# 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]). BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE c Société mathématique de France 0037–9484//193/$ 5.00 194 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 TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 195 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]. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 196 G. LETZTER, L. MAKAR-LIMANOV 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). TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 197 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 BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 198 G. LETZTER, L. MAKAR-LIMANOV 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 TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 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 . BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 200 G. LETZTER, L. MAKAR-LIMANOV 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. TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 201 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 BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 202 G. LETZTER, L. MAKAR-LIMANOV 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 TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 203 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. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 204 G. LETZTER, L. MAKAR-LIMANOV 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). TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 205 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 BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 206 G. LETZTER, L. MAKAR-LIMANOV 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 TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 207 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∂). BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 208 G. LETZTER, L. MAKAR-LIMANOV 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, , p. 538–540. TOME 118 — — N ◦ 2 RINGS OF DIFFERENTIAL OPERATORS 209 [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 Aﬃne Curve, Proc. London Math. Soc., t. 56, , p. 229–259. [10] STAFFORD (J.T.). — Endomorphisms of Right Ideals of the Weyl Algebra, Transactions of the AMS, t. 299, , p. 623–639. BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE