D-MODULES ON SMOOTH ALGEBRAIC VARIETIES EIVIND ERIKSEN Abstract. We consider algebraic varieties X de¯ned over C which are smooth, a±ne and irreducible. We study the ring D = D(X) of C-linear di®erential operators on X, and we explain Bernstein's theory of holonomic D-modules in this case. This is a generalization of Bernstein's original work, which covers the case when X is a±ne n-space and D is the n'th Weyl algebra. I shall follow the approach to this generalized theory given in chapter 3 of BjÄork[1]. 1. Differential operators on the variety X Let k = C denote the ¯eld of complex numbers. We consider an a±ne algebraic variety X ⊆ Cn such that X is smooth and irreducible. We denote by A = A(X) the a±ne coordinate ring of X, so A is a commutative k-algebra of ¯nite type and an integral domain. In particular, A is a Notherian ring. Let k(X) denote the quotient ¯eld of the integral domain A. Since A is of ¯nite type over k, it is clear that k ⊆ k(X) is a ¯eld extension of ¯nite degree of transcendence. We denote by d = dim(X) the degree of transcendence of this ¯eld extension. This is the classically de¯ned dimension of the variety X. It is clear that A is a regular Noetherian ring of pure dimension d. That is, the local Noetherian ring Am is a regular ring of dimension d for all maximal ideals m ⊆ A. So clearly, the Krull dimension dim A = d, and we have 0 · d · n with ² d = n if and only if X = Cn, ² d = 0 if and only if X is reduced to a single point. The vector ¯elds on X are given as θ(X) = Derk(A), where Derk(A) is the module of derivations Derk(A) = fD 2 Endk(A): D(xy) = D(x)y + xD(y) for all x; y 2 Ag: Let m ⊆ A be any maximal ideal, let Am be the corresponding local ring and let » t1; : : : ; td be a local system of parameters for Am. Then Derk(Am) = Am­ADerk(A) is a free Am-module of rank d, generated by derivations D1;:::;Dd such that Di(tj) = ±ij for 1 · i; j · d. We may present the ring A in the form A = S=I, where S = k[x1; : : : ; xn] is the a±ne coordinate ring of Cn and I = I(X) ⊆ S is the prime ideal in S consisting of all polynomials in S which vanish on X. It is clear that Derk(S) is the free S-module generated by @i = @=@xi for 1 · i · n. It is not di±cult to see that there is a canonical isomorphism » Derk(A) = fP 2 Derk(S): P (I) ⊆ Ig=I Derk(S); and that P (I) ⊆ I is satis¯ed if and only if P (fi) 2 I for any set of generators f1; : : : ; fr of the ideal I. So Derk(A) can be identi¯ed with the kernel of the A- n r linear map A ! A given by the matrix (@fi=@xj). Since A is a Noetherian ring, it follows that Derk(A) is a left A-module of ¯nite type. We de¯ne the ring D = D(X) of k-linear di®erential operators on X to be the sub-ring of Endk(A) generated by the multiplication operators induced by the ring A and the derivations in Derk(A). It follows that D(X) is a associative k-algebra. 1 2 EIVIND ERIKSEN Since A is a ¯nitely generated k-algebra and Derk(A) is a ¯nitely generated A- module, it follows that D is a k-algebra of ¯nite type. We denote by Dp ⊆ D the k-linear subspace of D generated by products of at most p derivations for any integer p. Then Dp = 0 when p < 0, D0 = A, and 1 p D = A © Derk(A). Moreover, we have that the subspaces D form an exhaustive, ascending ¯ltration of the ring D. This ¯ltration is called the order ¯ltration, and we say that a di®erential operator P 2 D has order p if P 2 Dp n Dp¡1 for some p ¸ 0, and that P = 0 has order ¡1. We shall write d(P ) for the order of the di®erential operator P . Note that the ¯ltered ring D coincides with the ring of di®erential operators on X=k de¯ned by Grothendieck in EGA IV [2]. Consider the associated graded ring gr D associated with the order ¯ltration of the ring D, de¯ned as gr D = ©Dp=Dp¡1: This is a graded k-algebra. We shall denote by grp D the p'th homogeneous part Dp=Dp¡1 of gr D for all integers p. Then we have grp D = 0 when p < 0, gr0 D = A 1 and gr D = Derk(A). Since we have d(PQ ¡ QP ) < d(P ) + d(Q) for all non-zero di®erential operators P; Q, we see that gr D is a commutative ring. Moreover, it is a ¯nitely generated k-algebra, and hence Noetherian, since it is generated by homogeneous elements of degree 1 considered as a gr0 D-algebra. In the following theorem, we summarize some properties of the rings D and gr D which will be useful. The proof of most of the statements in this theorem can be found in BjÄork[1], and references to the remaining parts can be found in Smith and Sta®ord [3]: Theorem 1. Let X be a smooth, irreducible a±ne algebraic variety of dimension d de¯ned over C, let D be the ring of di®erential operators and let gr D be the asso- ciated graded ring associated with the order ¯ltration on D. Then we have: i) D is an associative k-algebra of ¯nite type, ii) D is an integral domain, iii) D is a simple ring, iv) D has global homological dimension d, v) gr D is a commutative k-algebra of ¯nite type, vi) gr D is an integral domain, vii) gr D is a Noetherian regular ring of pure dimension 2d. 2. Modules on filtered rings Let D be any ¯ltered k-algebra with a ¯xed ascending ¯ltration fDpg of k-linear subspaces of D. We shall assume that the ¯ltration (Dp) is exhaustive and such that D0 contains the unit 1 2 D and such that Dp = 0 for all p < 0. Moreover, we assume that Dp is ¯nitely generated considered as a left and right D0-module for all integers p. Finally, let us consider the associated graded ring gr D, and assume that gr D is a commutative Noetherian ring. This last condition implies that D0 is a commutative, Noetherian k-algebra. Notice that when X is a smooth, irreducible a±ne algebraic variety over C and D = D(X) is the ring of k-linear di®erential operators with the order ¯ltration, then these conditions are ful¯lled. Moreover, the ring D0 = A, the a±ne coordinate ring of X. This example will motivate the constructions in this section. We refer to any element P 2 D as an operator, and we denote by d(P ) the order of the operator P , de¯ned as d(P ) = inf fp : P 2 Dpg. By convention, d(P ) = ¡1 when P = 0. When P 2 D, we denote by σ(P ) the image of P in Dp=Dp¡1 ⊆ gr D with p = d(P ). By convention, we have σ(P ) = 0 when P = 0. Let M be a left D-module. We denote by a ¯ltration of M any exhaustive, ascending ¯ltration fMig of M compatible with the given ¯ltration of D such that D-MODULES ON SMOOTH ALGEBRAIC VARIETIES 3 0 Mi is a ¯nitely generated D -module for all integers i and Mi = 0 for some integer i. For any such ¯ltration, we consider the associated graded gr D-module gr M = ©Mi=Mi¡1: For any element m 2 M, we denote by d(m) = inffi : m 2 Mig the order of the element m. By convention, d(m) = ¡1 when m = 0. We denote by σ(m) the image of m in Mi=Mi¡1 ⊆ gr M with i = d(m). By convention, σ(m) = 0 when m = 0. Proposition 2. Let M be a left D-module, and let (Mi) be a ¯ltration of M. If fm®g ⊆ M is a subset of M such that fσ(m®)g is a generating set for gr M as a left gr D-module, then fm®g is a generating set for M as a left D-module. Proof. Assume that fσ(m®)g is a generating set of gr M, and let M ⊆ M denote the left D-module generated by fm®g. It is enough to show that Mi ⊆ M for all integers i. Since Mi = 0 for some integer i, we can prove this by induction on i. So assume that Mi¡1 ⊆ M, and let m 2 Mi n Mi¡1. Then we have X σ(m) = σ(P®)σ(m®) P for operators P® of degree i ¡ d(m®). If follows that m ¡ P®m® 2 Mi¡1. By the induction hypothesis, Mi¡1 ⊆ M, so it follows that Mi ⊆ M. ¤ Assume that M is a ¯nitely generated left D-module, and choose a ¯nite set P i fm®g of generators for M. We de¯ne Mi = D m® for all integers i.