RINGS OF DIFFERENTIAL OPERATORS OVER RATIONAL AFFINE

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RINGS OF DIFFERENTIAL OPERATORS OVER RATIONAL AFFINE
Bull. Soc. math. France,
118, , p. 193–209.
RINGS OF DIFFERENTIAL OPERATORS OVER
RATIONAL AFFINE CURVES
BY
Gail LETZTER, Leonid MAKAR–LIMANOV (*)
RÉSUMÉ. — Soit X une courbe algébrique irréductible sur C dont la normalisée est la
droite affine et telle sur le morphisme de normalisation est injectif. Soit D(X) l’anneau
des opérateurs différentiels sur X. Nous étudions un invariant pour l’anneau D(X)
des opérateurs différentiels sur X, noté codim D(X). En particulier, nous montrons
que D(X) ∼
= D(Y ) implique codim D(X) = codim D(Y ). Cela permet de distinguer
dans certains cas les anneaux d’opérateurs différentiels de courbes non-isomorphes.
En outre, nous décrivons les sous-algèbres ad-nilpotentes maximales de D(X). Nous
montrons que si B est une sous-algèbre ad-nilpotente maximales de D(X), alors B est
un sous-anneau de type fini d’un C[b] où b désigne un élément du corps des fractions
de D(X) ; de plus, la clôture intégrale de B est C[b].
ABSTRACT. — Let X be an irreducible algebraic curve over the complex numbers such that its normalization is the affine line, and the normalization map is injective. Let D(X) denote its ring of differential operators. We find an invariant for
D(X) denoted as codim D(X). In particular, we show that D(X) ∼
= D(Y ) implies
codim D(X) = codim D(Y ). This allows us to distinguish certain rings of differential
operators of non-isomorphic curves. We also describe the maximal ad-nilpotent subalgebras of D(X). We show that if B is a maximal ad-nilpotent subalgebra of D(X), then
B is a finitely generated subring of C[b] for some element b of the quotient field of D(X)
and the integral closure of B is C[b].
1. Introduction
Let X and Y be irreducible algebraic curves over the complex numbers, C. Let D(X) and D(Y ) denote their ring of differential operators,
respectively. (For definition see [9]). This paper is motivated by the following open question.† Does D(X) ∼
= D(Y ) imply that X ∼
= Y ? Let
(*) Supported by a grand from the National Science Foundation. Texte reçu le 29 juin
, révisé le 11 mai .
G. LETZTER, L. MAKAR–LIMANOV, Dept. of. Mathematics, Wayne State University,
Detroit, Michigan 48202, USA
† G. LETZTER has now found nonisomorphic curves X and Y with isomorphic rings of
differential operators (see [4]).
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0037–9484//193/$ 5.00
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G. LETZTER, L. MAKAR-LIMANOV
denote the normalization of X. MAKAR–LIMANOV [5] shows that the
X
set of ad-nilpotent elements N (X) is exactly O(X) whenever O(X) is
not a subring of a polynomial ring in one variable over C. He thus answers the question affirmatively for these curves. Let A1 denote the affine
line. PERKINS [8] extends this result showing that D(X) ∼
= D(Y ) im1
1
∼
plies X = Y whenever X = A , or X = A but the normalization map
→ X is not injective. Thus, in the paper, we are interested in curves
π:X
∼
→ X is injective. STAFFORD [10] shows
X such that X
= A1 and π : X
the conjecture holds the following two examples of such curves : when X
is the affine line A1 , or when X is the cubic cusp y 2 = x3 .
For the remainder of the paper, assume that X is a curve such that
its normalization is isomorphic to the affine line A1 with an injective
normalization map. We may therefore assume that the coordinate ring
of X, denoted O(X), is a subring of a polynomial ring in one variable C[x]
is equal to C[x].
such that the integral closure of O(X), written O(X),
Furthermore D(X) is a subring of C(x)[∂] where [∂, x] = 1. Here ∂
is just ∂/∂x and the element fn (x)∂ n + · · · + f0 (x) of D(X) sends
g(x) ∈ O(X) to fn (x)g (n) (x) + · · · + f0 (x)g(x) where g (n) (x) denotes
the nth derivative of g(x).
PERKINS studies rings that satisfy these conditions in [8]. He shows
that in many cases, D(X) contains maximal commutative ad-nilpotent
subalgebras not isomorphic to O(X). Thus, for these curves, the set N (X)
of ad-nilpotent elements does not determine O(X).
In this paper, we obtain an invariant for D(X) and a nice description of
the maximal ad-nilpotent subalgebras of D(X). Set T = C(x)[∂] and set
∂-deg w = n where w = fn (x)∂ n + · · · + f0 (x) is an element of T . Define a
filtration on T by Ti = {w ∈ T | ∂-deg w ≤ i} and hence on any subring
R of T by Ri = R ∩ Ti . (Note that this is the same filtration on D(X) as
the one defined by the order of the differential
operator.) We may form
the associated graded ring ∂-gr R =
Ri /Ri−1 . We define codim R to
be equal to dimC ∂-gr C[x, ∂]/∂-gr R for those subrings R of T such that
∂-gr R ⊂ ∂-gr C[x, ∂].
Now assume that both X and Y are affine curves with normalization
equal to the affine line and injective normalization map. By [9], both
∂-gr D(X) and ∂-gr D(Y ) are subrings of ∂-gr C[x, ∂] and codim D(X)
and codim D(Y ) are finite numbers.
Our main results are :
THEOREM. — Suppose that B is a maximal ad-nilpotent subalgebra
of D(X). Then there exists elements x and ∂ in the quotient field of
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C(x)[∂] such that [∂ , x ] = 1, D(X) is a subring of C(x )[∂ ], D(X) ∩
C(x ) = B, and the integral closure of B is C[x ]. Furthermore, ∂ -gr D(X)
is a subring of ∂ -gr C[x , ∂ ] and
dimC ∂ -gr C[x , ∂ ]/∂ -gr D(X) = codim D(X).
COROLLARY. — If D(X) ∼
= D(Y ), then codim D(X) = codim D(Y ).
This result permits one to distinguish many rings of differential operators. For example, set O(Xn ) = C + xn C[x]. Then it will follow from the
COROLLARY, that D(Xn ) ∼
= D(Xm ) implies that n = m.
2. Graded Algebras of D(X)
In this section, α and β are nonnegative real numbers with α + β > 0.
Define valuations Vα,β on C(x)[∂] as follows. Set
Vα,β wn (x)∂ n + wn−1 (x)∂ n−1 + · · · + w0 (x)
equal to max{αdm + βm | n ≥ m ≥ 0} where dm = deg(wn (x)).
This extends the notion of valuations introduced by DIXMIER in [2] for
the Weyl algebra. For each valuation Vα,β we may define a filtration of
C(x)[∂], and hence on any subring R of C(x)[∂] as follows. Recall that
R ∩ Ti . We
T = C(x)[∂]. Set Ti = {z ∈ T | Vα,β (z) ≤ i} and Ri = may then define the associated graded algebra grα,β R =
Ri /Ri−1 .
Now the commutator [xi ∂ j , xk ∂ ] = (kj − i)xi+k−1 ∂ j+−1 + terms with
x-degree less than i + k − 1 and ∂-degree less than j + − 1. Therefore
Vα,β ([xi ∂ j , xk ∂ ]) < α(i + k) + β( + j). It follows that grα,β (C(x)[∂]) is
a commutative algebra.
Note that when α = 0 and β is positive, then the filtration defined
by V0,β on D(X) is the same filtration on D(X) as the one defined by
∂-deg in the introduction. We will write ∂-gr D(X) for gr0,β D(X) and
∂-deg for V0,β . Similarly, when β = 0 and α is positive the graded algebra
determined by Vα,0 is the same as x-gr R determined by x-deg defined in
[8].
is just the first Weyl
Set grα,β x = x and grα,β ∂ = y. Since D(X)
algebra, A1 , we have that ∂-gr D(X) = C[x, y] where ∂-gr x = x and
∂-gr ∂ = y. By [9, Proposition 3.11], it follows that ∂-gr D(X) is a subring
of C[x, y] and by [8, Lemma 2.3], x-gr D(X) is also a subring of C[x, y].
In the following lemma, we extend this to other gradings.
LEMMA 2.1. — Let R be a subring of C(x)[∂] such that ∂-gr R ⊂ C[x, y].
Then the graded algebra grα,β R is a subring of C[x, y].
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Proof. — If α = 0 then grα,β R = ∂-gr R. So we may assume
that α is positive. Let w be a typical element of D(X). Write w =
gm (x)∂ m + · · · + g0 (x) where gi (x) ∈ C(x) for 0 ≤ i ≤ m. Set degree
of gi (x) equal to di for 0 ≤ i ≤ m. Since ∂-gr R ⊂ C[x, y], it follows that
gm (x) ⊂ C[x] and thus dm ≥ 0. Set N = Vα,β (w). By the definition
of Vα,β , it follows
that N = max{di α + iβ | 0 ≤ i ≤ m}. Hence
ds s
ds s
grα,β (w) =
0≤s≤m γs x y where γs = 0 if Vα,β (x ∂ ) < N , and
γs xds is the leading term of gs (x) if Vα,β (xds ∂ s ) = N . We need to show
that whenever γs = 0, we have xds y s ∈ C[x, y]. In particular, since
0 ≤ s ≤ m, we need to show that ds ≥ 0 whenever γs = 0. Now
N = Vα,β (w) ≥ Vα,β (gm (x)∂ m ) = dm α+mβ. Hence ds α+sβ ≥ dm α+mβ.
Recall that m ≥ s, dm ≥ 0, and that α is positive. It follows that
ds ≥ dm ≥ 0. The lemma now follows.
Define a linear map φ : C(x)[∂] → C[x, ∂] as follows. Suppose that
w = gm (x)∂ m + · · · + g0 (x) is an element of C(x)[∂]. For each i such
that 1 ≤ i ≤ m, there exists a unique polynomial fi (x) such that
deg(gi (x) − fi (x)) < 0. Set
φ(w) = fm (x)∂ m + · · · + f0 (x).
Now consider two rational functions g1 (x) and g2 (x) such that φ(g1 (x)) =
f1 (x) and φ(g2 (x)) = f2 (x). Then clearly
deg λ1 g1 (x) + λ2 g2 (x) − (λ1 f1 (x) + λ2 f2 (x) < 0 and
φ(λ1 g1 (x) + λ2 g2 (x)) = λ1 f1 (x) + λ2 f2 (x).
It follows that φ is a well defined linear map from C(x)[∂] to C[x, ∂].
COROLLARY 2.2. — Let R be a subring of C(x)[∂] such that ∂-gr R ⊂
C[x, y]. If w is an element of R, then grα,β φ(w) = grα,β (w).
Proof. — This is clear since grα,β (w − φ(w)) does not contain any
monomials xds y s with ds ≥ 0.
Remark 2.3. — Note that φ(R) is a linear subspace of the first Weyl
algebra A1 = C[x, ∂], but, generally speaking, is not a subalgebra.
Nevertheless α, β gradings are defined on φ(R) and grα,β φ(R) = grα,β R.
Now
dimC C[x, y]/∂-gr D(X) < ∞ ([9, 3.12]) and
dimC C[x, y]/x-gr D(X) < ∞ ([8, Lemma 2.5])
In the next proposition, we will show that these two finite numbers are
equal. We will later show that this codimension is an invariant for D(X).
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PROPOSITION 2.4. — Suppose that R is a subring of C(x)[∂] such that
∂-gr R ⊂ C[x, y] and dimC C[x, y]/∂-gr R < ∞. Then grα,β R is a subring
of C[x, y] and dimC C[x, y]/ grα,β R = dimC C[x, y]/∂-gr R.
Using COROLLARY 2.2 and REMARK 2.3, we may replace R by φ(R)
and prove the following.
PROPOSITION 2.4 . — Suppose that R is a linear subspace of the Weyl
algebra C[x, ∂] and that dimC C[x, y]/∂-gr R < ∞. Then grα,β R is a linear subspace of C[x, y] and dimC C[x, y]/ grα,β R = dimC C[x, y]/∂-gr R .
Before proving PROPOSITION 2.4 , we need some additional notation
and lemmas. Set, for i ≥ 0,
Ei = C[x] + C[x]y + · · · + C[x]y i
Bi = {w ∈ R | ∂-gr w ∈ Ei }.
Note that
i≥0
Bi = R . Set E =
i≥0
and
Ei = C[x, y].
In PROPOSITION 2.4 , we assume that dimC E/∂-gr R < ∞. Since
∂-gr w ∈ Ei if and only if w ∈ Bi for any w ∈ R , it follows that
dimC Ei /∂-gr Bi < ∞ for all i ≥ 0, and that there exists an N > 0
such that dimC Ei /∂-gr Bi = dimC E/∂-gr R for all i ≥ N . Hence for
each i ≥ 0, there exists an integer Mi ≥ −1 such that for each m > Mi
there exists a monic polynomial pi,m (x) of degree m in C[x] such that
pi,m (x)y i is an element of ∂-gr Bi . Furthermore, for i ≥ N , we may assume
that Mi = −1.
We have the following lemmas.
LEMMA 2.5.
Suppose that R satisfies
the conditions of PROPOSITION 2.4 . Suppose
i+1
d
+ · · · + f0 (x) is an element of Bi+1 where
that w = αx + fi+1 (x) ∂
α ∈ C − {0} and deg fi+1 (x) < d. Then there exists a w ∈ Bi+1 such that
w = (αxd + gi+1 (x))∂ i+1 + gi (x)∂ i + · · · + g0 (x) and deg gk (x) ≤ Mk for
each k such that i + 1 ≥ k ≥ 0.
Proof. — Let us use the following induction. Set w−1 = w. Suppose
that
wk = axd + gi+1 (x) ∂ i+1 + · · · + gi−k (x)∂ i−k
+ fi−k−1 (x)∂ i−k−1 + · · · + f0 (x),
where deg gj (x) ≤ Mj , is defined. There exists b ∈ Bi−k−1 such that
∂-gr b = (fi−k−1 − gi−k−1 )y i−k−1 where deg gi−k−1 ≤ Mi−k−1 by the
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paragraph preceding the lemma. So we can define wk+1 as wk − b, and w
as wi .
Let Pi be the set of positive integers m such that there exists a nonzero
polynomial qi,m (x) of degree m in C[x] with qi,m (x)y i ∈ ∂-gr R . Note that
if n is an integer such that n > Mi , then n ∈ Pi . By LEMMA 2.5, it now
follows that for each m ∈ Pi there exists a monic polynomial pi,m (x) of
degree m ∈ C[x] such that bi,m = pi,m (x)∂ i + gi−1 (x)∂ i−1 + · · · + g0 (x) is
an element of Bi with deg gk (x) ≤ Mk for i − 1 ≥ k ≥ 0. Furthermore, for
i ≥ N , we may assume that pi,m (x) = xm . Note that the set
bi,m | i ≥ 0 and m ∈ Pi
forms a basis for R over C, and
pi,m (x)y i | i ≥ 0 and m ∈ Pi
forms a basis for ∂-gr R over C. Thus if w ∈ R , with ∂-gr w = f (x)y i ,
then for i > k ≥ 0, there exist fk (x) ∈ C[x] with deg fk (x) ≤ Mk , such
that f (x)∂ i + fi−1 (x)∂ i−1 + · · · + f0 (x) is an element of R .
Set M = max{Mk | N > k ≥ 0}. Then we may assume that
bi,m = pi,m (x)∂ i +wi,m with ∂-deg wi,m < min(i, N ) and x-deg wi,m ≤ M .
LEMMA 2.6.
Assume that R satisfies the conditions of PROPOSITION 2.4 . For each
m ≥ 0, there exists a positive integer Sm such that for all i ≥ Sm , there is
an element ci,m in R of the form pi,m (x)∂ i + ti,m with deg pi,m (x) = m
and ∂-deg ti,m < i and x-deg ti,m ≤ m. If m > M we may set Sm = 0.
Proof. — If m > M , then we may take ci,m = bi,m . So we may assume
that m ≤ M . Consider the subset {bi,m = pi,m (x)∂ i + wi,m | i ≥ 0} of R .
Let EM,N = {r ∈ E | x-deg r ≤ M and y-deg r ≤ N }, and let V be the
vector space spanned by {wi,m | i ≥ 0}. Set W = {x-gr w | w ∈ V } ∩ E.
Note that W is a subspace of EM,N . It is clear that EM,N and hence W
is a finite dimensional subspace of E. So there is an Sm > 0 such that W
is spanned by a subset of
x-gr w | w is in the span of the set wi,m | Sm ≥ i ≥ 0 ·
It follows that for i > Sm , there exist complex numbers αk,m for
Sm ≥ k ≥ 0 such that
Sm
αk,m wk,m < 0 and
x-deg wi,m −
k=0
Sm
∂-deg wi,m −
αk,m wk,m < 0.
k=0
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We may now set ci,m = bi,m −
Sm
k=0
199
αk,m bk,m .
The next corollary follows immediately from LEMMA 2.6.
COROLLARY 2.7. — We have dimC C[x, y]/x-gr R < ∞.
LEMMA 2.8.
Let W be a linear subspace of A1 . Then dimC W = dimC grα,β W .
Proof. — Suppose that W is a vector space and that
{Wi | i is an integer }
is a filtration
for W such that the vector spaces Wi = 0 for i < 0 and
and
Wi /Wi−1 are isomorphic as vector
W = i≥0 Wi . Then clearly W spaces. Hence dimC W = dimC Wi /Wi−1 . In particular if W is a linear
subspace of A1 , then dimC W = dimC grα,β W .
We are now ready to prove PROPOSITION 2.4 .
Proof of PROPOSITION 2.4 . — Note that R is a linear subspace of
C[x, ∂]. Hence, it follows from the definition of grα,β R that grα,β R is a
linear subspace of grα,β C[x, ∂]. Thus we only need to prove the statement
about dimensions.
Set Vn = {xi y j | αi+βj
≤ n} for all n ≥ 0. Note that each Vn has finite
Wn = {w ∈ R | grα,β w ∈ Vn }.
dimension and that n≥0 Vn = C[x, y]. Set
Since grα,β R ⊂ C[x, y], we have that n≥0 Wn = R . Suppose that
w ∈ Wn . We can write w = p(x)∂ k + c for some p(x) ∈ C[x] and k ≥ 0
such that ∂-deg(c) < k and α deg p(x) + βk ≤ n. So ∂-gr w = p(x)y k is
also in Vn . Thus ∂-gr Wn ⊂ Vn for all n ≥ 0.
Set L = αM +βN . We will show that ∂-gr Wn = ∂-gr R ∩Vn for all n ≥
L. Since ∂-gr Wn ⊂ Vn , it is clear that ∂-gr Wn ⊂ ∂-gr R ∩ Vn . Suppose
∂-gr w = p(x)y j is an element of ∂-gr R ∩ Vn . So α deg p(x) + βj ≤ n. By
LEMMA 2.5, we may find in R an element w = p(x)∂ j + gN (x)∂ N + · · · +
g0 (x) and deg gk (x) ≤ Mk for each k such that N ≥ k ≥ 0. Now
Vα,β (gN (x)∂ N + · · · + g0 (x)) ≤ αM + βN = L.
Hence Vα,β (w) ≤ max{α deg p(x) + βj, L}. If α deg p(x) + βj > L, then
Vα,β (w) = α deg p(x) + βj ≤ n since p(x)y j is an element of Vn . Hence
w ∈ Wn . If α deg p(x) + βj ≤ L, then Vα,β (w) ≤ L ≤ n, hence again
w ∈ Wn . Therefore ∂-gr Wn = ∂-gr R ∩ Vn for all n ≥ L.
Since Wn is a linear subspace of C[x, ∂], by LEMMA 2.8, we have that
dimC Wn = dimC ∂-gr Wn and
dimC Wn = dimC grα,β Wn .
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Furthermore, for all n ≥ L, we have that dimC ∂-gr R ∩ Vn = dimC Wn =
dimC grα,β Wn . Since dimC Vn is finite, it follows that dimC Vn /∂-gr R ∩
Vn = dimC Vn / grα,β Wn for all n ≥ L. Clearly
dimC C[x, y]/∂-gr R = lim dimC Vn /∂-gr R ∩ Vn
n→∞
dimC C[x, y]/ grα,β R = lim dimC
n→∞
and
Vn / grα,β Wn .
Therefore dimC C[x, y]/∂-gr R = dimC C[x, y]/ grα,β R .
By COROLLARY 2.7, we have that dimC C[x, y]/x-gr R < ∞. So we may
apply the first part of the proof with x replaced by ∂ and vice versa to
show that dimC C[x, y]/x-gr R = dimC C[x, y]/ grα,β R which completes
the proof of PROPOSITION 2.4 and therefore of PROPOSITION 2.4.
Recall that codim R is defined to be dimC C[x, y]/∂-gr R.
PROPOSITION 2.4 implies that codim R = dimC C[x, y]/ grα,β R for any
two nonnegative not both zero real numbers α and β. We will eventually
show that codim R is an invariant of R.
3. Ad-Nilpotent subalgebras of D(X)
∼ D(Y ). Then D(X) contains a maximal commuSuppose that D(X) =
tative ad-nilpotent subalgebra isomorphic to O(Y ). So it is interesting to
understand the maximal commutative ad-nilpotent subalgebras of D(X).
Let D denote the quotient field of the first Weyl algebra, A1 . In this
section, we show that if B is a maximal commutative ad-nilpotent subalgebra of D(X), then there exists an element b ∈ D such that B is a
subring of C[b].
LEMMA 3.1. — Suppose that R is a subalgebra of D so that the quotient
ring of R is D, and that u is an element of D − C that acts ad-nilpotently
on R. Then there exists a v ∈ D such that [u, v] = 1. Furthermore, for any
v ∈ D such that [u, v] = 1, we have R ⊂ CD (u)[v] where CD (u) denotes
the centralizer of u in D.
Proof. — Define
R0 = CD (u) and Ri = {z ∈ D | [z, u] ∈ Ri−1 }.
Now R ⊂ i≥0 Ri since u acts ad-nilpotently on R. Let a be a nonzero
element of R1 − R0 . (Note that R1 − R0 is nonempty since u ∈ C and C
is the center of R.) Then 0 = [u, a] = b ∈ R0 . So [u, b−1 a] = b−1 [u, a] = 1.
Set v = b−1 a.
Clearly R0 ⊂ CD (u). We will show by induction on i that
Ri ⊂ CD (u)v i + · · · + CD (u) for all i ≥ 0.
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i−1
+ · · · + CD (u) and choose z ∈ Ri . Then
Assume that Ri−1 ⊂ CD (u)v
[z, u] ∈ Ri−1 , hence [z, u] = 0≤m≤i−1 fm (u)v m . Then
z−
0≤m≤i−1
Hence z −
0≤m≤i−1
v m+1
, u = 0.
fm (u)
m+1
fm (u)v m+1 /(m + 1) ∈ CD (u). Therefore
z ∈ CD (u)v i + · · · + CD (u).
We may define the graded algebra v-gr CD (u)[v] by setting v-gr a =
ui wi where a = ui v i + · · · + u0 is an element of CD (u)[v] with uk ∈ CD (u)
for i ≥ k ≥ 0.
We will show that CD (u) is in fact a rational function field in one
variable.
The next lemma is well known. See for example [3, Corollary 3.2].
LEMMA 3.2. — If f ∈ D − C then CD (f ) is commutative.
LEMMA 3.3. — If u ∈ D acts ad-nilpotently on R, where R is a
subalgebra of D such that the quotient ring of R is D, then there exists
z ∈ D such that CD (u) is isomorphic to a rational function field C(z).
Proof. — Let us call an element a ∈ D ad-nilpotent if it acts adnilpotently on some subalgebra R(a) of D such that the quotient ring
of R(a) is D. By LEMMA 3.1, there exists an element v ∈ D such that
[v, u] = 1 and D = CD (u)(v).
We will first assume that there exists an ad-nilpotent element a of D
with v-deg a = 0. Now for each element c ∈ CD (u), there exists elements
c1 = c1 (c) and c2 = c2 (c) in R(a) such that c = c1 c−1
2 . It is clear that
v-gr a acts nilpotently by Poisson bracket action on v-gr c1 and v-gr c2 . Let
v-gr a = a0 wn , v-gr c1 = c1,0 wm , and v-gr c2 = c2,0 wm . (Since c ∈ CD (u),
it is clear that v-deg c1 = v-deg c2 .)
By the same arguments as in [5, Lemma 7], there exists an element b
in the algebraic closure of CD (u) such that c1,0 wm = (a0 wn )m/n p1 (b) and
c2,0 wm = (a0 wn )m/n p2 (b) where p1 (b) and p2 (b) are polynomials.
−1
−1
Since v-deg c = 0, we have that c = c1 c−1
.
2 = c1,0 c2,0 = p1 (b)(p2 (b))
Therefore CD (u) ⊂ C(b). By Luroth’s theorem, CD (u) is isomorphic to a
field of rational functions in one variable.
Now assume that v-deg a = 0 for all ad-nilpotent elements. Consider
the standard generators x and ∂ for D. These are ad-nilpotent elements
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of D since they act ad-nilpotently on C[x, ∂]. Therefore 1 = [∂, x] has
negative v-degree which is impossible.
4. Codim is an invariant of D(X)
In this section R = D(X) for a curve X satisfying the conditions of the
introduction. Suppose that u and v are elements of D with commutator
[v, u] = 1 such that D(X) ⊂ C(u)[v] and v-gr D(X) is a subring of the
polynomial ring in two generators, u = v-gr u and w = v-gr v. We may
define codimu,v D(X) as dimC C[u, w]/v-gr D(X). In this section, we will
show that codimu,v D(X) = codim D(X). So codim does not depend on
the embedding of D(X) inside of C(x)[∂].
Note that u-gr C[u, v] and v-gr C[u, v] are isomorphic polynomial rings.
We will identify these isomorphic rings and thus write u-gr u = v-gr u = u
and u-gr v = v-gr v = w.
LEMMA 4.1. — Suppose that R ⊂ C(u)[v] ⊂ D, where [v, u] = 1, such
that the quotient ring of R is D, the graded algebra v-gr R is a subset of
C[u, w], and codimu,v R is finite. Then there exist elements u and v of D
such that u-gr v = w and u-gr u = −u, the commutator [u , v ] is 1, and
the ring R is a subring of C(v )[u ]. Moreover, there is an isomorphism
from u -gr C[u , v ] to u-gr C[u, v] which restricts to an isomorphism from
the graded algebra u -gr R to u-gr R, and codimv ,u R = codimu,v R.
Proof. — Define subalgebras Ri of R for i ≥ 0 as follows :
Ri = z ∈ R | u-deg(z) ≤ 1 .
(The following argument is similar to [8, Theorem 2.7].) Now
for i ≥ 0.
u-gr f (v)ui , g(v) = u-gr −if (v)g (v)ui−1
Also u-gr R is a subset of C[u, w] by LEMMA 2.1. Hence, it is easy to
see that R0 is a maximal commutative ad-nilpotent subalgebra of R.
Furthermore the map which sends z to u-gr z is an isomorphism of R0
to u-gr R0 = u-gr R ∩ C[w]. By assumption, codimu,v R < ∞, hence
dimC C[w]/u-gr R0 < ∞. So the integral closure of u-gr R0 is C[w], and
thus the integral closure of R0 is C[v ] for some v ∈ D with u-gr v = w
and R0 = R ∩ C[v ] for some v ∈ D with u-gr v = w and R0 = R ∩ C[v ].
Note that u-gr p(v ) = p(w) for any polynomial p(t) ∈ C[t].
By LEMMA 3.3, CD (v ) is a rational function field in one variable. Let
us check that CD (v ) = C(v ). Let f ∈ CD (v ). Then u-deg f = 0, because
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otherwise [v , f ] = 0, and u-gr f = r(w) where r(w) ∈ C(w). Therefore
f = r(v ) + f1 where u-deg f1 < 0. But f1 ∈ CD (u) and can not have a
negative degree. Hence f1 is 0. Now, according to LEMMA 3.1, there exists
a u ∈ D such that [u , v ] = 1 and R ⊂ C(v )[u ].
Suppose that u-gr u = f (w)ui . Since u-gr v = w, we must have
u-gr[u , v ] = −if (w)ui−1 unless i = 0. If i = 0, then either [u , v ] =
0 or u-deg[u , v ] < −1. Since [u , v ] = 1, it follows that i = 0.
Hence −if (w)ui−1 must equal 1. Therefore i = 1 and f (w) = −1 and
u-gr u = −u.
Suppose that z is an element of R ⊂ C(v )[u ]. We may write
z = f (v )(u )j + e where u -deg e < j, and f (v ) is a polynomial,
and j ≥ 0. Since u-deg v = 0 and u-deg u = 1, we must have that
u-deg e < j and u-gr z = u-gr f (v )(u )j . Since u-gr f (v ) = f (w)
and u-gr u = −u, it follows that u-gr z = f (w)(−u)j . Hence the
isomorphism from u -gr C[u , v ] to u-gr C[u, v] which sends u -gr u to
u-gr u = −u and u -gr v to u-gr v = w restricts to an isomorphism
from u -gr R to u-gr R. Since codimu,v R is finite, by PROPOSITION 2.4, we
have that codimu,v R = dimC C[u, w]/u-gr R. It follow immediately that
codimu,v R = codimv ,u R.
For the next three lemmas, assume that R is a subring of C(u)[v] ⊂ D,
where u and v are elements of D whose commutator is 1, and that
v-gr R ⊂ C[u, w] with codimu,v R < ∞. Write R0 for the ad-nilpotent
subalgebra {z ∈ R | u-gr z = 0}. We may define valuations Vα,β and
corresponding graded algebras on R as in Section 1 using u and v instead
of x and ∂. For example, Vα,β (ui v j ) = αi + βj.
LEMMA 4.2. — Suppose that r is an ad-nilpotent element of R that is
not contained in C(u) and is not contained in R0 . Then there exist positive
integers n and m and complex numbers λ and γ such that u-gr r = (λu)n
and v-gr r = (γw)m . Furthermore, Vm,n (r) = mn.
Proof. — Since r is not an element of C(u) and is not an element
of R0 , it follows that u-deg r > 0 and v-deg r > 0. We will argue as in
[2, Lemma 8.7]. We may write
r=
σi,j ui v j + fk (u)v k + · · · + f0 (u)
i≥0,j≥0
where deg fj (u) < 0 for k ≥ j ≥ 0. Clearly, v-deg r > k. Let n be the
smallest nonnegative integer such that σj,0 = 0 for all j > n. Let m be
the smallest nonnegative integer such that σ0,k = 0 for all k > m. We
claim that σi,j = 0 for all pairs i, j such that mi + nj > mn.
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Assume the claim is false. Then there exist positive real numbers α
and β and a pair of positive integers i and j with σi,j = 0, such that
grα,β r = σi,j ui wj . Without loss of generality, σi,j = 1. First assume i ≥ j.
Now there exists a monic polynomial p(t) such that p(u) ∈ R. Since both
α and β are positive, we have that grα,β p(u) = ud where d = deg p(u).
Note that grα,β [r, p(u)] = djui−1+d wj−1 . Suppose that
grα,β adkr p(u) = αk uk(i−1)+d wk(j−1) .
Then
grα,β adk+1
p(u) =
r
αk k(i − 1) + d j − ik(j − 1) u(k+1)(i−1)+d w(k+1)(j−1) .
Now (k(i − 1) + d)j − ik(j − 1) = (i − j)k + dj > 0 for all k ≥ 0 since
i ≥ j. This contradicts the fact that r is ad-nilpotent.
Now assume that i < j. Consider a nonconstant element z ∈ R0 .
Recall that R0 sits inside a polynomial algebra C[v ] where v ∈ D where
u-gr v = w. So z = q(v ) for some nonconstant polynomial q(t). Since
both α and β are positive, it follows that grα,β z = wk where k = deg q(t).
The argument now follows as in the preceding paragraph.
We have shown that σi,j = 0 for all pairs of positive integers i and j
such mi + nj > nm. In particular, u-gr r = σn,0 un , and v-gr r = σ0,m wm ,
and Vm,n (r) = mn.
LEMMA 4.3. — Suppose that r is an ad-nilpotent element of R that
is not contained in C(u) and is not contained in R0 . Set n = u-deg r
and m = v-deg r. Then one of the following two statements hold where
λ, λ , γ, γ are elements of C, and i is an integer such that n ≥ i ≥ 0.
(1) If n ≥ m, then n is a multiple of m and
m
grm,n r = (λu)n/m + γw .
(2) If m > n, then m is a multiple of n and
n
grn,m r = λu + (γw)m/n .
Proof. — By LEMMA 4.2, both n and m are positive. So there exist
nonzero complex numbers σ1 and σ2 such that u-gr r = σ1 un and v-gr r =
σ2 wm . Now by LEMMA 2.1, grm,n R ⊂ C[u, w], and by PROPOSITION 2.4,
dimC C[u, w]/ grm,n R < ∞. Hence we may apply the arguments of [2,
Lemma 7.3] to the ad-nilpotent element r of R.
In the next lemma, we will show that codim R is independent of the
choice of generator for C(u).
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LEMMA 4.4. — Suppose that u1 and v1 are elements of D whose
commutator is 1 such that C(u) = C(u1 ), the ring R is a subring of
C(u1 )[v1 ] ⊂ D, and that v1 -gr R ⊂ C[u1 , w1 ] with codimu1 ,v1 R < ∞.
Then codimu,v R = codimu1 ,v1 R.
Proof. — Set B = R ∩ C(u) = R ∩ C[u]. Since C(u1 ) = C(u) and
v1 -gr R ⊂ C[u1 , w1 ], we have that B = R ∩ C(u1 ) = R ∩ C[u1 ]. By
assumption, both codimu,v R and codimu1 ,v1 R are finite. Hence both
dimC C[u]/B and dimC C[u1 ]/B are finite. Therefore the integral closure
of B in C(u) is C[u] and is also C[u1 ]. So C[u] = C[u1 ] and there exist
integers α and β such that u = αu1 + β. Since [v1 , u1 ] = 1, we have
that [αv − v1 , u] = 0. So v + g(u) = α−1 v1 for some g(u) ∈ C(u).
Set v2 = v + g(u). Note that [v2 , u] = 1 and R ⊂ C(u)[v2 ]. Now
f (u)v i = f (u)(v2 − g(u))i , hence v-gr R = v2 -gr R and codimu,v R =
codimu,v2 R. Without loss of generality, we may assume that v = v2 and
that v = α−1 v1 . The isomorphism of C[u, w] to C[u1 , w1 ] which sends u
to αu1 and w to α−1 w1 clearly induces an isomorphism from v-gr R to
u-gr R. The result now follows.
We are now ready to show that codim D(X) is an invariant of D(X).
THEOREM 4.5. — Suppose that X is an affine curve such that the
normalization of X is the affine line, with the normalization map π :
→ X injective. Then for any pair of elements u and v in D, such that
X
[v, u] = 1, the ring D(X) is a subring of C(u)[v], and v-gr D(X) is a
subring of the polynomial ring with generators v-gr u and v-gr v, we have
that codimu,v D(X) = codim D(X).
Proof. — Now D(X) is a subring of C(x)[∂] and codim D(X) =
codimx,∂ D(X). Assume that u and v are elements of D such that
[v, u] = 1, the ring D(X) is a subring of C(u)[v], and v-gr D(X) is a
subring of the polynomial ring C[u, w] where v-gr u = u and v-gr v = w.
Let r be a nonconstant ad-nilpotent element of D(X) contained inside
C(u). Set x-deg r = n and ∂-deg r = m. We will induct on t = m + n.
If m = 0, then r is an element of C(x) and the result now follows by
LEMMA 4.4.
If n = 0, then r is an element of {z ∈ D(X) | x-deg z = 0}, and the
result follows from LEMMA 4.1 and LEMMA 4.4. Hence the theorem holds
for t = 0.
So we may assume that both n and m are positive.
First assume that n ≥ m. By LEMMA 4.3, n is a multiple of m and
there exist elements λ, and γ of C such that grm,n r = ((λx)n/m + γy)m .
Hence
m
r = (λx)n/m + γ∂ + c
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where Vm,n (c) < mn and x-deg c < n and ∂-deg c < m. Set ∂1 =
∂ − (γ)−1 (λx)n/m and x1 = x. Note that ((λx)n/m + γ∂)m = (γ∂1 )m .
Furthermore (∂)i = (∂1 + (γ)−1 (λx1 )n/m )i . It follows that ∂1 -deg c < m
and ∂1 -deg r = m. Also x1 -deg c ≤ (m − 1)n/m < n. Since r =
(γ∂1 )m +c, we have that x1 -deg r < n. By LEMMA 4.4, codimx1 ,∂1 D(X) =
codimx,∂ D(X). Now ∂1 -deg r + x1 -deg r < t, hence the result now follows
by induction for this case.
Now assume that n < m. By LEMMA 4.1, there exist elements x1 and
∂1 in D such that D(X) ⊂ C(∂1 )[x1 ], [x1 , ∂1 ] = 1, x1 -gr ∂ = x1 -gr ∂1 ,
x1 -gr x = −x1 , x1 -gr R ∼
= x-gr R, and codim∂1 ,x1 R = codimx,∂ R. It
follows that x1 -deg r = x-deg r = n. If ∂1 -deg r < m, then the proof
follows by induction.
Otherwise ∂1 -deg r ≥ m > n and we may apply the methods used
above repeatedly to find elements ∂2 = ∂1 and x2 = x1 + g(∂1 ) where
g(∂1 ) ∈ C(∂1 ) such that x2 -deg r = n and ∂2 -deg r < m. The proof again
follows by induction.
We are now able to obtain a nice description of the maximal adnilpotent subalgebras of D(X).
COROLLARY 4.6. — Suppose that X is an affine curve such that the
normalization of X is the affine line, with the normalization map π :
→ X injective. Suppose that B is a maximal ad-nilpotent subalgebra of
X
D(X). Then there exists an element u in D such that B is a commutative
finitely generated algebra with integral closure C[u] and the centralizer of
B in D(X) is the rational function field C(u).
Proof. — By LEMMA 3.3 and LEMMA 3.4, there exists u in D such
that CD (B) = C(u) and B ⊂ C[u]. By LEMMA 3.1, there exists v in D
such that D(X) ⊂ C(u)[v]. Recall that the set of ad-nilpotent elements
of D(X) is strictly larger than the maximal commutative ad-nilpotent
subalgebra O(X) of D(X). Since B is commutative, B cannot contain all
the ad-nilpotent elements of D(X). Hence D(X) contains an ad-nilpotent
element s not contained in B. By [8, Lemma 1.7], v-gr s = λwn for some
λ ∈ C and n > 0. Since s acts ad-nilpotently on D(X), it is clear that
v-gr D(X) ⊂ C[u, w]. By THEOREM 4.5, dimC C[u]/B is finite hence the
integral closure of B is C[u]. By Eakin’s theorem [6, Section 35], B is
finitely generated.
The invariant codim D(X) can be used to distinguish rings of differential operators.
COROLLARY 4.7. — Suppose that X and Y are both affine curves with
normalization equal to the affine line and with injective normalization
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maps. If D(X) ∼
= D(Y ), then codim D(X) = codim D(Y ).
Proof. — Consider both D(X) and D(Y ) as subalgebras of C(x)[∂] using
the standard embedding. Let φ be an isomorphism which maps D(Y ) to
D(X). Set u = φ(x) and v = φ(∂). Clearly u and v satisfy the conditions of
THEOREM 4.5. Therefore codim D(Y ) = codimu,v D(X) = codim D(X).
5. Examples
In this section, we will consider two families of curves. We will calculate
codimensions to show that their rings of differential operators are mutually
nonisomorphic.
Recall that X is a monomial curve if O(X) is generated by monomials xk as an algebra over C. Let Λ be the subset {k | xk ∈ O(x)} of the
integers. Define the set Λ − i to be {k − i | k ∈ Λ} where i is an integer.
MUSSON gives a complete description of D(X) in [7]. In particular,
D(X) =
xk fk (x∂)C[x∂]
k∈Z
where
fk (x∂) =
(x∂ − α).
α∈Λ−(Λ−k)
Let Xn be the monomial curve with O(Xn ) = C+ xn C[x] as coordinate
ring, where n is a positive integer. Then by the previous paragraph, we
have
xk fk (x∂)C[x∂]
D(Xn ) =
k∈Z
where the polynomial fi is 1 for i = 0 and i ≥ n ; the polynomial fi is x∂
for 1 ≤ i ≤ n − 1 ; the polynomial fi is
(x∂)
(x∂ − k) for − 1 ≥ i ≥ −(n − 1)
n−i>k≥n
and the polynomial fi is
(x∂ − k)
(x∂)
n≤k<−i
(x∂ − k) for i ≤ −n.
−i<k<n−i
Note that if g(x∂) is a monic polynomial in C[x∂], then
∂-gr g(x∂) = xd ∂ d
where d = deg g(x∂).
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Hence ∂-gr D(Xn ) =
k∈Z
gk C[xy] where
g0 = 1 ;
for 1 ≤ i ≤ n − 1 ;
gi = xi+1 y
gi = xy
for − n + 1 ≤ i ≤ −1 ;
i+1
gi = xi
gi = y
i
for i ≥ n ;
for i ≤ −n.
A basis for C[x, y]/∂-gr D(Xn ) is just x, x2 , . . . , xn−1 , y, y 2 , . . . , y n−1 . Therefore codim D(Xn ) = 2(n − 1). By COROLLARY 4.7, D(Xn ) is isomorphic
to D(Xm ) if and only if O(Xn ) ∼
= O(Xm ).
Now set Y2n = C+Cx2 +· · ·+Cx2n C[x] for n ≥ 1. A similar calculation
shows that codim D(Y2n ) = n(n + 1). Therefore D(Y2n ) ∼
= D(Y2m ) if and
O(Y
).
only if O(Y2n ) ∼
=
2m
Consider just the curves X4 and Y4 . Now O(X4 ) = C + x4 C[x] and
O(Y4 ) = C+ Cx2 + x4 C[x]. Clearly O(X4 ) is not isomorphic to O(Y4 ). But
codim D(X4 ) = codim D(Y4 ) = 6. Therefore codim does not distinguish
between these two rings of differential operators. We should add that it
has now been shown that D(X4 ) and D(Y4 ) are actually isomorphic rings
even though O(X4 ) and O(Y4 ) are not isomorphic (see [4]).
BIBLIOGRAPHY
[1] AMITSUR (S.A.). — Commutative Linear Differential Operators,
Pacific Journal of Mathematics, t. 8, , p. 1–10.
[2] DIXMIER (J.). — Sur les algèbres de Weyl, Bull. Soc. Math. France,
t. 96, , p. 209–242.
[3] GOODEARL (K.). — Centralizers in Differential, Pseudodifferential, and
Fractional Differential Operator Rings, Rocky Mountain Journal of
Mathematics, t. 13, , p. 573–618.
[4] LETZTER (G.). — Non Isomorphic Curves with Isomorphic Rings of
Differential Operators, preprint.
[5] MAKAR -LIMANOV (L.). — Rings of Differential Operators on Algebraic Curves, J. of the London Mathematical Society, t. 21, ,
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[6] MATSAMURA (H.). — Commutative Algebra. — W.A. Benjamin Co,
New York, .
[7] MUSSON (I.M.). — Some Rings of Differential Operators which are
Morita Equivalent to the Weyl Algebra A1 , Proc. Amer. Math. Soc.,
t. 98, , p. 29–30.
[8] PERKINS (P.). — Commutative Subalgebras of the Ring of Differential
Operators on a Curve, Pacific J. Math., t. 139, , p. 279–302.
[9] SMITH (S.P.) and STAFFORD (J.T.). — Differential Operators on an
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