g : note that19or any function whose germ at x is non-zero the germ of all its powers is different from . B Proof of Proposition 1.2
Proof.
Recall first the following functorial presentation of a unital K -algebra by ‘generators andrelations’: for a given unital K -algebra R (with unit R ) there is a natural surjective algebrahomomorphism ˆ ǫ R : T K ( R ) → R where T K ( R ) is the free associative unital K -algebra (or tensoralgebra) generated by the K -module R . Note that the natural morphism ˆ ǫ R is just the R -componentof the counit ˆ ǫ of the adjunction given by the functor T K from the category K mod of all K -modulesto the category K Alg which is a left adjoint of the obvious forgetful functor K Alg to K mod . Themorphism ˆ ǫ R is determined by defining it to be the identity on the generating module R . The kernel κ ( R ) ⊂ T K ( R ) of ˆ ǫ R is a canonical -sided ideal in the free algebra T K ( R ) (containing for instance r ⊗ r ′ − rr ′ , r, r ′ ∈ R , and T − R ) for which T K Φ( κ ( R )) ⊂ κ ( R ′ ) for any morphism of unital K -algebras Φ : R → R ′ . Hence R is canonically presented by the ‘ K -module of generators R ’ andby the ‘ideal of relations κ ( R ) ’.Next, for any object ( R, S ) in K AlgMS let KS denote the free (!) K -module having basis S , andconsider the free K -algebra T K (cid:0) R ⊕ KS (cid:1) generated by the K -module R ⊕ KS . The natural K-linear injection i R : R → R ⊕ KS given by i R ( r ) = ( r, defines a natural injection T K i R : T K R → T K (cid:0) R ⊕ KS (cid:1) . Let κ ( R, S ) be the two-sided ideal in T K (cid:0) R ⊕ KS (cid:1) generated by T K i R (cid:0) κ ( R ) (cid:1) and bythe subsets { ( s, ⊗ (0 , s ) − T | s ∈ S } and { (0 , s ) ⊗ ( s, − T | s ∈ S } of T K (cid:0) R ⊕ KS (cid:1) where themultiplication ⊗ and the unit T are taken in the free algebra T K (cid:0) R ⊕ KS (cid:1) . Define the localized K -algebra with respect to S , L ( R, S ) =: R S , by the factor algebra R S := T K (cid:0) R ⊕ KS (cid:1) /κ ( R, S ) .Since a morphism Φ : (
R, S ) → ( R ′ , S ′ ) in K AlgMS clearly maps R to R ′ and KS to KS ′ , theinduced algebra morphism T K (cid:0) R ⊕ KS (cid:1) → T K (cid:0) R ′ ⊕ KS ′ (cid:1) maps κ ( R, S ) to κ ( R ′ , S ′ ) , and induceshence a morphism L Φ : R S → R ′ S ′ of unital K -algebras. It is readily checked that L is a covariantfunctor K AlgMS → K Alg . Denoting by π ( R,S ) : T K (cid:0) R ⊕ KS (cid:1) → R S the canonical projectionwe observe that –by construction– for every s ∈ S the image π ( R,S ) ( s, ∈ R S has the inverse π ( R,S ) (0 , s ) and is thus an invertible element of R S .Furthermore, for any ( R, S ) in K AlgMS there is a canonical map η ( R,S ) : R → R S determined bythe diagram η ( R,S ) ◦ ˆ ǫ R = π ( R,S ) ◦ T K i R , hence ∀ r ∈ R : η ( R,S ) ( r ) = π ( R,S ) ( r, , (B.1)which is a well-defined morphism of K -algebras since the right hand side of this equation vanisheson the kernel κ ( R ) of ˆ ǫ R thanks to T K i R ( κ ( R )) ⊂ κ ( R, S ) = ker( π ( R,S ) ) . It follows that for any s ∈ S we have η ( R,S ) ( s ) = π ( R,S ) ( s, which is invertible in R S , hence η ( R,S ) defines a morphism ( R, S ) → (cid:0) R S , U ( R S ) (cid:1) = U L ( R, S ) in the category K AlgMS .Moreover, for any R in K Alg we consider the canonical K -linear map j R : R ⊕ KU ( R ) → R givenby j R (cid:0) r, P Nn =1 λ n s n (cid:1) = r + P Nn =1 λ n s − n for any r ∈ R , λ , . . . , λ n ∈ K , and s , . . . , s n ∈ U ( R ) , andits induced morphism of K -algebras T K j R : T K ( R ⊕ KU ( R )) → T K R . Note that j R ◦ i R = id R .There is a canonical K -linear map ǫ R : R U ( R ) → R defined by the diagram ǫ R ◦ π ( R,U ( R )) = ˆ ǫ R ◦ T K j R . This is a well-defined morphism of K -algebras since the right hand side ˆ ǫ R ◦ T K j R vanishes on thegenerators of the ideal κ ( R, U ( R )) : this is clear for T K i R ( κ ( R )) , and ( s, ⊗ (0 , s ) − T is firstmapped to ( s ⊗ s − ) − T by T K j R , and then clearly annihilated by ˆ ǫ R .It is readily seen that the collection η of all the maps η ( R,S ) defines a natural transformation20 K AlgMS . −→ U L , and the collection ǫ of all the maps ǫ R defines a natural transformation LU . −→ I K Alg . Moreover, the identity j R ◦ i R = id R immediately shows the identity ǫ R ◦ η ( R,U ( R )) = id R by a combination of the above two diagrams. This implies the categorical equation ( U ǫ R ) ◦ ( η U R ) =id U R which is the first equation of eqn (8) in [13, p.82], and if R is replaced by some localizedalgebra R S w.r.t. some multiplicative S ⊂ R in the above equation we get the categorical equation ǫ L ( R,S ) ◦ L η ( R,S ) = id L ( R , S ) which is the second equation of equation (8) of [13, p.82]. It follows nowfrom [13, p.83, Thm 2.(v)] that L is a left adjoint of U with unit η and counit ǫ .Finally, for any r ∈ R and s ∈ U ( R ) we have ( i R ◦ j R )( r, s ) = ( r + s − , : since π ( R,S ) (0 , s ) is the inverse of π ( R,S ) ( s, as is π ( R,S ) ( s − , we have (0 , s ) − ( s − , ∈ ker( π ( R,S ) ) = κ ( R, S ) showing that η ( R,U ( R )) ◦ ǫ R = id R U ( R ) by a combination of the above first and second diagram. Itfollows in addition that ǫ R is a natural isomorphism R U ( R ) → R with inverse η ( R,U ( R )) . This provesthe the first statement of the Proposition since units and counits of adjunctions are automaticallyuniversal. In order to prove formula (1.5) we observe that each element of R S is a finite sumof elements of images (under π ( R,S ) ) of words in T K ( R ⊕ KS ) consisting of letters of the form ( r, or (0 , s ) with r ∈ R and s ∈ S . We clearly have (writing π = π ( R,S ) ) for all r, r ′ ∈ R : π (cid:0) ( r, ⊗ ( r ′ , (cid:1) = π ( r, π ( r ′ ,
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