aa r X i v : . [ m a t h . QA ] O c t PICARD GROUPS OF DIFFERENTIAL OPERATORS ONCURVES
GEORGE WILSON
Abstract.
These notes are a supplement to the first part of [CH2], concerningthe Picard group of D ( X ) , where X is an affine curve. The main new fact isthat the exact sequence of [CH2] describing Pic D is split. Introduction
Let X be a smooth irreducible complex affine curve. To make the statementsthat follow as simple as possible, we shall assume that X has no nontrivial au-tomorphisms, and also that the ring O ( X ) has no nonconstant units (that is the“case of general position”: we shall remove these assumptions in section 5 below).Let D ≡ D ( X ) be the ring of differential operators on O ( X ) , and let Pic D be the group of (isomorphism classes of) autoequivalences of the category of left D -modules. In [CH2], Cannings and Holland proved the following. Theorem 1.1.
There is an exact sequence of groups (1.1) 0 → Ω ( X ) → Pic
D →
Pic X → . Here (as usual) Ω ( X ) is the additive group of regular differentials on X , and Pic X is the group of (algebraic) line bundles.Theorem 1.1 almost determines Pic D ; however, there remains the question ofthe group extension in (1.1). Proposition 1.2.
The sequence (1.1) is split.
This sequence takes on a more familiar appearance if we rephrase some of thematerial of [CH2] in terms of line bundles with connection. We denote by
Pic ♭ X the group of isomorphism classes of line bundles with (flat algebraic) connectionover X , so that we have an exact sequence of abelian groups(1.2) 0 → Ω ( X ) → Pic ♭ X p −→ Pic X → p assigns to a line bundle with connection the underlying line bundle,and Ω ( X ) is considered as the space of connections on the trivial bundle. Now, if L is a line bundle on X , with space of global sections Γ L , a connection on L maybe viewed as a structure of left D -module on Γ L , extending the given structure of O ( X )-module. The functor Γ L ⊗ O ( X ) − then defines an element of Pic D , so wehave a natural homomorphism(1.3) χ : Pic ♭ X → Pic D . It is easy to see that this map χ is injective: Cannings and Holland proved (ineffect) the following. Date : October 16, 2010.
Theorem 1.3.
The map (1.3) is an isomorphism.
Using this isomorphism, the exact sequence (1.1) becomes the sequence (1.2),which is quite amenable to study. The proof of Theorem 1.3 consists in combiningthe main results of [CH1] and of [ML]: we give the outline in section 6 below.Most of the discussion above still holds, mutatis mutandis , if X is a complete(that is, projective) curve: in that case we have to replace the ring D ( X ) by the sheaf D X of differential operators on X . I do not know whether Theorem 1.3still holds in the complete case; however, we certainly still have the exact sequence(1.2), provided we replace Pic X by the group Pic X of line bundles of degreezero (since only these admit a connection). Our proof that the sequence is splitstill holds in the complete case. That contrasts with the known fact (see [M]) that Pic ♭ X is the universal extension of the Jacobian Pic X by a vector group, thusin some sense as far as possible from being split. The explanation is that thislast statement considers (1.2) as an extension of complex algebraic groups, and thedistinguished splitting described below is a splitting only of real Lie groups. In theaffine case, there is no natural algebraic structure on the groups in (1.2): indeed,
Pic X is typically a quotient of a torus by a countably infinite subgroup, so theonly possibility seems to be to regard it just as a huge abstract group. However, if X is obtained from a complete curve Σ by removing just one point, then Pic X iscanonically identified with Pic Σ , so we are in an awkward intermediate situation.The paper is organized as follows. In section 2 I review the main technical deviceused in the proof of Proposition 1.2, namely the description of
Pic ♭ X in terms ofdifferentials of the third kind on X (see [M]). I give a self-contained account ofthis which is less sophisticated than the one in [M]; it uses arguments that will befamiliar to readers of [CH2]. Section 3 gives two constructions of splittings of thesequence (1.2). The first is purely algebraic, but involves an arbitrary choice ofbasis for an infinite-dimensional vector space. The second construction gives thedistinguished splitting mentioned above; however, it involves analytic considera-tions. From an algebraic point of view there seems to be no natural splitting ofthe sequence (1.1), which is no doubt why none was found in [CH2]. In section 4I explain very briefly the claim above that for a complete curve our distinguishedsplitting is a splitting of real Lie groups; and section 5 gives the small changesneeded to treat the case of a general affine curve (possibly with automorphismsand units). Finally, in section 6 we sketch the proof of the basic Theorem 1.3.2. Differentials and divisors
Let Div X be the group of divisors on a curve X , and let K be the field ofrational functions on X . Then we have the homomorphism K × → Div X assigningto a rational function its divisor of zeros and poles: its image is the subgroup P of principal divisors and (as is very well-known) the quotient Div X/P is canonicallyidentified with
Pic X .Slightly less well-known is the fact that Pic ♭ X has a similar description. LetΩ ( X ) be the (additive) group of differentials of the third kind on X (that is,rational differentials with only simple poles), and let Ω Z ( X ) be the subgroup of ICARD GROUPS OF DIFFERENTIAL OPERATORS ON CURVES 3 differentials with integer residues at each pole. There is an obvious map(2.1) res : Ω Z ( X ) → Div X which assigns to a differential ω the divisor P x ∈ X (res x ω ) x : its kernel is Ω ( X ) .We have also the map dlog : K × → Ω Z ( X ) : we shall identify its image with P , sothat the map (2.1) restricts to the identity on P . Proposition 2.1.
The quotient Ω Z ( X ) /P can be canonically identified with Pic ♭ X . The identification is such that the diagram(2.2) 0 ✲ Ω ( X ) ✲ Ω Z ( X ) res ✲ Div X ✲ ✲ Ω ( X ) Id ❄ ✲ Pic ♭ X ❄ p ✲ Pic X ❄ ✲ X is affine. Let us first review the notion of a (necessarilyflat) connection on a line bundle L : roughly speaking it is a way of making D ( X )act on (sections of) L . To be precise, let D L be the ring of differential operatorson L : it contains O ( X ) as the subalgebra of operators of degree 0 , and we maydefine a connection on L to be an isomorphism ϕ : D ( X ) → D L such that therestriction of ϕ to O ( X ) is the identity. Let us look at the special case where L is the trivial bundle X × C , so that D L ≡ D ( X ) . Then a connection is just an au-tomorphism of D ( X ) which fixes O ( X ) . Since D ( X ) is generated by O ( X ) andits derivations, such an automorphism ϕ is determined by its action on derivations ∂ ; this action necessarily takes the form(2.3) ϕ ( ∂ ) = ∂ + h ω, ∂ i where ω ∈ Ω ( X ) and h , i is the natural pairing between 1-forms and vectorfields. In this way, connections on the trivial bundle are in 1-1 correspondence withregular 1-forms.For a general line bundle L , a connection is usually described in terms of locallydefined 1-forms as above, using local trivializations of L ; however, in our algebraicsituation, we can use the fact that L always has a rational trivialization to describea connection by a single rational differential, much as above. More precisely, letus fix a divisor D = P n x x in the class of L . Corresponding to D we have thefractional ideal of O ( X )(2.4) I D := { f ∈ K : ν x ( f ) ≥ − n x ∀ x ∈ X } (as usual ν x ( f ) is the order to which f vanishes (or minus the order of pole) at x ). Then I D is isomorphic to the O ( X )-module Γ L of sections of L ; indeed,choosing a divisor in the class of L is equivalent to choosing a (fractional) ideal I ⊂ K isomorphic to Γ L . We may now identify D L with the algebra D ( I D ) := { θ ∈ D ( K ) : θ.I D ⊆ I D } . Some authors include this in the definition of “third kind”; others call any rational differential“of the third kind”.
GEORGE WILSON
So a connection on L is an isomorphism ϕ : D ( X ) → D ( I D ) . This extendsuniquely to an automorphism ϕ of D ( K ) (restricting to the identity on K ), whichmust have the form (2.3), with ω ∈ Ω rat ( X ) now a rational differential. If wechange the choice of ideal I by a factor f ∈ K × , the corresponding differentialchanges by the gauge transformation ω ω + dlog f . Thus so far we have seenthat Pic ♭ X embeds into the space Ω rat ( X ) / dlog K × . To complete the explanationof Proposition 2.1, we have only to see what is the image of this embedding; thatis, which rational differentials give rise to automorphisms of K that map D ( X )onto D ( I D ) . Proposition 2.2.
Let ω be a rational differential, ϕ the corresponding automor-phism of D ( K ) . Then ϕ maps D ( X ) onto D ( I D ) if and only if (i) ω ∈ Ω Z ( X ) and (ii) res ω = D .Proof. Note first that if ω and ω are two differentials such that the correspondingautomorphisms ϕ and ϕ both map D ( X ) onto D ( I D ) , then ϕ − ϕ restricts toan automorphism of D ( X ) : it follows that ω − ω is a regular differential on X .Thus it is enough to find just one automorphism ϕ which maps D ( X ) onto D ( I D ) ,and such that the corresponding differential ω has the principal parts specified bythe properties (i) and (ii) in Proposition 2.2. For this, let I ∗ D be the fractionalideal inverse to I D : it corresponds to the divisor − D . Since I D I ∗ D = O ( X ) ,we can choose α i ∈ I D and β i ∈ I ∗ D such that P α i β i = 1 . We claim that thedifferential ω := P α i dβ i has the required properties. Indeed, the automorphism ϕ corresponding to ω acts on derivations ∂ of K by ϕ ( ∂ ) = ∂ + h X α i dβ i , ∂ i = ∂ + X α i h dβ i , ∂ i = ∂ + X α i ∂ ( β i )= X α i ∂β i , where the last step used that P α i β i = 1 . If now ∂ is a derivation of O ( X ) , then( α i ∂β i ) .I D ⊆ ( α i ∂ ) . O ( X ) ⊆ α i O ( X ) ⊆ I D ;that is, ϕ ( ∂ ) ∈ D ( I D ) , whence ϕ maps D ( X ) to D ( I D ) . Similarly, ϕ − maps D ( I D ) to D ( X ) , so ϕ does indeed give an isomorphism between these two rings.It remains to check that ω has the properties (i) and (ii). Fix a point x in thesupport of D , and let z be a local parameter near x . Then near x the α i and β i have the form α i = a i z − n x + . . . , β i = b i z n x + . . . , where a i and b i are constants and the . . . denote higher order terms. Thus dβ i = n x b i z n x − dz + . . . , so ω := X α i dβ i = ( X a i b i ) n x z − dz + . . . . But since P α i β i = 1 , we have P a i b i = 1 ; it follows that ω has a simple pole at x with residue n x . That finishes the proof of Proposition 2.2. (cid:3) ICARD GROUPS OF DIFFERENTIAL OPERATORS ON CURVES 5 Splitting
Let us return to the diagram (2.2). The bottom map p , which we want toshow is split, is obtained from the top map res by dividing out by the (common)subgroup P of principal divisors. Thus it is enough to construct a splitting s : Div X → Ω Z ( X )of the top map which extends the identity map on P , for this will then descend tothe quotient to give a splitting of the bottom map.That is almost trivial, but not quite, since P is not a direct factor in Div X (thequotient Pic X has elements of finite order, while Div X is a free abelian group).However, we can consider the map of larger groupsres : Ω ( X ) → (Div X ) ⊗ Z C . These are now vector spaces, so we can certainly choose a C -linear splitting s with s ( p ⊗
1) = p for p ∈ P . The very fact that s is a splitting implies that it mapsDiv X ⊗ Z ( X ) , so we are finished.It is more satisfactory to describe a “natural” splitting of our sequence. Let usconsider first the case of a complete curve X : in that case we interpret Div X tobe group of divisors of degree zero on X . To define s , for each such divisor D we have to choose a differential with principal parts as prescribed by (i) and (ii) inProposition 2.2. There are several ways to normalize a differential with prescribedprincipal parts: the one we need is to make all its periods pure imaginary . It isclear that the resulting map s is additive; further, if D is the divisor of a rationalfunction f , then we have s ( D ) = dlog( f ) ; that is, s extends the identity map onthe group P of principal divisors, so again we are finished.If X is obtained from a complete curve Σ by removing a single point, thesituation is equally good. Indeed, if D ∈ Div X , then D has a unique extensionto a divisor D of degree zero on Σ ; if we take the above normalized differential s ( D ) and restrict it to X , we again get a distinguished splitting of the sequence(1.2).To extend this construction to the general case, when X is obtained by removingseveral points from Σ , we need to choose some way of extending divisors on X to divisors of degree zero on Σ . For example, we could single out one of thepoints “at infinity” and let the others have multiplicity 0 in the extension; or, moredemocratically, we could let the extended divisor have the same multiplicity at eachof the points at infinity. I leave the choice to the reader.4. The complete case
Let us return to the case where X is complete, and explain the claim thatin that case we have a splitting of (finite dimensional) real Lie groups. We takean analytic point of view. Recall that (as for any complex manifold) there is acanonical identification Pic ♭ X ≃ H ( X, C × ) . From this point of view (1.2) comesfrom the cohomology sequence of the exact sequence of analytic sheaves0 → C × → O × dlog −−−→ Ω → . These sheaves could be interpreted algebraically, but then the map dlog would not besurjective.
GEORGE WILSON
The map π : Ω Z → Pic ♭ X also has a very simple description from this point ofview: if we identify Pic ♭ X ≃ H ( X, C × ) ≃ Hom( H ( X, Z ) , C × ) , then π sends a differential ω to the homomorphism π ( ω ) : [ c ] exp Z c ω , where [ c ] is the homology class of a 1-cycle c . Clearly, ω is normalized (to haveimaginary periods) if and only if π ( ω ) takes values in the unit circle S ⊂ C × .Now, as a real Lie group we have the polar decomposition C × ≃ R × S , inwhich a pair ( λ, e iθ ) ∈ R × S corresponds to the complex number e λ + iθ . Ourdecomposition of Pic ♭ X is just the product of 2 g copies of this one.5. The general affine case
We have assumed so far that O ( X ) has no nontrivial units, and that X has nonontrivial automorphisms; however, neither of these assumptions is essential. In thegeneral case, let U be the group of units in O ( X ) . If u ∈ U , then dlog u ∈ Ω ( X ) ;set Ω := Ω ( X ) / dlog U .
Recall that
Pic ♭ X is the group of isomorphism classes (that is, gauge equivalenceclasses) of lines bundles with connection. Each u ∈ U gives an isomorphism ofconnections on the trivial bundle, changing the corresponding differential ω by thegauge transformation ω ω + dlog u ; thus the space of isomorphism classes ofconnections on the trivial bundle is Ω , so in the exact sequence (1.2) we have toreplace Ω ( X ) by Ω . Similarly, in the diagram (2.2), we have to change Ω ( X )to Ω , and also Ω Z ( X ) has to be replaced by the quotient Ω Z ( X ) / dlog U . Themain part of our discussion is then unaffected by the presence of U .The automorphism group Aut X equally easy to deal with, but more interesting,since it gives us extra elements of Pic D . This group acts compatibly on all thegroups in the split exact sequence (1.2), so we get a split exact sequence(5.1) 0 → Ω → Pic ♭ X ⋊ Aut X → Pic X ⋊ Aut X → . Further, there is a natural inclusion(5.2) χ : Pic ♭ X ⋊ Aut X → Pic D generalizing (1.3): Cannings and Holland show that it is an isomorphism. Insertingthis isomorphism into (5.1), we get the exact sequence of [CH2], Theorem 1.15. Remark.
It follows from Theorem 1.3 that if the group
Aut X is trivial, then Pic D is abelian. On the other hand, a nontrivial automorphism of X cannot act triviallyon Ω (because it does not act trivially on the subgroup d O ( X ) ). So from theexact sequence (5.1) we get the following curious fact: Pic D is abelian if and onlyif Aut X is trivial. At this point we have to exclude the case where X is isomorphic to the affine line. ICARD GROUPS OF DIFFERENTIAL OPERATORS ON CURVES 7 Outline of proof of Theorem 1.3
We first translate Theorem 1.3 into the language of bimodules used in [CH2].Recall that any autoequivalence T of the category of left D -modules is given bytensoring with the invertible D -bimodule T ( D ) (that is the case for any algebra;see, for example [B], p. 60 et seq.). Given a line bundle L with connection, choosean ideal I ⊆ O isomorphic to Γ L ; then as in section 2, the connection can beregarded as an isomorphism ϕ : D → D ( I ) , and the corresponding bimodule is I ⊗ O D = I D with the obvious right D -module structure and the left D -modulestructure defined via ϕ . Note that the algebra D ( I ) = I D I ∗ can be identified withthe endomorphism ring of the right ideal I D ⊆ D . Now (as for any Noetheriandomain D ) if we are given an invertible D -bimodule; we may consider it first justas a right D -module; it can then be embedded as a right ideal M in D , and thestructure of left D -module is given by some isomorphism ϕ : D →
End D M . SoTheorem 1.3 amounts to the claim that (in our case) we can always choose M tobe of the form I D , and furthermore that the isomorphism ϕ then restricts to theidentity map on O . It is in this form that the theorem is proved in [CH2].In broad outline, the proof goes as follows. By [St], Lemma 4.2, we may assumethat M is fat , that is, M ∩O 6 = 0 . The main result of [CH1] is that the assignment M V := M. O defines a bijection between the fat right ideals in D and certainsubspaces V ⊆ O (called “primary decomposable”); and furthermore that End D M then gets identified with the algebra D ( V ) := { D ∈ D ( K ) : D.V ⊆ V } . Thus themap defining the left D -module structure on M can be regarded as an isomorphism ϕ : D → D ( V ) . Concerning V , we need only know that the subalgebra D ( V ) := D ( V ) ∩ K is contained in O , and that the inclusion D ( V ) ⊆ O is a birationalisomorphism. Now we use the main result of [ML], which states that O is the unique maximal abelian ad-nilpotent (mad) subalgebra of D . Since D ( V ) is amad subalgebra of D ( V ) , that implies that ϕ must map O isomorphically onto D ( V ) , so that we have a birational isomorphism O → D ( V ) ⊆ O . Because X issmooth, this must be a genuine isomorphism, so under our assumption that Aut X is trivial, it must be the identity. Thus D ( V ) = O , and V is an ideal I of O .Under the bijection mentioned above, the fat right ideal of D corresponding to I is I D . That completes the proof. Acknowledgments . This work was completed during a visit to Cornell University; it is a pleasure tothank Cornell Mathematics Department, and especially Yuri Berest, for their hospitality. The supportof NSF grant DMS 09-01570 is gratefully acknowledged.
References [B] H. Bass.
Algebraic K-theory , Benjamin, New York-Amsterdam, 1968.[CH1] R. C. Cannings and M. P. Holland,
Right ideals of rings of differential operators , J. Algebra (1994), 116–141.[CH2] R. C. Cannings and M. P. Holland,
Etale covers, bimodules and differential operators ,Math. Z. (1994), 179–194.[M] W. Messing,
The universal extension of an abelian variety by a vector group , SymposiaMathematica, Vol. XI (Convegno di Geometria, INDAM, Rome) 1972, 359–372.[ML] L. Makar-Limanov,
Rings of differential operators on algebraic curves , Bull. London Math.Soc. (1989), 538–540.[St] J. T. Stafford, Endomorphisms of right ideals of the Weyl algebra , Trans. Amer. Math. Soc. (1987), 623–639.
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