Embeddings of quotient division algebras of rings of differential operators
aa r X i v : . [ m a t h . R A ] N ov EMBEDDINGS OF QUOTIENT DIVISION ALGEBRAS OF RINGS OFDIFFERENTIAL OPERATORS
JASON P. BELL, COLIN INGALLS, AND RITVIK RAMKUMAR
Abstract.
Let k be an algebraically closed field of characteristic zero, let X and Y besmooth irreducible algebraic curves over k , and let D ( X ) and D ( Y ) denote respectively thequotient division rings of the ring of differential operators of X and Y . We show that if thereis a k -algebra embedding of D ( X ) into D ( Y ) then the genus of X must be less than or equalto the genus of Y , answering a question of the first-named author and Smoktunowicz. Introduction
One of the central results in algebraic geometry is the birational classification of surfaces.In particular, this work shows that the most crucial birational invariants of a surface arethe Kodaira dimension, the geometric genus, the irregularity, and the plurigenera. A relatedquestion concerns when there exists a dominant rational map from a smooth irreducibleprojective surface X to another smooth surface Y . Algebraically, this corresponds to theexistence of an embedding of the field of rational functions of Y into the corresponding fieldfor X . Many of these birational invariants give restrictions upon when such a map can exist.For example, if X is a smooth irreducible surface over a base field of characteristic zero andthere is dominant rational map to another smooth surface Y then all of the above listedbirational invariants of X must be at least those of Y (see Lemma 2.5 for details). In thecase of curves, the Riemann-Hurwitz theorem [11, IV, Corollary 2.4] gives that if X and Y are smooth irreducible projective curves and there is surjective morphism from X to Y thenthe genus of X must be at least that of Y .One has noncommutative analogues of these questions, which we now describe. If A is agraded noetherian domain of GK dimension 3 that is generated in degree one, and we let C denote the category of finitely generated graded right A -modules modulo the subcategory oftorsion modules, then we can create a noncommutative analogue of Proj( A ) by taking thetriple ( C , O , s ), where O is the image of the right module A in C and s is the autoequivalenceof C defined by the shift operator on graded modules, as described in [1]. Furthermore, A hasa graded quotient division ring, which we denote Q gr ( A ), which is formed by inverting thenonzero homogeneous elements of A . Then the graded form of Goldie’s theorem [10] givesthat there is a division ring D and automorphism σ of D such that Q gr ( A ) = D [ t, t − ; σ ] . We can then view D as being the “function field” of X = Proj( A ) and in the commutativesetting this construction coincides exactly with the ordinary field of rational functions on an Mathematics Subject Classification.
Key words and phrases.
Rings of differential operators, genus, division rings, noncommutative surfaces, Oreextensions, embeddings, birational invariants.The authors thank NSERC for its generous support. irreducible projective variety. We can then view Proj( A ) and Proj( B ) as being birationallyisomorphic if their corresponding division rings are isomorphic.Unlike in the commutative case, however, much less is known in the noncommutativesetting and in general there is a dearth of noncommutative birational invariants outside ofvarious analogues for transcendence degree [9, 18, 19, 20]. Despite the lack of noncommutativeinvariants, there are large classes of well-studied examples of noncommutative surfaces andArtin [1] has given a proposed birational classification of graded k -algebra domains A of GKdimension 3 (possessing additional homological properties which we do not mention here). Inthis case, we have Q gr ( A ) = D [ x, x − ; σ ] for some division ring D and, in analogy with thebirational classification of surfaces mentioned earlier, one would like to understand the typeof division rings that can occur. In particular, Artin claims that under the hypotheses on A from his paper, the division ring D must satisfy at least one of the following properties:(1) D is finite-dimensional over its centre, which is a finitely generated extension of k oftranscendence degree 2;(2) D is isomorphic to a Skylanin division ring;(3) D is isomorphic to the quotient division ring of K [ t ; σ ] where K is a finitely generatedfield extension of k of genus 0 or 1 and σ is a k -algebra automorphism of K ;(4) D is isomorphic to the quotient division ring of K [ t ; δ ] where K is a finitely generatedfield extension of k and δ is a k -linear derivation of K .We point out that the first class encompasses all of the fields of rational functions on algebraicsurfaces and has lumped them together as part of a single class. In this sense, the classificationis much less specific than in the commutative setting, but from the noncommutative pointof view one can see the first case as being essentially “understood”. (Work of Chan and thesecond-named author [4] provides a birational classification in this case. In particular, itshows that such division rings are either “ruled”, “del Pezzo”, or have a minimal model whichis unique up to Morita equivalence.) We also note that division rings of types (2), (3), and (4)can be finite over their centres, and so when we talk about these latter types we specificallymean only the division rings of this type that are infinite-dimensional over their centres. Ringsof the form K [ t ; δ ] with K a finitely generated field extension of k of transcendence degreeone and δ a derivation of K are birationally isomorphic to rings of differential operators ofa smooth curve. The most important ring of this type is the first Weyl algebra, which hasgenerators x and y and the relation xy − yx = 1.Just as one would like to understand conditions which must be met to have a dominantrational map of surfaces in terms of invariants, one would also like to understand when onecan have an embedding of division rings between two division rings on Artin’s list. In fact, insome cases it is not hard to see that embeddings cannot occur except for trivial reasons (see § F of a smooth curve, and by this we simply mean the geometric genus of a smooth irreducibleprojective curve that has F as its field of rational functions. Theorem 1.1.
Let k be an algebraically closed field of characteristic zero, let F and K befinitely generated field extensions of k of transcendence degree one, and let µ and δ be nonzeroderivations of F and K respectively. If F ( t ; µ ) embeds in K ( x ; δ ) then the genus of F is lessthan or equal to the genus of K . We note that if either µ or δ is the zero derivation then the theorem holds trivially (see theremarks in § K [ t ; δ ] can be viewed asnoncommutative deformations of the polynomial ring K [ t ] and have an associated gradedring of this form. This fact often simplifies the study of their ring theoretic properties. Onthe other hand, for K of characteristic zero, when one forms the quotient division ring byinverting nonzero elements, the resulting division ring is unwieldy and contains a copy ofthe free algebra on two generators unless δ is zero (this can be seen from the fact that thedivision ring K ( t ; δ ) then contains a copy of the Weyl algebra and one can then use a result ofMakar-Limanov [14]). Thus the problem of understanding embeddings of F [ t ; µ ] into K [ x ; µ ] isconsiderably simpler than the problem of understanding embeddings of their quotient divisionrings. Our approach in proving Theorem 1.1 is to use reduction modulo primes. Here oneagain must be careful about what we even mean. It is known that rings of the form F [ t ; µ ]satisfy a polynomial identity when F is a a field of positive characteristic and thus one canhope that commutative methods might apply. The one subtlety, however, is that while onecan expect to obtain information about F [ t ; µ ] by “reducing modulo primes” one cannot ingeneral expect to gain understanding about its quotient division ring in this manner. Thereason for this is that if one has a finitely generated noncommutative Z -algebra then reducingmod a prime p and then localizing is generally straightforward, but trying to reduce mod p after localizing will in many cases yield the zero ring.The strategy to get around these difficulties is to suppose that we have an embedding F ( t ; µ ) into K ( x ; δ ) for fields of rational functions F and K of curves X and Y . Then thisembedding gives an embedding of C [ t ; µ ] into K ( x ; δ ) where C is a finitely generated Z -algebra, closed under the derivation µ , such that C contains a set of generators for F asa field extension of k . Although C [ t ; µ ] is mapping into K ( x ; δ ), we in fact show that itmaps into well-behaved localization of a ring of the form A [ x ; δ ] where A is again a finitelygenerated Z -algebra. In particular, we show that we are still able to reduce modulo primesin Spec( A ) in our localization. We now reduce modulo various prime ideals and then localizeto obtain embeddings of division rings that are finite over their centres and we also showthat these embeddings yield embeddings of the respective centres. We then use properties offlat morphisms of schemes to show that the centres of these division rings are isomorphic tofunction fields of ruled surfaces and we finally use results from algebraic geometry to showthat the embedding of the centres can occur only if we have the desired inequality for thegenera of X and Y .The outline of this paper is as follows. In §
2, we prove Theorem 1.1 and then in §
3, wegive general remarks about the embedding question for other division rings on Artin’s list.2.
Proof of Theorem 1.1
We begin with a lemma, which we will use to construct a suitable model for our ring ofdifferential operators that will allow us to reduce mod p . JASON P. BELL, COLIN INGALLS, AND RITVIK RAMKUMAR
Lemma 2.1.
Let k be an algebraically closed field of characteristic zero, let F be a finitelygenerated extension of k of transcendence degree one and genus g equipped with a nonzero k -linear derivation δ , and let B ⊆ F be a finitely generated Z -algebra. Then there exist finitelygenerated Z -subalgebras C and C of k and F respectively such that the following propertieshold:(1) δ ( C ) ⊆ C ;(2) there exists s ∈ C such that δ ( s ) and s are units in C ;(3) C ⊇ B ;(4) each maximal ideal P of C has the property that P C is a prime ideal of C and C/P C ⊗ C /P C /P is an integral domain whose field of fractions is a field extension F P of C /P of transcendence degree one and genus g ;(5) there is some fixed d ≥ such that for a Zariski dense set of maximal ideals P of C ,the field F P is an extension of the C /P -subfield of F P generated by the image of s offixed degree d = [ F P : C /P ( s )] . Proof.
We can view F as the field of rational functions on some smooth irreducible projectivecurve X . We fix an embedding f : X → P mk , and we may assume without loss of generalitythat m is minimal. We have that f ( X ) is the zero set of a homogeneous ideal I in k [ x , . . . , x m ].We choose a finite set of generators for I and we let S denote the set of coefficients of thepolynomials in our finite generating set. Then the homogeneous coordinates induce rationalfunctions f i,j = ( x i /x j ) (cid:12)(cid:12) X for i = j and by minimality of m , the f i,j can be extended to regular and non-constant mapson a dense neighbourhood of X . Since δ is nonzero there is some s among the f i,j such that δ ( s ) = 0. Then for k = ℓ we have δ ( f k,ℓ ) = P k,ℓ /Q k,ℓ for some polynomials P k,ℓ and Q k,ℓ inthe f i,j . We also note that the inclusion of fields k ( s ) → F induces a non-constant morphism s : X → P and we let d denote the degree of this map.We then take A to be a finitely generated Z -algebra that contains the elements of S , thecoefficients of the P k,ℓ and Q k,ℓ , and the inverses of the nonzero coefficients of the Q k,ℓ .Then we obtain a model X for X over Spec( A ), by viewing our generating set as lyingin A [ x , . . . , x m ] and taking the zero set of the ideal in A [ x , . . . , x m ] generated by theseelements. We then have a morphism ˜ s : X → P A , and a morphism χ : P A → Spec( A ), and welet φ = χ ◦ ˜ s : X →
Spec( A ). By generic flatness [7, Th´eor`eme 6.9.1] there is some nonzero f ∈ A such that if we replace A by A f and replace X accordingly, the morphisms φ and ˜ s areflat. Then for each prime ideal t ∈ Spec( A ), we have a fibre, which we consider as a closedsubscheme X t of P m Frac(
A/t ) , and we have that X t has constant arithmetic genus [11, III, Cor.9.10]. Since the generic fibre X η has the property that if we extend the base field we obtain X , and since X is smooth and irreducible, we see that the arithmetic genus of X η is equal tothe geometric genus of X and so the arithmetic genus of each X t is equal to g .Now the set of t in Spec( A ) for which the fibre X t is geometrically irreducible is constructible[12, p. 36, Th´eoreme 4.10]; that is, it is a finite union of sets of the form U ∩ V where U isopen and V is closed. Similarly, since the generic fibre is smooth, the fibres X t are smoothon a Zariski open set of Spec( A ) [8, Proposition 17.7.11(ii)]. Since the generic fibre, X η , isalso geometrically irreducible, we then see that there is some nonzero h ∈ A such that for t ∈ Spec( A h ) ⊆ Spec( A ), the fibres are geometrically irreducible and smooth. Since the arithmetic genera of X t are all equal to g for t in the open subset Spec( A h ), by smoothnesswe then have that the geometric genus of each X t is g for t ∈ Spec( A h ). We now replace A by A h and again adjust our model accordingly. We pick an affine open subset U of X such that the finite set of generators for B , s and s − , δ ( s ) and δ ( s ) − and the functions f i,j are regular on U . Now the collection of regular functions on U is a finitely generated A h algebra and if u , . . . , u k are generators then there is some b that is regular on U suchthat δ ( u i ) ∈ O X ( U )[1 /b ]. Since derivations behave well with respect to localization, it is theneasy to check that O X ( U )[1 /b ] is closed under application of δ . In particular, we can refine U if necessary and assume that δ preserves the ring of functions that are regular on U . Thenif we let C = A h and C = O X ( U ) then C ⊆ F is a C -algebra and by construction forevery maximal ideal P of C we have that CP is a prime ideal of C since the correspondingfibre X P is geometrically irreducible (in the scheme theoretic sense); and C/CP is C /P -algebra of Krull dimension one. Moreover, by geometric irreducibility of the fibres, we have C/CP ⊗ C /P C /P is an integral domain and its field of fractions is a transcendence degreeone extension of C /P of genus g , by the remarks above. We also note that our constructiongives that C ⊇ B and s, s − ∈ C and δ ( C ) ⊆ C . It only remains to show (5).Let O (1) denote a very ample invertible sheaf on P A . Then since ˜ s is a finite map, weobtain an invertible sheaf L := ˜ s ∗ O (1) on X , which is ample over A . This sheaf L then givesus a second embedding of X into some P mA . By generic flatness, we again have that on anopen subset of Spec( A ) the degree (for this embedding induced by L ) of X t is constant [11,III, Cor. 9.10] and this must be equal to the degree of the generic fibre, X η , which (again byextending the base and obtaining X ) is the limit as n tends to infinity of h (˜ s ∗ ( O ( n ))) /n =deg(˜ s ) = d . This says that the degree of the field extension of Frac( C/P ) is d -dimensionalover the extension generated by the image of s and this gives (5) and completes the proof. (cid:3) Once we reduce modulo p , our ring of differential operators will satisfy a polynomial identity.The embedding questions will then reduce to embedding questions for division rings that arefinite-dimensional over their centres. Our first result shows that under certain circumstances,an embedding of division rings will induce an embedding of the centres. Given a ring R , welet Z ( R ) denote its centre. Lemma 2.2.
Let D and D be two division rings both containing a central subfield k andsuppose that [ D : Z ( D )] = [ D : Z ( D )] . If there is an injective k -algebra homomorphism φ : D → D , then φ ( Z ( D )) ⊆ Z ( D ) .Proof. Suppose that this does not hold and let K ⊃ Z ( D ) denote the subalgebra of D generated by Z ( D ) and φ ( Z ( D )). We note that K is commutative since Z ( D ) commuteswith everything in φ ( Z ( D )) and φ ( Z ( D )) is commutative. Moreover, since D is finite-dimensional over its centre, we see that K is an integral domain that is finite-dimensional over Z ( D ) and hence must be a field. By assumption, K = Z ( D ). Notice that K centralizes E := φ ( D ) ⊆ D . Let L be the division subring of D generated by E and K . Then[ L : K ] ≤ [ D : K ] < [ D : Z ( D )]. It follows that the dimension of L over its centre, whichcontains K , is less than the order of D over its centre. In particular, the PI degree of L isstrictly less than the PI degree of D . But D ∼ = E embeds in L and so the PI degree of D is strictly less than the PI degree of D , a contradiction. The result follows. (cid:3) To apply the preceding lemma, we need a characterization of the centre of a divisionring obtained from the skew polynomial rings over the function field of a curve in positive
JASON P. BELL, COLIN INGALLS, AND RITVIK RAMKUMAR characteristic. The following two lemmas give this characterization. We recall that K h p i = { a p | a ∈ K } . Lemma 2.3.
Let p be a prime number and suppose that K is a finitely generated transcendencedegree one extension of an algebraic extension E of F p and that K is degree < p over a purelytranscendental extension of E . Then [ K : K h p i ] = p .Proof. By assumption, we have that K is a degree d extension of E ( s ) for some s ∈ K andsome d < p . Thus K is separable and so K is generated by two elements s, t ∈ K by theprimitive element theorem, and there is some irreducible polynomial P ( x, y ) of degree d in y with P ( s, t ) = 0. Let F = E ( s, t p ). Then [ K : F ] ∈ { , p } since t ∈ K has annihilatingpolynomial x p − t p over F , and this is either irreducible over F or is the p -th power of a linearpolynomial over F . But [ K : F ] ≤ [ K : E ( s )] ≤ deg y ( P ) < p and so we see that K = F . Thenwe have K = E ( s, t p ) and since K h p i ⊇ E ( s p , t p ), we see that [ K : K h p i ] ≤ [ E ( s ) : E ( s p )] = p .Thus it suffices to show that [ K : K h p i ] ≥ p . Since K is finitely generated and not algebraicover E , K is not perfect and so we have that there is some u ∈ K that is not in K h p i . Then u is a root of the polynomial x p − u p ∈ K h p i [ x ] and this is irreducible since u K h p i . Inparticular, [ K : K h p i ] ≥ p and the result follows. (cid:3) Lemma 2.4.
Let p be a prime number and suppose that K is a finitely generated transcendencedegree one extension of an algebraic extension E of F p and that K is degree < p over a purelytranscendental extension of E . If δ is a nonzero E -linear derivation of K then Z ( K ( x ; δ )) = K h p i ( x p ) ∼ = K ( t ) .Proof. Pick s ∈ K such that [ K : E ( s )] < p . Then δ ( s ) = β for some β ∈ K . Notice that the E -linear derivation µ of E ( s ) given by differentiation with respect to s extends uniquely toa derivation of K since K is a finite separable extension of E ( s ). In particular, we see that δ = βµ and K ( x ; δ ) = K ( x ; µ ). Thus we may assume without loss of generality that δ = µ and δ ( s ) = 1. Let Z = Z ( K ( x ; δ )). For α ∈ K we have δ ( α p ) = 0 and so K h p i ⊆ Z . Also for α ∈ K , since K has characteristic p , we have [ x p , α ] = ad px ( α ) = δ p ( α ). Notice that δ p = ad x p is a derivation of K and it annihilates E ( s ). Since K is a separable extension of E ( s ), we seethat δ p ( K ) = 0 and so x p ∈ Z . Now we claim that [ Z ( x ) : Z ] = p . To see this, observe that x satisfies the polynomial equation t p − x p = 0 in Z [ t ] and this is irreducible unless x ∈ Z . Itfollows that [ Z ( x ) : Z ] ∈ { , p } . Since δ is nonzero, we see that x Z and so we obtain theclaim.We have [ K : K h p i ] = p by Lemma 2.3 and so [ K ( x : δ ) : K h p i ( x p )] = p . Thus [ K ( x : δ ) : Z ] ≤ p . But whenever F is a maximal subfield of K ( x ; δ ) we have [ K ( x ; δ ) : Z ] = [ F : Z ] .In particular, if we pick a maximal subfield containing Z ( x ), we see that [ F : Z ] ≥ [ Z ( x ) : Z ] = p . Thus [ K ( x : δ ) : Z ] = p and so Z = K h p i ( x p ). To get the final isomorphism, noticethat the map f ( t ) f ( t ) p gives an isomorphism from K ( t ) to K h p i ( t p ). (cid:3) We next prove a few results that are well-known, but for which we are unaware of properreferences. We first prove Lemma 2.5, which gives many of the claimed inequalities on bi-rational invariants for X and Y when there is a dominant rational map from Y to X . Wethen give a non-embedding result (Lemma 2.6) that can apply to centres. We point out thatLemma 2.6 immediately follows form Lemma 2.5 in the separable case, but we require a moregeneral version. Lemma 2.5.
Let k be a base field and suppose that F ⊆ K are finitely generated fields over k and that the extension F ⊆ K is separable. Suppose that the transcendence degrees of F, K are two or the characteristic of k is zero. Let X, Y be smooth models for
F, K respectively. Then h ( X, Ω jX ) ≤ h ( Y, Ω jY ) and h ( X, ω ⊗ nX ) ≤ h ( X, ω ⊗ nX ) for n ≥ , and the Kodaira dimensionof Y is at least that of X .Proof. We have a dominant rational map Y X . Due to our assumptions on characteristicand dimension, we may resolve indeterminacies of the map to obtain a regular map π : e Y → X where e Y is smooth. Since the above numbers are all birational invariants we may replace Y with e Y .
Now since the extension F ⊆ K is separable, π is generically ´etale so π ∗ Ω jX isisomorphic to Ω jY generically. Since π ∗ Ω jX , Ω j e Y are locally free, the natural map π ∗ Ω jX → Ω j e Y is injective. Now applying the global section functor and the projection formula yields theresult. The second inequality follows by the same argument, and since the Kodaira dimensionis the growth of the plurigenera, we are done. (cid:3) Lemma 2.6.
Let k be an algebraically closed field and let F and K be function fields ofsmooth projective irreducible curves over k of genera g F and g K respectively. If there is a k -algebra embedding of F ( t ) into K ( t ) then g F ≤ g K .Proof. Choose a smooth minimal model X for F ( t ). Note that ρ : X → C is ruled over acurve C with k ( C ) = F . If g F = 0 we are done, so let us suppose that g F >
0. We can choosea smooth model Y of K ( t ) and we will have a dominant rational map Y X . By resolvingthe singularities of the map we may replace Y with a smooth model where we have a regularmap π : Y → X . Now Y is birational to a surface ruled over a curve D with k ( D ) = K , so wehave a map Y → D with fibres that are trees of rational curves. By Tsen’s Theorem [17], wehave a section s : D → Y . Let us now consider the map ψ = ρ ◦ π ◦ s : D → X . If the imageof ψ is C we are done, otherwise the image ψ must be a point p in C . So we see that D mapsto the fibre ρ − ( p ) = F ≃ P . Now consider a fibre F ′ in Y over a point q ∈ D . Now π ( F ′ )must be connected and since all the components of F ′ are rational curves, the map ρ ◦ π mustbe constant on F ′ . Since F ′ meets D we see that the image of F ′ in X is contained in F . Sothe map Y → X is not dominant. (cid:3) We are now ready to prove our main result.
Proof of Theorem 1.1.
Let g F and g K denote the genera of F and K respectively.By Lemma 2.1, there exist finitely generated Z -algebras C and C such that C [ t ; µ ] is asubring of F ( t ; µ ), C is a finitely generated C -algebra, and C and C satisfy the followingconditions:(1) δ ( C ) ⊆ C ;(2) there exists s ∈ C such that µ ( s ) = 1 and s is a unit in C ;(3) each maximal ideal P of C has the property that P C is a prime ideal of C and C/P C ⊗ C /P C /P is an integral domain whose field of fractions is a field extension F P of C /P of transcendence degree one and genus g F ;(4) there is some fixed d ≥ P of C ,the field F P is an extension of the C /P -subfield of F P generated by the image of s of fixed degree d .Then the embedding of F ( t ; µ ) into K ( x ; δ ) gives an embedding ι of C [ t ; µ ]. Let c , . . . , c r be generators for C as a Z -algebra. Then there exist elements p i ( x ) , q i ( x ) ∈ F [ x, δ ] such that ι ( c i ) = p i ( x ) q i ( x ) − and there exist elements r ( x ) , s ( x ) ∈ F [ x, δ ] such that ι ( t ) = r ( x ) s ( x ) − . JASON P. BELL, COLIN INGALLS, AND RITVIK RAMKUMAR
Let B denote the Z -algebra generated by the coefficients of p i ( x ) , q i ( x ) , r ( x ) , s ( x ) as wellas the inverses of all coefficients of leading monomials. Then by Lemma 2.1, there exists afinitely generated Z -subalgebra A of k and a finitely generated A -algebra A such that:(i) δ ( A ) ⊆ A ;(ii) there exists s ′ ∈ A such that δ ( s ′ ) = 1 and s ′ is a unit in A ;(iii) A ⊇ B ;(iv) each maximal ideal P of A has the property that P A is a prime ideal of A and A/P A ⊗ A /P A /P is an integral domain whose field of fractions is a field extension K P of A /P of transcendence degree one and genus g K ;(v) there is some fixed d ′ ≥ P of A ,the field K P is an extension of the A /P -subfield of F P generated by the image of s ′ of fixed degree d ′ .Since A is noetherian, the set S of monic polynomials in A [ x ; δ ] is an Ore set [6, Lemma1.5.1]. Then since A contains B and the leading coefficients of the q i ( x ) and s ( x ) are unitsin A we see that the embedding of C [ t ; µ ] into K ( x ; δ ) sends C [ t ; µ ] into S − A [ x ; δ ], since agenerating set for C [ t ; µ ] is sent into this ring. We also note that prime ideals of A that areclosed under application of the derivation δ lift to prime ideals of A [ x ; δ ] and, moreover, theysurvive when we invert S since the elements of S are all regular modulo these prime ideals.Then for each maximal ideal P of A , the composition of maps C [ t ; µ ] → S − A [ x ; δ ] → ¯ S − ( A/P A )[ x ; δ ] , where ¯ S is the monic polynomials in ( A/P A )[ x ; δ ], gives a map φ P from C [ t ; µ ] to¯ S − ( A/P A )[ x ; δ ]. Since ¯ S − ( A/P A )[ x ; δ ] is a domain, the kernel must be a completely primeideal of C [ t ; µ ].Moreover, since the embedding is the identity on k , we see that the embedding ι sends C ⊆ C ∩ k into A ∩ k and so φ P maps C to ( A ∩ k ) / ( P A ∩ k ). We claim that ( A ∩ k ) / ( P A ∩ k )is an algebraic extension of a finite field. This will then give that Q := ker( φ ) ∩ C is a maximalideal of C since φ P must then map C into a finite field since C is finitely generated. Toobtain the claim, we note that by construction A is a finitely generated A -algebra whoseKrull dimension is one greater than that of A . It follows that A cannot contain a polynomialring in two variables over A . Moreover, A is not algebraic over k and so there exists some z ∈ A that is transcendental over k . We now claim that if α ∈ A ∩ k then A + A α + · · · is not direct; if it were, then since A cannot contain a polynomial ring in two variables over A , the infinite sum A [ α ] + A [ α ] z + · · · could not be direct, and this would then give that z is algebraic over k . It follows that every element of A ∩ k is algebraic over A and hence( A ∩ k ) / ( P A ∩ k ) is algebraic over the finite field A /P , thus giving the claim.By property (4), we have that QC is a prime ideal of C and since µ ( Q ) = 0 we see that thisprime ideal is µ -invariant and is in the kernel of φ . Since C /Q and A /P are finite fields ofthe same characteristic, they have isomorphic algebraic closures and thus we get an inducedmap ¯ φ P : ( C/QC ⊗ C /Q C /Q )[ t ; µ ] → ( ¯ S − ( A/P A ) ⊗ A /P A /P )[ x ; δ ] . We claim that ¯ φ P is injective. We observe that once we have this, we are done, because¯ φ P will induce an injection from the division ring F P ( t ; µ ) into K P ( x ; δ ), by localizing. If wechoose a maximal ideal P such that A /P has characteristic p > max( d, d ′ ) and such that F P is degree d over the subfield generated by s and K P is degree d ′ over the subfield generated by s ′ (this is possible since we can invert the set of primes p ≤ max( d, d ′ ) in A and we will still have an infinite spectrum and for a Zariski dense set of P we will get the desired degrees), thenby Lemmas 2.3 and 2.4, F P ( t ; µ ) and K P ( x ; δ ) are both p -dimensional over their respectivecentres and so by Lemma 2.2 we have that this embedding restricts to an embedding of theircentres and by Lemma 2.4, this then gives an embedding of F P ( y ) into K P ( y ), where y is anindeterminate, which gives that the genus of F P is at most the genus of K P by Lemma 2.6.Since we have that the genera of F P and K P are respectively g F and g K , we see that g F ≤ g K , as desired.Thus it only remains to show that the map ¯ φ P is injective. Let R = ( C/QC ⊗ C /Q C /Q ).Then R is a finitely generated commutative C /Q -algebra of Krull dimension one. It followsthat R has Gelfand-Kirillov dimension one [13, Theorem 4.5 (a)], and so since R is finitelygenerated, R [ x ; µ ] has Gelfand-Kirillov dimension two [13, Proposition 3.5]. Since R is adomain, we see that if ¯ φ P is not injective then there is some nonzero prime ideal I of R [ t ; µ ]such that I is equal to the kernel of ¯ φ P . Moreover, I must be a completely prime idealsince ¯ φ maps into a domain. Now the Gelfand-Kirillov dimension of R [ t ; µ ] /I is at most oneas a C /Q -algebra if I is nonzero [13, Proposition 3.15]. But this then gives that R [ t ; µ ] /I is commutative, as it is a domain of Gelfand-Kirillov dimension one over an algebraicallyclosed field. (This is a now well-known observation that uses Tsen’s theorem [17] and theSmall-Stafford-Warfield theorem [16].) Now by (2), we have [ t, s ] = δ ( s ) and s and δ ( s ) areunits. But if I is nonzero then t and s commute modulo I and so [ t, s ] = δ ( s ) ∈ ker( φ ) P , acontradiction. The result follows. (cid:3) Additional remarks about embeddings
The general question as to when there exists an embedding of D into D when D and D are two division rings on Artin’s list has been looked at before and there are many folkloreresults in this area. We can divide Artin’s list into four types of division rings:(1) those that are finite-dimensional over their centres;(2) the Sklyanin division rings not finite over their centres;(3) Skew field extensions of automorphism type (not finite over the centre);(4) Skew field extensions of derivation type (not finite over the centre).Theorem 1.1 then addresses embeddings for division rings of Type 4. In fact, for embeddingsof some of the other types much more is already known. For example, maximal subfields ofdivision rings of Types 2–4 have transcendence degree one over k (cf. [2, Theorem 1.4]) andhence no division ring of Type 1 can embed into one of another type; conversely, divisionrings of Types 2–4 all contain free algebras on two generators and hence cannot embed intodivision rings of Type 1.In the case of embedding division rings of Type 1, we have the following easy observations. Proposition 3.1.
Let D , D be division algebras that are finite-dimensional over centralfields K, F which are finitely generated of transcendence degree 2 over k . Then if there is anembedding of D into D then the period of D is at least that of D . If furthermore, theirperiods are equal, then the plurigenera, irregularity, geometric genus and Kodaira dimensionof Z ( D ) are at least those of Z ( D ) .Proof. Since the period of D i is its PI degree we immediately obtain the first statement. Thelast statement follows from Lemmas 2.2 and 2.5. (cid:3) Question 1.
With the above hypotheses, can we conclude that the Kodaira dimension of D is at least that of D ?We note that the division ring of Type 3 given by k ( t )( x ; σ ) where σ ( t ) = t + 1 is in factisomorphic to the division ring k ( y )( z ; δ ) where δ is differentiation with respect to y . Here theautomorphism is given by t = yz and x = z . This division ring embeds into every division ringof Type 4 since, by working up to isomorphism, we may assume that there is always a solutionto δ ( s ) = 1 when our field is a finitely generated transcendence degree one field extension of k .Other than these trivial cases, it is known in some cases that some division rings of Type 4 donot embed into division rings of Type 3 and it is known that some division rings of Type 3 donot embed into division rings of Type 4. Perhaps the easiest such example is given by taking k ( t )( x ; σ ) where σ is not conjugate to an automorphism of the form t t + 1. In this case,one can perform a change of variables and assume that σ ( t ) = qt for some nonzero element of k . Moreover, since we are assuming we are not finite over our centre, we have that q is not aroot of unity. The quotient division ring of a ring of the form K [ t ; δ ] embeds in a skew powerseries ring K (( t − ; δ )) and a simple computation by looking at leading terms shows that thereare no solutions to the equation xt = qtx in this ring with q = 1. In particular, k ( t )( x ; σ )cannot embed into K (( t − ; σ )) and hence cannot embed into any division ring of type 3.Finally, while not explicitly written down in the literature, Artin [1] points out that aSklyanin division algebra is a ring of invariants of a division ring of Type 3—specifically thereis an elliptic curve E and an infinite-order translation σ of E such that a Sklyanin divisionring is the ring of invariants of a Z / Z -action on k ( E )( t ; σ ), where the action on k ( E )( t ; σ )comes from the induced map on k ( E ) from the negation map on E and then by extendingthe action by sending t to t − . ∗ This gives an embedding of a division ring of Type 2 into oneof Type 3. We are not aware of any additional results involving embeddings from or into theSklyanin division rings and a systematic study of the possible embeddings of the division ringson Artin’s list would make an interesting topic for future study. We conclude by asking abouta generalization of Theorem 1.1, which would give a complete understanding of embeddingsbetween division rings of Type 4 if the question were answered affirmatively.
Question 2.
Let k be an algebraically closed field of characteristic zero, let X and Y besmooth irreducible algebraic curves over k , and let D ( X ) and D ( Y ) denote respectively thequotient division rings of the ring of differential operators of X and Y . If d is a naturalnumber, is it the case that there is an embedding of D ( X ) into D ( Y ) of degree d if and onlyif there is a degree d surjective morphism from Y to X ?While it is known that for division rings D and D from Artin’s list with D ⊆ D , wehave that D is finite-dimensional as a left and right D -vector space [2, Theorem 1.4] (seealso Schofield [15, Corollary 35]), it is not known that these two dimensions coincide. Ingeneral, examples where these two quantities are different exist (see Cohn [5, Section 5.9]),although there are no known counter-examples for division rings from Artin’s list. We thusdefine the degree for an embedding of D ( X ) into D ( Y ) to be the minimum of the dimensionsof D ( Y ) as a left and right D ( X )-vector space. We point out that one direction is trivial, butthe other direction would have important implications beyond the inequality between generagiven in Theorem 1.1. For example, it would show that the gonality of the curve X boundsthe degree of the embedding of the Weyl division algebra into D ( X ). ∗ Artin attributes this (non-trivial) observation to Michel Van den Bergh. References [1] M. Artin, Some problems on three-dimensional graded domains.
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Jason Bell and Ritvik Ramkumar, Department of Pure Mathematics, University of Waterloo,Waterloo, ON, N2L 3G1, Canada
E-mail address : [email protected] E-mail address : [email protected] Colin Ingalls, Department of Mathematics, University of New Brunswick, Fredricton, NB,,Canada
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