Philippe Maisonobe Claude Sabbah ASPECTS OF THE THEORY OF D-MODULES

Philippe Maisonobe Claude Sabbah ASPECTS OF THE THEORY OF D-MODULES LECTURE NOTES (KAISERSLAUTERN 2002) REVISED VERSION: JULY 2011

Ph. Maisonobe UMR 6621 du CNRS, Laboratoire J.A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice cedex 2, France. E-mail : phm@math.unice.fr C. Sabbah UMR 7640 du CNRS, Centre de Mathématiques, École polytechnique, F 91128 Palaiseau cedex, France. E-mail : sabbah@math.polytechnique.fr Url : http://www.math.polytechnique.fr/~sabbah/sabbah.html

ASPECTS OF THE THEORY OF D-MODULES LECTURE NOTES (KAISERSLAUTERN 2002) REVISED VERSION: JULY 2011 Philippe Maisonobe, Claude Sabbah

CONTENTS 1. Basic constructions........................................................ 1 1.1. The sheaf of holomorphic differential operators........................... 1 1.2. Left and right............................................................ 5 1.3. Examples of D-modules.................................................. 10 1.4. de Rham and Spencer.................................................... 13 1.5. Filtered objects: the Rees construction................................... 17 2. Coherence.................................................................. 19 2.0. A reminder on coherence................................................. 19 2.1. Coherence of D......................................................... 20 2.2. Coherent D -modules and good filtrations............................... 21 2.3. Support.................................................................. 26 2.4. Characteristic variety.................................................... 27 2.5. Involutiveness of the characteristic variety............................... 29 2.6. Non-characteristic restrictions............................................ 30 3. Differential complexes and local duality................................ 33 3.1. Induced D-modules and differential morphisms.......................... 33 3.2. Differential complexes.................................................... 37 3.3. Differential complexes of finite order..................................... 42 3.4. A prelude to local duality................................................ 46 3.5. Local duality............................................................. 51 3.6. Local duality and external tensor product................................ 53 4. Direct images of D -modules............................................ 57 4.1. Example of computation of a Gauss-Manin differential equation......... 58 4.2. Inverse images of left D-modules......................................... 60 4.3. Direct images of right D-modules........................................ 62 4.4. Direct images of differential complexes................................... 66 4.5. Direct image of currents.................................................. 68 4.6. The Gauss-Manin connection............................................. 69 4.7. Coherence of direct images............................................... 73 4.8. Kashiwara s estimate for the behaviour of the characteristic variety...... 74 5. Holonomic D -modules................................................... 77 5.1. Motivation: division of distributions..................................... 77 5.2. First properties of holonomic D -modules............................... 79

vi CONTENTS 5.3. Vector bundles with integrable connections............................... 81 5.4. The local duality theorem for holonomic D -modules.................... 82 5.5. Direct images of holonomic D -modules................................. 82 5.6. The de Rham complex of a holonomic D -module....................... 83 5.7. Recent advances.......................................................... 84 6. Computational aspects in D-module theory............................ 87 6.1. Review on the Gröbner basis of an ideal in a polynomial ring............ 87 6.2. Gröbner basis of a noncommutative algebra.............................. 92 6.3. Some applications........................................................ 97 7. Specializable D -modules................................................. 99 7.1. The V -filtration.......................................................... 100 7.2. Coherence................................................................ 102 7.3. Y -specializable D -modules.............................................. 104 7.4. Localization and restriction of specializable D -modules................. 108 7.5. V -filtration and direct images............................................ 111 Bibliography................................................................... 119

LECTURE 1 BASIC CONSTRUCTIONS In this first lecture, we introduce the sheaf of differential operators and its (left or right) modules. Our main concern is to develop the relationship between two a priori different notions: (1) the classical notion of a O -module with a flat connection, (2) the notion of a left D -module. Both notions are easily seen to be equivalent. However, the extension of the equivalence to complexes (or to the derived category) is less clear. Later on, we will introduce the notion of differential complex to express this equivalence. The main result in this direction will be Theorem 3.2.8. The relationship between left and right D -modules, although simple, is also somewhat subtle, and we insist on the basic isomorphisms. We also develop the same theory with filtration, by explaining a recipe (Rees construction) to treat the filtered analogue along the same lines. The notion of strictness plays a major role here. The results in this lecture are mainly algebraic, and do not involve any analytic property. They could be translated easily to the algebraic situation. One can find many of these notions in the classical books [Kas95, Bjö79, Bor87, Meb89, MS93a, MS93b, Bjö93, Cou95, Kas03]. Some of them are also directly inspired from the work of M. Saito [Sai88, Sai89b, Sai89a] about Hodge D-modules. In this lecture, we denote by a complex manifold of dimension dim = n. 1.1. The sheaf of holomorphic differential operators We will denote by Θ the sheaf of holomorphic vector fields on. This is the O -locally free sheaf generated in local coordinates by x1,..., xn. It is a sheaf of O -Lie algebras which is locally free as a O -module, and vector fields act (on the left) on functions by derivation, in a way compatible with the Lie algebra structure:

2 LECTURE 1. BASIC CONSTRUCTIONS given a local vector field ξ acting on functions as a derivation g ξ(g), and a local holomorphic function f, fξ is the vector field acting as f ξ(g), and given two vector fields ξ, η, their bracket as derivations [ξ, η](g) := ξ(η(g)) η(ξ(g)) is still a derivation, hence defines a vector field. Dually, we denote by Ω 1 the sheaf of holomorphic 1-forms on. We will set Ω k = k Ω 1. We denote by d : Ωk Ωk+1 the differential. Exercise 1.1.1. Let E be a locally free O -module of rank d and let E be its dual. Show that, given any local basis e = (e 1,..., e d ) of E with dual basis e, the section d i=1 e i e i of E O E does not depend on the choice of the local basis e and extends as a global section of E O E. Show that it defines, up to a constant, a O -linear section O E O E of the natural duality pairing E O E O. Conclude that we have a natural global section of Ω 1 O Θ given, in local coordinates, by i dx i xi. Let ω denote the sheaf Ω dim of forms of maximal degree. Then there is a natural right action (in a compatible way with the Lie algebra structure) of Θ on ω : the action is given by ω ξ = L ξ ω, where L ξ denotes the Lie derivative, equal to the composition of the interior product ι ξ by ξ with the differential d, as it acts on forms of maximal degree. The action is on the right since applying the vector field fξ (as defined above) to ω consists in multiplying first ω by f, and then applying ξ. The choice of the sign above makes this definition compatible with bracket. Exercise 1.1.2 (The sheaf Hom ). Let be a topological space and let F and G be two sheaves of A -modules on, A being a sheaf of rings on. We denote by Hom A (F, G ) the Γ(, A )-module of morphisms of sheaves of A -modules from F to G. An element φ of Hom A (F, G ) is a collection of morphisms φ(u) Hom A (U) (F (U), G (U)), on open subsets U of, compatible with the restrictions. Show that the presheaf Hom A (F, G ) defined by Γ(U, Hom A (F, G )) = Hom A U (F U, G U ) is a sheaf (notice that U Hom A (U) (F (U), G (U)) is not a presheaf, because there are no canonical morphisms of restriction). Definition 1.1.3 (The sheaf of holomorphic differential operators) For any open set U of, the ring D (U) of holomorphic differential operators on U is the subring of Hom C (O U, O U ) generated by multiplication by holomorphic functions on U, derivation by holomorphic vector fields on U. The sheaf D is defined by Γ(U, D ) = D (U) for any open set U of. By construction, the sheaf D acts on the left on O, i.e., O is a left D -module.

LECTURE 1. BASIC CONSTRUCTIONS 3 Definition 1.1.4 (The filtration of D by the order). The increasing family of subsheaves F k D D is defined inductively: F k D = 0 if k 1, F 0 D = O (via the canonical injection O Hom C (O, O )), the local sections P of F k+1 D are characterized by the fact that [P, ϕ] is a local section of F k D for any holomorphic function ϕ. Exercise 1.1.5. Show that a differential operator P of order 1 satisfying P (1) = 0 is a derivation of O, i.e., a section of Θ. Exercise 1.1.6 (Local computations). Let U be an open set of C n with coordinates x 1,..., x n. Denote by x1,..., xn the corresponding vector fields. (1) Show that the following relations are satisfied in D(U): [ xi, ϕ] = ϕ, x i ϕ O(U), i {1,..., n}, [ xi, xj ] = 0 i, j {1,..., n}, x α ϕ = α! (α β)!β! α β x (ϕ) x β, ϕ α x = 0 β α 0 β α α! (α β)!β! ( 1) α β x β x α β (ϕ), with standard notation concerning multi-indices α, β. (2) Show that any element P D(U) can be written in a unique way as α a α x α or α α x b α with a α, b α O(U). Conclude that D is a locally free left and right module over O. (3) Show that max{ α ; a α 0} = max{ α ; b α 0}. It is denoted by ord x P. (4) Show that ord x P does not depend on the coordinate system chosen on U. (5) Show that P Q = 0 in D(U) P = 0 or Q = 0. (6) Identify F k D with the subsheaf of local sections of D having order k (in some or any local coordinate system). Show that it is a locally free O -module of finite rank. (7) Show that the filtration F D is exhaustive (i.e., D = k F kd ) and that it satisfies F k D F l D = F k+l D. (The left-hand term consists by definition of all sums of products of a section of F k D and a section of F l D.) (8) Show that the bracket [P, Q] := P Q QP induces for each k, l a C-bilinear morphism F k D C F l D F k+l 1 D. (9) Conclude that the graded ring gr F D is commutative.

4 LECTURE 1. BASIC CONSTRUCTIONS The sheaf D is not commutative. The lack of commutativity of D is analyzed in Exercise 1.1.7. Exercise 1.1.7 (The graded sheaf gr F D ). The goal of this exercise is to show that the sheaf of graded rings gr F D may be canonically identified with the sheaf of graded rings Sym Θ. If one identifies Θ with the sheaf of functions on the cotangent space T which are linear in the fibres, then Sym Θ is the sheaf of functions on T which are polynomial in the fibres. In particular, gr F D is a sheaf of commutative rings. (1) Identify the O -module Sym k Θ with the sheaf of symmetric C-linear forms ξ : O C C O on the k-fold tensor product, which behave like a derivation with respect to each factor. (2) Show that Sym Θ := k Sym k Θ is a sheaf of graded O -algebras on and identify it with the sheaf of functions on T which are polynomial in the fibres. (3) Show that the map F k D Hom C ( k C O, O ) which sends any section P of F k D to ϕ 1 ϕ k [ [[P, ϕ 1 ]ϕ 2 ] ϕ k ] induces an isomorphism of O -modules gr F k D Sym k Θ. (4) Show that the induced morphism gr F D := k gr F k D Sym Θ is an isomorphism of sheaves of O -algebras. On the other hand, it has no non-trivial two-sided ideals (see Exercise 1.1.8), hence it is simple. Exercise 1.1.8 (The sheaf of rings D has no non-trivial two-sided ideals) Let I be a non-zero two-sided ideal of D. (1) Let x and 0 P I x. Show that there exists f O,x such that [P, f] 0. [Hint: use local coordinates to express P ]. (2) Conclude by induction on the order that I x contains a non-zero g O,x. (3) Show that I x contains all iterated differentials of g, and conclude that I x contains h O,x such that h(x) 0. (4) Conclude that I x 1, hence I x = D,x. This leads us to consider left or right D -modules (or ideals), and the theory of two-sided objects is empty.

LECTURE 1. BASIC CONSTRUCTIONS 5 Exercise 1.1.9 (The universal connection) (1) Show that the natural left multiplication of Θ on D can be written as a connection : D Ω 1 O D, i.e., as a C-linear morphism satisfying the Leibniz rule (fp ) = df P + f P, where f is any local section of O and P any local section of D. [Hint: (1) is the global section of Ω 1 O Θ considered in Exercise 1.1.1.] (2) Extend this connection for any k 1 as a C-linear morphism satisfying the Leibniz rule written as (k) : Ω k O D Ω k+1 O D (k) (ω P ) = dω P + ( 1) k ω P. (3) Show that (k+1) (k) = 0 for any k 0 (i.e., is flat). (4) Show that the morphisms (k) are right D -linear (but not left O -linear). Exercise 1.1.10. More generally, show that a left D -module M is nothing but a O -module with an integrable connection : M Ω 1 O M. [Hint: to get the connection, tensor the left D -action D O M M by Ω 1 on the left and compose with the universal connection to get D M Ω 1 M ; compose it on the left with M D M given by m 1 m.] Define similarly the iterated connections (k) : Ω k O M Ω k+1 O M. Show that (k+1) (k) = 0. In conclusion: Proposition 1.1.11. Giving a left D -module M is equivalent to giving a O -module M together with an integrable connection. Proof. Exercises 1.1.1, 1.1.9 and 1.1.10. 1.2. Left and right The categories of left (resp. right) D -modules are denoted by l M(D ) (resp. r M(D ). We analyze the relations between both categories in this section. Let us first recall the basic lemmas for generating left or right D-modules. We refer for instance to [Cas93, 1.1] for more details. Lemma 1.2.1 (Generating left D -modules). Let M l be a O -module and let ϕ l : Θ C M l M l be a C-linear morphism such that, for any local sections f of O, ξ, η of Θ and m of M l, one has (1) ϕ l (fξ m) = fϕ l (ξ m), (2) ϕ l (ξ fm) = fϕ l (ξ m) + ξ(f)m, (3) ϕ l ([ξ, η] m) = ϕ l (ξ ϕ l (η m)) ϕ l (η ϕ l (ξ m)).

6 LECTURE 1. BASIC CONSTRUCTIONS Then there exists a unique structure of left D -module on M l such that ξm = ϕ l (ξ m) for any ξ, m. Lemma 1.2.2 (Generating right D -modules). Let M r be a O -module and let ϕ r : M r C Θ M r be a C-linear morphism such that, for any local sections f of O, ξ, η of Θ and m of M r, one has (1) ϕ r (mf ξ) = ϕ r (m fξ) (ϕ r is in fact defined on M r O Θ ), (2) ϕ r (m fξ) = ϕ r (m ξ)f mξ(f), (3) ϕ r (m [ξ, η]) = ϕ r (ϕ r (m ξ) η) ϕ r (ϕ r (m η) ξ). Then there exists a unique structure of right D -module on M r such that mξ = ϕ r (m ξ) for any ξ, m. Example 1.2.3 (Most basic examples) (1) D is a left and a right D -module. (2) O is a left D -module (Exercise 1.2.4). (3) ω := Ω dim is a right D -module (Exercise 1.2.5). Exercise 1.2.4 (O is a simple left D -module) (1) Use the left action of Θ on O to define on O the structure of a left D - module. (2) Let f be a nonzero holomorphic function on C n. Show that there exists a multi-index α N n such that ( α f)(0) 0. (3) Conclude that O is a simple left D -module, i.e., does not contain any proper non trivial D -submodule. Is it simple as a left O -module? Exercise 1.2.5 (ω is a simple right D -module) (1) Use the right action of Θ on ω to define on ω the structure of a right D -module. (2) Show that it is simple as a right D -module. Exercise 1.2.6 (Tensor products over O ) (1) Let M and N be two left D -modules. (a) Show that the O -module M O N has the structure of a left D - module by setting, by analogy with the Leibniz rule, ξ (m n) = ξm n + m ξn. (b) Notice that, in general, m n (ξm) n (or m n m (ξn)) does not define a left D -action on the O -module M O N. (c) Let ϕ : M M and ψ : N N be D -linear morphisms. Show that ϕ ψ is D -linear.

LECTURE 1. BASIC CONSTRUCTIONS 7 (2) Let M be a left D -module and N be a right D -module. Show that N O M has the structure of a right D -module by setting (n m) ξ = nξ m n ξm. Remark: one can define a right D -module structure on M O N by using the natural involution M O N N O M, so this brings no new structure. (3) Assume that M and N are right D -modules. Does there exist a (left or right) D -module structure on M O N defined with analogous formulas? Exercise 1.2.7 (Hom over O ) (1) Let M, N be left D -modules. Show that Hom O (M, N ) has a natural structure of left D -module defined by (ξ ϕ)(m) = ξ (ϕ(m)) + ϕ(ξ m), for any local sections ξ of Θ, m of M and ϕ of Hom O (M, N ). (2) Similarly, if M, N are right D -modules, then Hom O (M, N ) has a natural structure of left D -module defined by (ξ ϕ)(m) = ϕ(m ξ) ϕ(m) ξ. Exercise 1.2.8 (Tensor product of a left D -module with D ) Let M l be a left D -module. Notice that M l O D has two commuting structures of O -module. Similarly D O M l has two such structures. The goal of this exercise is to extend them as D -structures and examine their relations. (1) Show that M l O D has the structure of a left and of a right D -module which commute, given by the formulas: { f (m P ) = (fm) P = m (fp ), (left) ξ (m P ) = (ξm) P + m ξp, (right) { (m P ) f = m (P f), (m P ) ξ = m (P ξ), for any local vector field ξ and any local holomorphic function f. Show that a left D -linear morphism ϕ : M l 1 M l 2 extends as a bi-d -linear morphism ϕ 1 : M l 1 O D M l 2 O D. (2) Similarly, show that D O M l also has such structures which commute and are functorial, given by formulas: { f (P m) = (fp ) m, (left) ξ (P m) = (ξp ) m, (right) { (P m) f = P (fm) = (P f) m, (P m) ξ = P ξ m P ξm.

8 LECTURE 1. BASIC CONSTRUCTIONS (3) Show that both morphisms M l O D D O M l D O M l M l O D m P (1 m) P P m P (m 1) are left and right D -linear, induce the identity M l 1 = 1 M l, and their composition is the identity of M l O D or D O M l, hence both are reciprocal isomorphisms. Show that this correspondence is functorial (i.e., compatible with morphisms). (4) Let M be a left D -module and let L be a O -module. Justify the following isomorphisms of left D -modules and right O -modules: M O (D O L ) (M O D ) O L (D O M ) O L D O (M O L ). Assume moreover that M and L are O -locally free. Show that M O (D O L ) is D -locally free. Exercise 1.2.9 (Tensor product of a right D -module with D ) Let M r be a right D -module. (1) Show that M r O D has two structures of right D -module denoted r (right) and t (tensor; the latter defined by using the left structure on D and Exercise 1.2.6(2)), given by: { (m P ) r f = m (P f), (right) r (right) t (m P ) r ξ = m (P ξ), { (m P ) t f = mf P = m fp, (m P ) t ξ = mξ P m (ξp ). (2) Show that there is a unique involution ι : M r O D M r O D which exchanges both structures and is the identity on M r 1, given by (m P ) r (m 1) t P. [Hint: show first the properties of ι by using local coordinates, and glue the local constructions by uniqueness of ι.] (3) For each p 0, consider the pth term F p D of the filtration of D by the order (see Exercise 1.1.4) with both structures of O -module (one on the left, one on the right) and equip similarly M r O F p D with two structures t and r of O -modules. Show that, for each p, ι induces an isomorphism of O -modules (M r O F p D ) t (M r O F p D ) r. Definition 1.2.10 (Right-left transformation). Any left D -module M l gives rise to a right one M r by setting (see [Cas93] for instance) M r = ω O M l and, for any vector field ξ and any function f, (ω m) f = fω m = ω fm, (ω m) ξ = ωξ m ω ξm.

LECTURE 1. BASIC CONSTRUCTIONS 9 Conversely, set M l = Hom O (ω, M r ), which also has in a natural way the structure of a left D -module (see Exercise 1.2.7(2)). Exercise 1.2.11 (Compatibility of right-left transformations) Show that the natural morphisms M l Hom O (ω, ω O M l ), ω O Hom O (ω, M r ) M r are isomorphisms of D -modules. Exercise 1.2.12 (Compatibility of left-right transformation with tensor product) Let M l and N l be two left D -modules and denote by M r, N r the corresponding right D -modules (see Definition 1.2.10). Show that there is a natural isomorphism of right D -modules (by using the right structure given in Exercise 1.2.6(2)): N r O M l M r O N l (ω n) m (ω m) n and that this isomorphism is functorial in M l and N l. Exercise 1.2.13 (Local expression of the left-right transformation) Let U be an open set of C n. (1) Show that there exists a unique C-linear involution P t P from D(U) to itself such that ϕ O(U), t ϕ = ϕ, i {1,..., n}, t xi = xi, P, Q D(U), t (P Q) = t Q tp. (2) Let M be a left (resp. right) D -module and let t M be M equipped with the right (resp. left) D -module structure Show that t M M r (resp. t M P m := t P m. M l ). Exercise 1.2.14 (The left-right transformation is an isomorphism of categories) To any left D -linear morphism ϕ l : M l N l is associated the O -linear morphism ϕ r = Id ω ϕ l : M r N r. (1) Show that ϕ r is right D -linear. (2) Define the reverse correspondence ϕ r ϕ l. (3) Conclude that the left-right correspondence l M(D ) r M(D ) is a functor, which is an isomorphism of categories, having the right-left correspondence r M(D ) l M(D ) as inverse functor.

10 LECTURE 1. BASIC CONSTRUCTIONS 1.3. Examples of D-modules We list here some classical examples of D-modules. examples by applying various operations on D-modules. One may get many other 1.3.a. Let I be a sheaf of left ideals of D. We will see in Lecture 2 that, locally on, I is generated by a finite set {P 1,..., P k } of differential operators (this follows from the noetherianity and coherence properties of D ). Then the quotient M = D /I is a left D -module. Locally, M is the D -module associated with P 1,..., P k. Notice that different choices of generators of I give rise to the same D -module M. It may be sometime difficult to guess that two sets of operators generate the same ideal. Therefore, it is useful to develop a systematic procedure to construct from a system of generators a division basis of the ideal in order to have a decision algorithm (see Lecture 6 on Gröbner bases). Exercise 1.3.1. Show that the two sets of differential operators { x1,..., xn } and { x1, x 1 x2 + + x n 1 xn } generate the same ideal of D C n. 1.3.b. Let L be a O -module. There is a very simple way to get a right D -module from L : consider L O D equipped with the natural right action of D. This is called an induced D -module. Although this construction is very simple, it is also very useful to get cohomological properties of D -modules, as we will see in Section 3.1. One can also consider the left D -module D O L (however, this is not the left D -module attached to the right one L O D by the left-right transformation of Definition 1.2.10). 1.3.c. One of the main geometrical examples of D -modules are the vector bundles on equipped with an integrable connection. Recall that left D -modules are O - modules with an integrable connection. Among them, the coherent D -modules are particularly interesting. We will see (see Exercise 2.4.6), that such modules are O - locally free, i.e., correspond to holomorphic vector bundles of finite rank on. It may happen that, for some, such a category does not give any interesting geometric object. Indeed, if for instance has a trivial fundamental group (e.g. = P 1 (C)), then any vector bundle with integrable connection is isomorphic to the trivial bundle O r with the connection d. However, on Zariski open sets of, there may exist interesting vector bundles with connections. This leads to the notion of meromorphic vector bundle with connection. Let D be a divisor in and denote by O ( D) the sheaf of meromorphic functions on with poles along D at most. This is a sheaf of left D -modules, being a O - module equipped with the natural connection d : O ( D) Ω 1 ( D). By definition, a meromorphic bundle is a locally free O ( D) module of finite rank. When it is equipped with an integrable connection, it becomes a left D -module.

LECTURE 1. BASIC CONSTRUCTIONS 11 1.3.d. One may twist the previous examples. Assume that ω is a closed holomorphic form on. Define : O Ω 1 by the formula = d + ω. As ω is closed, is an integrable connection on the trivial bundle O. Usually, there only exist meromorphic closed form on, with poles on some divisor D. Then is an integrable connection on O ( D). If ω is exact, ω = df for some meromorphic function f on, then may be written as e f d e f. More generally, if M is any meromorphic bundle with an integrable connection, then, for any such ω, + ω Id defines a new D -module structure on M. 1.3.e. Denote by Db the sheaf of distributions on : given any open set U of, Db (U) is the space of distributions on U, which is by definition the week dual of the space of C forms with compact support on U, of type (dim U, dim U). By Exercise 1.2.5, there is a right action of D on such forms. The left action of D on distributions is defined by adjunction: denote by ϕ, u the natural pairing of a compactly supported C -form ϕ with a distribution u on U; let P be a holomorphic differential operator on U; define then P u such that, for any ϕ, on has ϕ, P u = ϕ P, u. Given any distribution u on, the subsheaf D u Db is the D -module generated by this distribution. Saying that a distribution is a solution of a family P 1,..., P k of differential equation is equivalent to saying that the morphism D D u sending 1 to u induces a surjective morphism D /(P 1,..., P k ) D u. Similarly, the sheaf C of currents of maximal degree on, dual to Cc,, is a right D -module. In local coordinates x 1,..., x n, a current of maximal degree is nothing but a distribution times the volume form dx 1 dx n dx 1 dx n. As we are now working with C forms or with currents, it is natural not to forget the anti-holomorphic part of these objects. Denote by O the sheaf of antiholomorphic functions on and by D the sheaf of anti-holomorphic differential operators. Then Db (resp. C ) are similarly left (resp. right) D -modules. Of course, the D and D actions do commute, and they coincide when considering multiplication by constants. It is therefore natural to introduce the following sheaves of rings: O, := O C O, D, := D C D, and consider Db (resp. C ) as left (resp. right) D, -modules. 1.3.f. One may construct new examples from old ones by using various operations.

12 LECTURE 1. BASIC CONSTRUCTIONS Let M be a left D -module. Then Hom D (M, D ) has a natural structure of right D -module. Using a resolution N of M by left D -modules which are acyclic for Hom D (, D ), one gets a right D -module structure on the Ext k D (M, D ). Given two left (resp. a left and a right) D -modules M and N, the same argument allows one to put on the various Tor i,o (M, N ) a left (resp. a right) D -module structure. We will see in Lecture 4 the geometric operations direct image and inverse image of a D -module by a holomorphic map. 1.3.g. Solutions. Let M, N be two left D -modules. Definition 1.3.2. The sheaf of solutions of M in N is the sheaf Hom D (M, N ). Remark 1.3.3 (1) The sheaf Hom D (M, N ) has no structure more than that of a sheaf of C- vector spaces in general, because D is not commutative. (2) According to Exercise 1.2.7(1), Hom O (M, N ) is a left D -module, that is, a O -module with an integrable connection (Proposition 1.1.11). Then Hom D (M, N ) is the subsheaf of Hom O (M, N ) consisting of local morphisms M N which commute with the connections on M and N, in other words local sections which are annihilated by the connection on Hom O (M, N ). Example 1.3.4. Let U be a coordinate chart and let P D (U). Let I = D U P be the left ideal of D U generated by P and let M = D U /I. We have a canonical isomorphism Hom D (M, N ) Ker[P : N N ], and this explains the terminology solutions of M in N. If N = O, we get the sheaf of holomorphic solutions of P. If N = Db, we get the sheaf of distributions solutions of P. If N = D (with its standard left structure), then P : D D is injective (Exercise 1.1.6), so Hom D (M, D ) = 0. It maybe therefore interesting to consider higher Hom, namely, Ext sheaves. We consider the free resolution of M defined as 0 D U P D U M 0. The map P is injective (same argument as for P ), so this is indeed a resolution. By definition, Ext 1 (M, N ) is the cokernel of Hom DU (D U, N ) Hom DU (D U, N ) ϕ( ) ϕ( P ). If one identifies Hom DU (D U, N ) with N by ϕ ϕ(1), the previous morphism reads N P N,

LECTURE 1. BASIC CONSTRUCTIONS 13 so we recover that Hom D (M, N ) = Ker[P : N N ], and we find that Ext 1 (M, N ) = Coker[P : N N ]. In other words, Ext 1 (M, N ) measures the obstruction to the solvability of the differential equation P m = n for n N. Notice that, in this example, since the free resolution of M has length two, we have Ext k D (M, N ) = 0 for k 2, for any N. When N has a supplementary structure which commutes with its left D - structure, then Hom D (M, N ) and the Ext k D (M, N ) inherit this supplementary structure. Example 1.3.5 (1) Assume N = D. Then the definition of Hom D (M, N ) and the Ext k D (M, N ) uses the left D -module structure of D, which commutes with the right one, so these solution sheaves are right D -modules. (2) Assume that N = Db. Then the definition of Hom D (M, N ) and the Ext k D (M, N ) uses the left D -module structure of Db, which commutes with the left D -structure, so these solution sheaves are left D -modules. 1.4. de Rham and Spencer Let M l be a left D -module and let M r be a right D -module. Definition 1.4.1 (de Rham). The de Rham complex Ω n+ (M l ) of M l is the complex having as terms the O -modules Ω n+ O M l and as differential the C-linear morphism ( 1) n defined in Exercise 1.1.10. Notice that the de Rham complex is shifted n = dim with respect to the usual convention. The shift produces, by definition, a sign change in the differential, which is then equal to ( 1) n. The previous definition produces a complex since = 0, according to the integrability condition on, as remarked in Exercise 1.1.10. Definition 1.4.2 (Spencer). The Spencer complex (Sp (M r ), δ) is the complex having as terms the O -modules M O Θ (with 0) and as differential the C-linear map δ given by m ξ 1 ξ k δ k ( 1) i 1 mξ i ξ 1 ξ i ξ k i=1 + i<j( 1) i+j m [ξ i, ξ j ] ξ 1 ξ i ξ j ξ k. Exercise 1.4.3. Check that (Sp (M r ), δ) is indeed a complex, i.e., that δ δ = 0. Of special interest will be, of course, the de Rham or Spencer complex of the ring D, considered as a left or right D -module. Notice that, in Ω n+ (D ), the differentials are right D -linear, and in Sp (D ) they are left D -linear.

14 LECTURE 1. BASIC CONSTRUCTIONS Exercise 1.4.4 (The Spencer complex is a resolution of O as a left D -module) Let F D be the filtration of D by the order of differential operators. Filter the Spencer complex Sp (D ) by the subcomplexes F p (Sp (D )) defined as δ F p k D k Θ δ F p k+1 D k 1 Θ δ (1) Show that, locally on, using coordinates x 1,..., x n, the graded complex gr F Sp (D ) := p gr F p Sp (D ) is equal to the Koszul complex of the ring O [ξ 1,..., ξ n ] with respect to the regular sequence ξ 1,..., ξ n. (2) Conclude that gr F Sp (D ) is a resolution of O. (3) Check that F p Sp (D ) = 0 for p < 0, F 0 Sp (D ) = gr F 0 Sp (D ) is isomorphic to O and deduce that the complex gr F p Sp (D ) := { [δ] gr F p kd k Θ is acyclic (i.e., quasi-isomorphic to 0) for p > 0. [δ] gr F p k+1d k 1 Θ [δ] } (4) Show that the inclusion F 0 Sp (D ) F p Sp (D ) is a quasi-isomorphism for each p 0 and deduce, by passing to the inductive limit, that the Spencer complex Sp (D ) is a resolution of O as a left D -module by locally free left D -modules. Exercise 1.4.5. Similarly, show that the complex Ω n+ (D ) is a resolution of ω as a right D -module by locally free right D -modules. Exercise 1.4.6. Let M r be a right D -module (1) Show that the natural morphism M r D (D O k Θ ) M r O k Θ defined by m P ξ mp ξ induces an isomorphism of complexes M r D Sp (D ) Sp (M r ). (2) Similar question for Ω n+ (D ) D M l Ω n+ (M l ). Let M be a left D -module and let M r the associated right module. We will now compare Ω n+ (M ) and (M r ). Consider the function Sp Z ε {±1}, a ε(a) = ( 1) a(a 1)/2, which satisfies in particular ε(a + 1) = ε( a) = ( 1) a ε(a), Given any k 0, the contraction is the morphism (1.4.7) ε(a + b) = ( 1) ab ε(a)ε(b). ω O k Θ Ω n k ω ξ ε(n k)ω(ξ ).

LECTURE 1. BASIC CONSTRUCTIONS 15 Exercise 1.4.8. Show that the isomorphism of right D -modules ( ω O D O k ) ι Θ Ω n k O D [ ] ( ) ω (1 ξ) P ε(n k)ω(ξ ) P (where the right structure of the right-hand term is the natural one and that of the left-hand term is nothing but that induced by the left structure after going from left to right) induces an isomorphism of complexes of right D -modules ι : ω O ( Sp (D ), δ ) ( Ω n+ O D, ). Exercise 1.4.9. Similarly, if M is any left D -module and M r = ω O M is the associated right D -module, show that there is an isomorphism M r ( D Sp (D ), δ ) ( ω O M O Θ, δ ) given on ω O M O k Θ by ( Ω n+ O M, ) ( Ω n+ O D, ) D M ω m ξ ε(n k)ω(ξ ) m. Exercise 1.4.10. Using Exercise 1.4.9, show that there is a functorial isomorphism Sp (M r ) Ω n+ (M ) for any left D -module M, which is termwise O -linear. Remark 1.4.11. We will denote by p DR (M r ) the Spencer complex Sp (M r ) and by p DR (M l ) the de Rham complex Ω n+ (M l ). The previous exercise gives an isomorphism p DR (M r ) p DR (M l ) and justifies this convention. We will use this notation below. Exercise 1.4.6 clearly shows that p DR is a functor from the category of right (resp. left) D -modules to the category of complexes of sheaves of C-vector spaces. It can be extended to a functor between the corresponding derived categories. However, in 3.1, we will introduce a functor DR to a subcategory, in order to keep the information that the differentials in such a complex are of a special kind, i.e., are differential operators. We will then extend this functor as a functor between suitable localized categories. Then p DR will be the composition of DR and the forgetful functor Forget. Denote by (E (n+,0), ( 1) n d ) the complex C O Ω n+ with the differential induced by ( 1) n d (here, we assume n + 0). More generally, let E (n+p,q) = E (n+p,0) E (0,q) and let d be the antiholomorphic differential. For any p, the complex (E (n+p, ), d ) is a resolution of Ω n+p n+. We therefore have a complex (E, ( 1)n d), which is the single complex associated to the double complex (E (n+, ), ( 1) n d, d ). In particular, we have a natural quasi-isomorphism of complexes of right D - modules: (Ω n+ O D, ) (E n+ O D, )

16 LECTURE 1. BASIC CONSTRUCTIONS by sending holomorphic k-forms to (k, 0)-forms. complexes are flat over O. Remark that the terms of these Exercise 1.4.12. Define a sheaf E k,l Sp (D ) by fine sheaves. for k, l 0 and find a Dolbeault resolution of Exercise 1.4.13. Let L be a O -module. (1) Show that, for any k, we have a (termwise) exact sequence of complexes 0 L O F k 1 (Sp (D )) L O F k (Sp (D )) L O gr F k (Sp (D )) 0. [Hint: use that the terms of the complexes F j (Sp (D )) and gr F k (Sp (D )) are O -locally free.] (2) Show that L O gr F Sp (D ) is a resolution of L as a O -module. (3) Show that L O Sp (D ) is a resolution of L as a O -module. Definition 1.4.14 (Godement resolution) (1) The Godement functor C 0 (see [God64, p. 167]) associates to any sheaf L the flabby sheaf C 0 (L ) of its discontinuous sections and to any morphism the corresponding family of germs of morphisms. Then there is a canonical injection L C 0 (L ). (2) Set inductively (see [God64, p. 168]) Z 0 (L ) = L, Z k+1 (L ) = C k (L )/Z k (L ), C k+1 (L ) = C 0 (Z k+1 (L )) and define δ : C k (L ) C k+1 (L ) as the composition C k (L ) Z k+1 (L ) C 0 (Z k+1 (L )). This defines a complex (C (L ), δ), that we will denote as (God L, δ). (3) Given any sheaf L, (God L, δ) is a resolution of L by flabby sheaves. For a complex (L, d), we view God L as a double complex ordered as written, i.e., with differential (δ i, ( 1) i d j ) on God i L j, and therefore also as the associated simple complex. The following exercise will be useful when computing direct images of D-modules in Lecture 4 Exercise 1.4.15 (Compatibility with the Godement functor) (1) Show that, if L and F are O -modules and if F is locally free, then we have a natural inclusion C 0 (L ) O F C 0 (L O F ), which is an equality if F has finite rank. More generally, show by induction that we have a natural morphism C k (L ) O F C k (L O F ), which is an equality if F has finite rank. (2) With the same assumptions, show that both complexes God (L ) O F and God (L O F ) are resolutions of L O F. Conclude that the natural morphism of complexes God (L ) O F God (L O F ) is a quasi-isomorphism. (3) Let M be a D -module. Show that p DR God M = God p DR M.

LECTURE 1. BASIC CONSTRUCTIONS 17 1.5. Filtered objects: the Rees construction Definition 1.5.1 (of a filtered D -module). A filtration F M of a D -module M will mean an increasing filtration satisfying (for left modules for instance) F k D F l M F k+l M k, l Z. We usually assume that F l M = 0 for l 0 and that the filtration is exhaustive, i.e., l F lm = M. Definition 1.5.2 (of the de Rham complex of a filtered D -module) Let F M be a filtered D -module. The de Rham complex p DR M is filtered by sub-complexes F p p DR M defined by δ F p F p k M r k Θ p DR M = δ F p k+1 M r k 1 Θ δ Ω n+k F p+km l Ω n+k+1 F p+k+1 M l and the filtered de Rham complex is denoted by p DR F M. Exercise 1.5.3. Show that the isomorphisms in Exercises 1.2.8 and 1.2.9 are isomorphisms of filtered objects M l O F D, F D O M l and M r O F D. It is possible to apply the techniques of the previous sections to filtered objects. A simple way to do that is to introduce the Rees object associated to any filtered object. Introduce a new variable z. We will replace the base field C with the polynomial ring C[z]. Definition 1.5.4 (Rees ring and Rees module). If (A, F ) is a filtered ring, we denote by A (or R F A if we want to insist on the dependence with respect to the filtration) the subring k F k A z k of A C C[z, z 1 ]. For instance, if F k A = 0 for k 1 and F k A = A for k 0, we have A = A C C[z]. Any filtered module (M, F ) on the filtered ring (A, F ) gives rise similarly to a module M on A. Notice that A is a graded ring with a central element z and that M is a graded module on this graded ring. Notice also that, as M is contained in M C C[z, z 1 ], the multiplication by z is injective on M. Exercise 1.5.5 (1) Show that the Rees construction gives an equivalence between the category of filtered (A, F )-modules (the morphisms should preserve the filtrations up to some fixed shift) and the subcategory of the category of graded A -modules (the morphisms are homogeneous), the object of which have no z-torsion. [Hint: recover M from M by setting M = M /(z 1) M.] (2) Recover gr F M as M /zm.

18 LECTURE 1. BASIC CONSTRUCTIONS (3) Show that, if one defines the filtration F km = F j M z j F k M z j, j k j>k then gr F M can be identified with gr F M C C[z], where the grading in the previous term is the sum of the grading on gr F M and of the grading of C[z] by the degree in z. Applying this construction to the filtered ring (D, F ) and its (left or right) modules, we obtain the following properties: Õ = O [z]; in local coordinates, any local section of D may be written in a unique way as α a α(x)ð α x = α ðβ xb α (x), where we set ð xi := z xi ; Θ is the locally free sheaf locally generated by ð x1,..., ð xn and we have [ð xi, ϕ] = z ϕ/ x i for any local section ϕ of Õ ; the sheaf Ω 1 is defined as z 1 C[z] C Ω 1, and Ω k = k Ω1 ; the differential d is induced by the differential d on Ω k ; the local basis ( dx i = z 1 dx i ) i is dual to the basis (ð xi ) i of Θ. Exercise 1.5.6 (1) Extend the results of 1.1-1.4 to graded D -modules. (2) Show that the same results hold for unnecessarily graded D -modules and unnecessarily homogeneous morphisms. Definition 1.5.7 (z-connection). Let M be a Õ -module. A z-connection on M is a C[z]-linear morphism : M Ω1 M which satisfies the Leibniz rule Exercise 1.5.8 ( f m) = f m + d f m. (1) Show that D has a universal z-connection for which (1) = i dx i ð xi. (2) Show the equivalence between left D -modules and Õ-modules equipped with an integrable z-connection. Definition 1.5.9. Let M be a left D -module. The de Rham complex p DRM is n+ the complex having as terms the Õ-modules Ω M and as differentials the z-connection ( 1) n. Right analogues (in particular, the Spencer complex) are defined similarly as well as the right-left correspondence. All properties of 1.4 extend in this setting. Exercise 1.5.10. Using Exercise 1.4.4, show that Sp( D ) is a resolution of Õ.

LECTURE 2 COHERENCE Although it would be natural to develop the theory of coherent D -modules in a way similar to that of O -modules, some points of the theory are not known to extend to D -modules (the lemma on holomorphic matrices). The approach which is therefore classically used consists in using the O -theory, and the main tools for that purpose are the good filtrations. This lecture is much inspired from [GM93]. 2.0. A reminder on coherence Let us begin by recalling the definition of coherence. Let A be a sheaf of rings on a space. Definition 2.0.1 (1) A sheaf of A -modules F is said to be A -coherent if it is locally of finite type: x, V x, q, A q V x F Vx, and if, for any open set U of and any A -linear morphism ϕ : A r U F U, the kernel of ϕ is locally of finite type. (2) The sheaf A is a coherent sheaf of rings if it is coherent as a (left and right) module over itself. Lemma 2.0.2. Assume A coherent. Let F be a sheaf of A -module. Then F is A - coherent if and only if F is locally of finite presentation: x, V x, p, q and an exact sequence A p V x A q V x F Vx 0. Classical theorems of Cartan and Oka claim the coherence of O.

20 LECTURE 2. COHERENCE 2.1. Coherence of D Let K be a compact subset of. We say K is a compact polycylinder if there exist a neighbourhood Ω of K, an analytic chart φ : Ω W of, and (ρ 1,..., ρ n ) (R + ) n such that φ(k) = {(x 1,..., x n ) C n i {1,..., n}, x i ρ i }. In particular a point x is a compact polycylinder. Let F be a sheaf on and K a polycylinder. We know by [God64], that lim F (U) F K (K) U K Uopen denoted by F (K). We have D Ω D Cn W and this isomorphism is compatible with the filtrations. Thus, to study local properties of gr F D or of D in the neighbourhood of a polycylinder K we can assume that K C n is a usual polycylinder. We have D (K) Hom C (O, O )(K) and any element of D (K) can be written in a unique way as α I c α α, with c α O (K) and I N n finite. The relations in Exercise 1.1.6 remain true when we replace U by K. We also have lim F k D (U) = { P D (K) P = α k c α α with c α O (K) }. U K Uopen Let F k D (K) be this O(K)-submodule of D(K). We get a filtration of D (K) having the same properties as that of D (U). Finally, we deduce from Exercise 1.1.7 the existence of a canonical ring isomorphism We thus have an isomorphism gr F (D (K)) (gr F D )(K). gr F (D (K)) O C n(k)[ξ 1,..., ξ n ] by an inductive limit on U K. By a theorem of Frisch [Fri67], O C n(k) is a Noetherian ring and, for any x K, the ring O C n,x is flat over O C n(k). We therefore get: Proposition 2.1.1. If K is a compact polycylinder, gr F D (K) is a Noetherian ring. Proposition 2.1.2. The ring D (K) is Noetherian. Proof. Let I D (K) be a left ideal. We have to prove that it is finitely generated. Set F k I = I F k D (K). Then gr F I = k N F k I/F k 1 I is an ideal in gr F D (K), thus is of finite type. Let e 1,..., e l be homogeneous generators of gr F I, of degrees d 1,..., d l and P 1,..., P l elements of I with σ(p j ) = e j. It is easy to prove, by induction on the order of P I, that I = l i=1 D (K) P i (left to the reader). Theorem 2.1.3. The sheaf of rings D is coherent.

LECTURE 2. COHERENCE 21 Proof. If U is open and φ : (D U ) q (D U ) p is a morphism of left D U -modules, we have to prove that Ker φ is locally of finite type. We may assume that U is an open chart, thus in fact an open subset of C n. Let ε 1,..., ε q be the canonical base of D (U) q and k N be such that, for all i {1,..., q}, φ(ε i ) F k D (U) p. We then have φ(f l D q U ) F k+ld p U, and Ker φ F l D q U is the kernel of a morphism between locally free O U -modules of finite type F l D q U F k+ld p U. Thus Ker φ F l D q U is O U coherent, and Ker φ is the union of these O U -modules. Let K U be a compact polycylinder. By Theorem A of Cartan, for any x K, the sheaf [Ker φ F l D q U ] x is generated by Γ(K, Ker φ F l D q U ), which is included in Γ(K, Ker φ). Thus for any x K, (Ker φ) x is generated by Γ(K, Ker φ), i.e., any germ of section of Ker φ at x is a linear combination with coefficients in O,x of sections of Ker φ over K. By left exactness of Γ(K, ) we have an exact sequence of left D (K)-modules: 0 Γ(K, Ker φ) Γ(K, D q U ) = D (K) p Γ(K, φ) D (K) p. Because D (K) is Noetherian, Γ(K, Ker φ) is then of finite type as a left D (K)- module. It is then easy to build a surjective morphism of left D K -modules (D K ) r (Ker φ) K 0 using the two properties above. This proves that Ker φ is locally of finite type. Exercise 2.1.4 (1) Prove similarly the coherence of the sheaf of rings gr F D and that of the Rees sheaf of rings R F D (see Definition 1.5.4). (2) Let D be a hypersurface and let O ( D) be the sheaf of meromorphic functions on with poles on D at most (with arbitrary order). Prove similarly that O ( D) is a coherent sheaf of rings. (3) Prove that D ( D) := O ( D) O D is a coherent sheaf of rings. (4) Let i : Y denote the inclusion of a smooth submanifold. Show that i D := O Y O D is a coherent sheaf of rings on Y. 2.2. Coherent D -modules and good filtrations Let M be a D -module. From Theorem 2.1.3 and the preliminary reminder on coherence, we know that M is D -coherent if it is locally finitely presented, i.e., if for any x there exists an open neighbourhood U of x an an exact sequence D q U D p U M U.

22 LECTURE 2. COHERENCE Exercise 2.2.1 (1) Let M N be a D -submodule of a coherent D -module N. Show that, if M is locally finitely generated, then it is coherent. (2) Let φ : M N be a morphism between coherent D -modules. Show that Ker φ and Coker φ are coherent. Definition 2.2.2 (Good filtrations). Let F M be a filtration of M (see 1.5). We say that the filtration is good if the Rees module R F M is coherent over the coherent sheaf R F D (i.e., locally finitely presented). It is useful to have various criteria for a filtration to be good. Exercise 2.2.3 (Characterization of good filtrations). Show that the following properties are equivalent: (1) F M is a good filtration; (2) for any k Z, F k M is O -coherent, and, for any x, there exists a neighbourhood U of x and k 0 Z such that, for any k 0, F k D U F k0 M U = F k+k0 M U ; (3) the graded module gr F M is gr F D -coherent. Conclude that, if F M, G M are two good filtrations of M, then, locally on, there exists k 0 such that, for any k, we have F k k0 M G k M F k+k0 M. Proposition 2.2.4 (Local existence of good filtrations). If M is D -coherent, then it admits locally on a good filtration. Proof. Exercise 2.2.5. Exercise 2.2.5 (Local existence of good filtrations) (1) Show that, if M has a good filtration, then it is D -coherent and gr F M is gr F D -coherent. In particular, a good D -module is coherent. [Hint: use that the tensor product C[z]/(z 1) C[z] is right exact.] (2) Conversely, show that any coherent D -module admits locally a good filtration. [Hint: choose a local presentation D q ϕ U D p U M U 0, and show that the filtration induced on M U by F D p U is good by using Exercise 2.2.3: Set K = Im ϕ and reduce the assertion to showing that F j D K is O -coherent; prove that, up to shrinking U, there exists k o N such that ϕ(f k D q U ) F k+ko D p U for each k; deduce that ϕ(f k D q U ), being locally of finite type and contained in a coherent O -module, is O -coherent for each k; conclude by using the fact that an increasing sequence of coherent O -modules in a coherent O -module is locally stationary.] (3) Show that, locally, any coherent D -module is generated over D by a coherent O -submodule.

LECTURE 2. COHERENCE 23 (4) Let M be a coherent D -module and let F be a O -submodule which is locally finitely generated. Show that F is O -coherent. [Hint: choose a good filtration F M and show that, locally, F F k M for some k; apply then the analogue of Exercise 2.2.1(1) for O -modules.] Good filtrations are the main tool to get results on coherent D -modules from theorems on coherent O -modules. This justifies the following definition: Definition 2.2.6 (Good D-modules, see [SS94]). A D -module is good if, for any compact set K, there exists, on some neighbourhood U of K, a finite filtration of M U by D U -submodules such that each successive quotient has a good filtration. Remark 2.2.7. It is not known whether any coherent D -module has globally a good filtration, or even whether it is good. Nevertheless, it is known that any holonomic D - module (see Definition 5.2.1) has a good filtration (see [Mal94a, Mal94b, Mal96]); in fact, if such is the case, there even exists a coherent O -submodule F of M which generates M, i.e., such that the natural morphism D O F M is onto (this is a little stronger than the existence of a good filtration, if the manifold is not compact). The main results concerning coherent D -modules are obtained from the theorems of Cartan and Oka for O -modules. Theorem 2.2.8 (Theorems of Cartan-Oka for D -modules). Let M be a left D -module and let K be a compact polycylinder contained in an open subset U of, such that M has a good filtration on U. Then, (1) Γ(K, M ) generates M K as a O K -module, (2) For any i 1, H i (K, M ) = 0. Proof. This is easily obtained from the theorems A and B for O -modules, by using inductive limits (for A it is obvious and, for B, see [God64, Th. 4.12.1]). Theorem 2.2.9 (Characterization of coherence for D -modules) (1) Let M be a left D -module. Then, for any small enough compact polycylinder K, we have the following properties: (a) M (K) is a finite type D(K)-module, (b) For any x K, O x O(K) M (K) M x is an isomorphism. (2) Conversely, if there exists a covering {K α } by polycylinders K α such that = K α and that on any K α the properties (1a) and (1b) are fulfilled, then M is D - coherent.