An introduction to D-modules

An introduction to -modules Pierre Schapira raft, v6, March 2013 (small corrrection March 2017) http://www.math.jussieu.fr/ schapira/lectnotes schapira@math.jussieu.fr

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Contents 1 The ring X 7 1.1 Construction of X........................ 7 1.2 Filtration on X......................... 12 1.3 Characteristic variety....................... 14 1.4 e Rham and Spencer complexes................ 18 1.5 Homological properties of X.................. 21 1.6 erived category and duality................... 27 2 Operations on -modules 31 2.1 External product......................... 31 2.2 Transfert bimodule........................ 32 2.3 Inverse images........................... 35 2.4 Holomorphic solutions of inverse images............ 40 2.5 irect images........................... 42 2.6 Trace morphism.......................... 46 2.7 -modules associated with a submanifold............ 49 Exercises................................. 58 3 Appendix 59 3.1 Symplectic geometry....................... 59 3.2 Coherent sheaves......................... 66 3.3 Filtered sheaves.......................... 68 3.4 Almost commutative filtered rings................ 71 3.5 O-modules............................. 75 3

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Introduction The aim of these Notes is to introduce the reader to the theory of -modules in the analytical setting. This text is a short introduction, not a systematic study. In particular many proofs are skipped and the reader is encouraged to consult the literature. To our opinion, the best reference to -modules is [Ka03], and, in fact, most of the material of these Notes are extracted from this book. Indeed, although we do not mention it in the course of the notes, almost all the results and proofs exposed here are due to Masaki Kashiwara. References for -modules. Some classical titles are [Ka70, Ka83, Bj93, Ka03] and, in the algebraic setting, [Bo87]. An elementary introduction may also be found in [Co85]. Applications to -modules to representation theory are studied in [HTT08]. Related theories to -modules. Microdifferential operators are the natural localization of differential operators. References are made to [SKK73, Ka83, Sc85]. In fact, microdifferential operators may also be considered as an avatar of rings of deformation quantization for which there exists an enormous literature. See [KS12] and the references therein. References for categories, homological algebra and sheaves. The reader is assumed to be familiar with sheaf theory as well as homological algebra, including derived categories. An exhaustive treatment may be found in [KS06] and a pedagogical treatment is provided in [Sc08]. Among numerous other references, see [GM96], [KS90, Ch. 1, 2] [We94]. Recent develoments. Chiral algebras are built upon -module theory and are exposed in [B04, FG10]. A theory of non linear differential equations, in the spirit of -module theory, is sketched in [KM99] as well as in [B04]. History. An outline of -module theory, including holonomic systems, was proposed by Mikio Sato in the early 60 s in a series of lectures at Tokyo 5

6 CONTENTS University (see [Sc07]). However, it seems that Sato s vision has not been understood until his student, Masaki Kashiwara, wrote his thesis in 1970 (see [Ka70]). Independently and at the same time, J. Bernstein, a student of I. Gelfand at Moscow s University, developed a very similar theory in the algebraic setting (see [Be71]).

Chapter 1 The ring X In all these Notes, all rings are associative and unital. If R is a ring, an R-module means a left R-module and we denote by Mod(R) the abelian category of such modules. We denote by R op the opposite ring. Hence, Mod(R op ) denotes the category of right R-modules. If a, b belong to R, their bracket [a, b] is given by [a, b] = ab ba. We use similar conventions and notations for a sheaf of rings R on a topological space X. In particular, Mod(R) denotes the category of sheaves of left R-modules on X. 1.1 Construction of X O-modules Let X denote a complex manifold, O X its structural sheaf, that is, the sheaf of holomorphic functions on X. Unless otherwise specified, we denote by d X the complex dimension of X. We denote by Ω p X the sheaf of holomorphic p-forms and one sets Ω X = Ω d X X. One also sets (1.1) Ω = p Ω p X. We denote by Mod(C X ) the abelian category of sheaves of C-vector spaces on X, and we denote by Hom and the internal Hom and tensor product in this category. For F Mod(C X ), we set End(F ) = Hom CX (F, F ). Similarly, we denote by Mod(O X ) the abelian category of sheaves of O X - modules, and we denote by Hom O and O the internal Hom and tensor product in this category. We denote by Mod coh (O X ) the full abelian subcategory consisting of coherent sheaves. One denotes by Θ X the sheaf of Lie algebras of holomorphic vector fields. Hence, Θ X = Hom O (Ω 1 X, O X). 7

8 CHAPTER 1. THE RING X The sheaf Θ X has two actions on Ω, that we recall. Let v Θ X. The interior derivative i v End(Ω X ) is characterized by the conditions i v (a) = 0, a O X (1.2) i v (ω) = v, ω, ω Ω 1, i v (ω 1 ω 2 ) = (i v ω 1 ) ω 2 + ( ) p ω 1 (i v ω 2 ), ω 1 Ω p X. Note that i v : Ω p X Ωp 1 X is of degree 1. On the other-hand, the Lie derivative L v End(Ω X ) is characterized by the conditions L v (a) = v(a) = v, da, a O X, (1.3) d L v = L v d, L v (ω 1 ω 2 ) = (L v ω 1 ) ω 2 + ω 1 (L v ω 2 ), The Lie derivative is of degree 0 and satisfies (1.4) [L u, L v ] = L [u,v], u, v Θ X. One has the relations (1.5) L v = d i v + i v d. Using v L v, one may regard Θ X as a subsheaf of End(O X ). The ring X efinition 1.1.1. One denotes by X the subalgebra of End(O X ) generated by O X and Θ X. If (x 1,..., x n ) is a local coordinate system on a local chart U of X, then a section P of X on U may be uniquely written as a polynomial (1.6) P = a α α and we use the classical notations for multi- where a α O X, i = xi indices: = xi α m α = (α 1,..., α n ) N n, α = α 1 + + α n, if X = (X 1,..., X n ), then X α = X α 1... X αn. Proposition 1.1.2. Let R be a sheaf of C X -algebras and let ι : O X R and ϕ : Θ X R be C X -linear morphisms satisfying (here, a, b O X and u, v Θ X ):

1.1. CONSTRUCTION OF X 9 (i) ι: O X R is a ring morphism, that is, ι(ab) = ι(a)ι(b), (ii) ϕ : Θ X R is left O X -linear, that is, ϕ(av) = ι(a)ϕ(v), (iii) ϕ : Θ X R is a morphism of Lie algebras, that is, [ϕ(u), ϕ(v)] = ϕ([u, v]), (iv) [ϕ(v), ι(a)] = ι(v(a)) for any v Θ X and a O X. Then there exists a unique morphism of C X -algebras Ψ : X R such that the composition O X X R coincides with ι and the composition Θ X X R coincides with ϕ. The proof is straightforward. Corollary 1.1.3. Let M be an O X -module and let µ: O X End(M) be the action of O X on M. Let ψ : Θ X End(M) be a C X -linear morphism satisfying: (i) µ(a) ψ(v) = ψ(av) (resp. ψ(v) µ(a) = ψ(av)). (ii) [ψ(v), ψ(w)] = ψ([v, w]) (resp. [ψ(v), ψ(w)] = ψ([v, w])), (iii) [ψ(v), µ(a)] = µ(v(a)), (resp. [ψ(v), µ(a)] = µ(v(a))). Then there exists one and only one structure of a left (resp. right) X -module on M which extends the action of Θ X. Proof. For the structure of a left module, apply Proposition 1.1.2 to R = End(M). The case of right modules follows since the bracket [a, b] op in op X is [a, b], where [a, b] is the bracket in X. Examples 1.1.4. (i) The sheaf O X is naturally endowed with a structure of a left X -module and 1 O X is a generator. Since the anihilator of 1 is the left ideal generated by Θ X, we find an exact sequence of left X -modules X Θ X X O X 0. Note that if X is connected and f is a section of O X, f 0 (i.e., f is not identically zero), then f is also a generator of O X over X. This follows from the Weierstrass Preparation Lemma. Indeed, choosing a local coordinate system (x 1,..., x n ), one may write f = m j=0 a j(x )x j 1, with a m 1. Then 1 m (f) = m!. (ii) The sheaf Ω X is naturally endowed with a structure of a right X -module, by v(ω) = L v (ω), v Θ X, ω Ω X.

10 CHAPTER 1. THE RING X (iii) Let F be an O X -module. Then X O F is a left X -module. (iv) Let Z be a closed complex submanifold of X of codimension d. Then HZ d (O X) is a left X -module. (v) Let X be a complex manifold and let P be a differential operator on X. The differential equation P u = v may be studied via the left X -module X / X P. (See below.) (vi) Let X = C n and consider the differential operators P = n j=1 2 j, Q ij = x i j x j i. Consider the left ideal J of X generated by P and the family {Q ij } i<j. The left X -module X /J is naturally associated to the operator P and the orthogonal group O(n; C). Internal hom and tens The sheaf X is a sheaf of non commutative rings and C X is contained (in fact, is equal, but we have not proved it here) in its center. It follows that we have functors: Hom : (Mod( X )) op Mod( X ) Mod(C X ), : Mod( op X ) Mod( X) Mod(C X ). We shall now study hom and tens over O X. Let M, N and P be left X -modules and let M and N be right X - modules. (a) One endows M O N with a structure of a left X -module by setting v(m n) = v(m) n + m v(n), m M, n N, v Θ X. (b) One endows Hom O (M, N ) with a structure of a left X -module by setting v(f)(m) = v(f(m)) f(v(m)), m M, f Hom O (M, N ), v Θ X. (c) One endows N O M with a structure of a right X -module by setting (n m)v = nv m n vm, m M, n N, v Θ X. (d) One endows and Hom O (M, N ) with a structure of a left X -module by setting v(f)(m) = f(mv) f(m)v m M, f Hom O (M, N ), v Θ X.

1.1. CONSTRUCTION OF X 11 (e) One endows and Hom O (M, N ) with a structure of a right X -module by setting (fv)(m) = f(m)v + f(vm) m M, f Hom O (M, N ), v Θ X. There are isomorphisms of C X -modules; Hom (M O N, P) Hom (M, Hom O (N, P)), Hom (M O M, N ) Hom (M, Hom O (M, N )), (M O M) N M (M O N ). To summarize, we have functors O : Mod( X ) Mod( X ) Mod( X ), O : Mod( op X ) Mod( X) Mod( op X ), Hom O : Mod( X ) op Mod( X ) Mod( X ), Hom O : Mod( op X )op Mod( op X ) Mod( X), Hom O : Mod( X ) op Mod( op X ) Mod(op X ). Remark 1.1.5. Following [HTT08] who call it the Oda s rule, one way to memorize the left an right actions is to use the correspondence left = 0, right = 1, a b = a + b and Hom (a, b) = a + b. Twisted X -modules Let L be a holomorphic line bundle, that is, a locally free O X -module of rank one. One sets There are a natural isomorphisms L 1 = Hom O (L, O X ). O X Hom O (L, L) Hom O (L, O X ) O L. If s is a section of L 1 and t a section of L, their product will be denoted by s, t, a section of O X. Let R be a O X -ring, that is, a sheaf of rings together with a morphism of rings O X R. One can define a new O X -ring L R L 1 by setting (with obvious notations) (s m t) (s m t ) = s m t, s m t. If M is a left R-module, then L O M is a left L O R O L 1 -module. Clearly:

12 CHAPTER 1. THE RING X Proposition 1.1.6. The functor M L O M is an equivalence of categories from Mod(R) to Mod(L O R O L 1 ). Proposition 1.1.7. There is an isomorphism of O X -rings op X X O Ω 1 X. Ω X O Proof. The right X -module structure of Ω X defines the morphism of rings op X End(Ω X). On the other-hand, the morphism X End(O X ) defines the morphism of rings Ω X O X O Ω 1 X End(Ω X ). Both these morphisms are monomorphisms, and to check that their images in End(Ω X ) are the same, one remark that both rings are generated by O X and Θ X. Corollary 1.1.8. The functor M Ω O M induces an equivalence of categories Mod( X ) Mod( op X ) Remark 1.1.9. Suppose to be given a volume form dv on X. Then f fdv gives an isomorphism O X Ω X and we get an isomorphism X op X. The image of a section P X by this isomorphism is called its adjoint with respect to dv and is denoted by P. Hence, for a left X -module M and a section u of M, we have P u = (u dv) P. Clearly (Q P ) = P Q. If (x 1,..., x n ) is a local coordinate system on X and dv = dx 1 dx n, one checks that x i = x i and x i = xi. 1.2 Filtration on X Total symbol of differential operators Assume X is affine, that is, X is open in a finite dimensional complex vector space E. Let P be a section of X. One defines its total symbol (1.7) σ tot (P )(x; ξ) := exp x, ξ P (exp x, ξ ) = a α (x)ξ α. α m Using (1.6), one gets that σ tot (P ) is a function on X E, polynomial with respect to ξ E. This function highly depends on the affine structure, but

1.2. FILTRATION ON X 13 its order (a locally constant function on X) does not. It is called the order of P and denoted ord(p ). If Q is another differential operator with total symbol σ tot (Q), it follows easily from the Leibniz formula that the total symbol σ tot (R) of R = P Q is given by: (1.8) By this formula, one gets that σ tot (R) = α N n 1 α! α ξ (σ tot (P )) α x (σ tot (Q)). ord(p Q) = ord(p ) + ord(q), ord([p, Q]) ord(p ) + ord(q) 1. The ring X is now endowed with the filtration by the order, Fl m ( X ) = {P X ; ord(p ) m}. One can give a more intrinsic definition of the filtration. Filtration on X efinition 1.2.1. The filtration Fl X on X is given by Fl 1 X = {0}, Fl m X = {P X ; [P, O X ] Fl m 1 X }. Note that { Fl 0 X = O X, Fl 1 X = O X Θ X, (1.9) Fl m X Fl l X Fl m+l X, [Fl m X, Fl l X ] Fl m+l 1 X. One denotes by gr X the associated graded ring, by σ : Fl X gr X the principal symbol map and by σ m : Fl m X gr m X the map symbol of order m. One shall not confuse the total symbol, which is defined on affine charts, and the principal symbol, which is well defined on manifolds. It follows from (1.8) that σ(p )σ(q) = σ(q)σ(p ) = σ(p Q). Hence, gr ( X ) is a commutative graded ring. Moreover, gr 0 ( X ) O X and gr 1 ( X ) Θ X. enote by S O (Θ X ) the symmetric O X -algebra associated with the locally free O X -module Θ X. By the universal property of symmetric algebras, the morphism Θ X gr ( X ) may be extended to a morphism of symmetric algebra (1.10) S O (Θ X ) gr X.

14 CHAPTER 1. THE RING X Proposition 1.2.2. The morphism (1.10) is an isomorphism. Proof. Choose a local coordinate system (x 1,..., x n ) on X. Then Θ X n i=1 O X i and the correspondence i ξ i gives the isomorphism S O (Θ X ) α O X α O X [ξ 1,..., ξ n ] gr X. enote by π : T X X the projection. There is a natural monomorphism Θ X π O T X. Indeed, a vector field on X is a section of the tangent bundle T X, hence defines a linear function on T X. By the universal property of symmetric algebra, we get a monomorphism S O (Θ X ) π O T X. Applying Proposition 1.2.2, we get an embedding of C X -algebras: gr X π O T X. In the sequel, we shall still denote by σ : X π O T X and σ m : Fl m X π O T X, the maps obtained by applying the inverse of the isomorphism (1.10) to σ and σ m. Theorem 1.2.3. The sheaf of rings X is right and left Noetherian. Proof. This follows from Proposition 1.2.2 and general results of [Ka03, Th. A.20] on filtered ring with associated commutative graded ring (see Theorem 3.3.5). 1.3 Characteristic variety We shall use here the results of 3.4.

1.3. CHARACTERISTIC VARIETY 15 Poisson s structures The graded ring gr ( X ) is endowed with a natural Poisson bracket induced by the commutator in X. On the other hand, the sheaf O T X (hence, the sheaf π O T X) is endowed with the Poisson bracket induced by the symplectic structure of T X. Recall that if (x 1,..., x n ; ξ 1,..., ξ n ) is a local symplectic coordinate system on T X, this Poisson bracket is given by n {f, g} = ξi f xi g xi f ξi g. Proposition 1.3.1. The Poisson bracket on π O T X bracket on gr ( X ). i=1 induces the Poisson Proof. Let P Fl m ( X ) and Q Fl l ( X ). Then [P, Q] Fl m+l 1 ( X ) and it follows from (1.8) that n ( (1.11) σ m+l 1 ([P, Q]) = ξi σ m (P ) xi σ l (Q) ξi σ l (Q) xi σ m (P ) ). i=1 Hence, σ m+l 1 ([P, Q]) = {σ m (P ), σ l (Q)}. Good filtration We shall recall some notions also introduced in 3.3, 3.4. Recall that a good filtration on a coherent X -module M is a filtration which is locally the image of a finite free filtration. Hence, a filtration Fl M on M is good if and only if, locally on X, Fl j M = 0 for j 0, Fl j M is O X -coherent, (1.12) locally on X, (Fl k X ) (Fl j M) = Fl k+j M for j 0 and all k 0. Applying Corollary 3.3.6, we get: Lemma 1.3.2. Let M be a coherent X -module, N M a coherent submodule. Assume that M is endowed with a good filtration Fl M. Then the induced filtration on N defined by Fl j N = N Fl j M is good. enote by Mod gr coh (gr X) the abelian category of coherent graded gr X - modules and consider the functor : Mod gr coh (gr X) Mod coh (π O T X), gr M π O T X gr X gr M.

16 CHAPTER 1. THE RING X This functor is exact and faithful. If M is a coherent X -module endowed with a good filtration, the π O T X-module gr M = π O T X gr X gr M is thus coherent and its support satisfies: supp( gr M) = {p T X; σ(p )(p) = 0 for any P Icar(M)}. In the sequel, we shall often confuse gr M and gr M. efinition 1.3.3. The characteristic variety of M, denoted char(m), is the closed subset of T X characterized as follows: for any open subset U of X such that M U is endowed with a good filtration, char(m) T U is the support of gr M U. Theorem 1.3.4. (i) char(m) is a closed C -conic analytic subset of T X. (ii) char(m) is involutive for the Poisson structure of T X, and in particular, codim(char(m)) d X. (iii) If 0 M M M 0 is an exact sequence of coherent X - modules, then char(m) = char(m ) char(m ). Proof. (i) is obvious, (ii) follows from Gabber s theorem and (iii) follows from Lemma 1.3.2. Note that the involutivity theorem has first been proved by Sato, Kashiwara and Kawai [SKK73] using analytical tools, before Gabber gave is purely algebraic proof. Suppose that a coherent X -module M is generated by a single section u. Then M X /I, where I is the anihilator of u. There is a natural filtration on M, the image of Fl X. Put Fl j I = I Fl j X. It follows from Corollary 3.3.6 that the graded ideal gr I is coherent. Moreover, since gr M = gr X /gr I, we get (1.13) char(m) = {p T X; σ j (P )(p) = 0 for all P Fl j (I)}. If {P 0,..., P N } generates I it follows that char(m) j σ(p j ) 1 (0). In general the equality does not hold, since the family of the P j s may generate I although the family of the σ mj (P j ) s does not generate gr I.

1.3. CHARACTERISTIC VARIETY 17 Example 1.3.5. If X = A 1 (C), the affine line, the ideal generated by and x is X, but the ideal generated by their principal symbols is not O T X. Corollary 1.3.6. Let M be a coherent X -module, let p T X and assume that p / char(m). Let u M. Then there exists a section P X defined in a neighborhood of π(p) with P u = 0 and σ(p )(p) 0. Proof. Consider the sub- X -module X u generated by u. It is coherent and its characteristic variety is contained in that of M. Let I denotes the anihilator ideal of u in X and let P 1,..., P N denotes sections of this ideal such that σ(p 1 ),..., σ(p N ) generate the graded ideal gr I. Such a finite family exists since gr I is coherent. Since p / char( X u), there exists j with σ(p j )(p) 0. Example 1.3.7. (i) char(o X ) = T X X, the zero-section of T X. (ii) char( X / X P ) = {p T X; σ(p )(p) = 0}. Multiplicities By the result of Proposition 3.5.2, one sees that if M is a coherent X - module and V is an irreducible component of char(m) V, then mult V ( gr M) depends only on M. efinition 1.3.8. Let V be a closed analytic subset of T X and let M be a coherent X -module such that V is an irreducible component of char(m) V. The number mult V ( gr M) is called the multiplicity of M along V and denoted mult V (M). If 0 M M M 0 is an exact sequence of cherent X -modules with V irreducible in char(m) V, then mult V (M) = mult V (M ) + mult V (M ). Involutive basis efinition 1.3.9. Let I be a coherent ideal of X and let {P 1,..., P N } be a family of sections of I, with P j of order m j. One says that this family is an involutive basis of I if the family {σ(p 1 ),..., σ(p N )} generates gr I. Proposition 1.3.10. Assume (i) N j=1σ mj (P j ) 1 (0) is of codimension N,

18 CHAPTER 1. THE RING X (ii) there exist Q jkl Fl mj +m k m l 1 X such that for all j, k [P j, P k ] = l Q jkl P l Then {P 1,..., P N } is an involutive basis. Proof. Set p j = σ(p j ). Let a j gr l mj X with a j p j = 0. j By Proposition 3.4.9, it is enough to find A j X with σ(a j ) = a j and such that A j P j = 0. j By the hypothesis, the sequence {p 1,..., p N } is a regular sequence. Hence, we may find r ij gr l mi m j X satisfying a j = i r ij p i, r ij = r ji. Next we choose R ij Fl l mi m j X with σ(r ij ) = r ij and R ij = R ji. Set A j = i R ijp i. Then σ l mj (A j ) = a j and A j P j = R ij P i P j = R ij [P i, P j ] j i,j i<j = R ij Q ijk P k. i<j k Set S k = i<j R ijq ijk. Then S k has order l m k 1, j (A j S j )P j = 0 and σ l (A j S j ) = a j. 1.4 e Rham and Spencer complexes If A is a ring, M is an A-module, and ϕ := (ϕ 1,..., ϕ n ) are n-commuting endomorphisms of M, one can define the Koszul complex K (M; ϕ) and the co-koszul complex K (M; ϕ). We refer to [Sc08] for an exposition. Also recall the e Rham complex (1.14) R X (O X ) := 0 Ω 0 X d Ω 1 d X Ω d X X 0,

1.4. E RHAM AN SPENCER COMPLEXES 19 where d is the differential. Let M be a left X -module. One defines the differential d: M Ω 1 X O M as follows. In a local coordinate system (x 1,..., x dx ) on X, the differential d is given by M Ω 1 X O M, m i dx i i m and one checks easily that this does not depend on the choice of the local coordinate system. One defines the e Rham complex of M, denoted R X (M), as the complex (1.15) R X (M) := 0 Ω 0 X O M d Ω d X X O M 0, where Ω 0 X O M is in degree 0 and the differential d is characterized by: d(ω m) = dω m + ( ) p ω dm, ω Ω p X, m M. Note that R X ( X ) C b (Mod( op X )), the category of bounded complexes of right X -modules, and (1.16) R X (M) R X ( X ) M. Recall that there is a natural right -linear morphism Ω X O X Ω X. Moreover, one checks easily that the composition Ω d X 1 X O X Ω d X X O X Ω X is zero. Hence, we get a morphism in the derived category b ( op X ) (1.17) R X ( X ) Ω X [ d X ]. Proposition 1.4.1. The morphism (1.17) induces an isomorphism in b ( op X ). Proof. Since the morphism is well defined on X, we may argue locally and choose a local coordinate system. In this case, there is an isomorphism of complexes (1.18) R X ( X ) K ( X ; 1,..., dx ) where the right hand side is the Koszul complex of the the sequence 1,..., n acting on the left on X. Since this sequence is clearly regular, the result follows.

20 CHAPTER 1. THE RING X Applying Proposition 1.4.1 and isomorphism (1.16), we get: Corollary 1.4.2. Let M be a left X -module. Then R X (M) Ω X L M [ d X ]. Let us apply the contravariant functor Hom op(, X ) to the complex R X ( X ). One sets (1.19) SP X ( X ) := Hom (R X ( X ), X ), and calls SP X ( X ) the Spencer complex. (1.20) SP X ( X ) := d x 0 X d O ΘX X O Θ X X 0, One deduces from (1.18) the isomorphism of complexes (1.21) SP X ( X ) K ( X ; 1,..., dx ) where the right hand side is the co-koszul complex of the sequence 1,..., dx acting on the right on X. Since this sequence is clearly regular, we obtain: Proposition 1.4.3. The left -linear morphism X O X induces an isomorphism SP X ( X ) O X in b ( X ). Corollary 1.4.4. Let M be a left X -module. There is an isomorphism in b (C X ) RHom (O X, M) R X (M). Proof. Since SP X ( X ) is a complex of locally free X -modules of finite rank, one has RHom (O X, M) Hom (SP X ( X ), M) Hom (SP X ( X ), X ) M R X ( X ) M R X (M). Proposition 1.4.5. One has the isomorphism RHom (O X, X )[d X ] Ω X RHom op(ω X, X )[d X ] O X RHom (O X, O X ) C X.

1.5. HOMOLOGICAL PROPERTIES OF X 21 Proof. (i) One has the chain of isomorphisms RHom (O X, X )[d X ] RHom (SP X ( X ), X )[ d X ] Hom (SP X ( X ), X )[ d X ] R( X )[ d X ] Ω X. (ii) The proof is similar. (iii) The canonical morphism C X Hom (O X, O X ) induces the morphism C X RHom (O X, O X ) Hom (SP X ( X ), O X ) Ω X. The isomorphism C X Ω X is the classical Poincaré lemma. 1.5 Homological properties of X Vanishing theorems and dimension There is a corresponding theorem to Theorem 3.5.6 for -modules. Theorem 1.5.1. Let M be a coherent X -module. Then (i) Ext k (M, X) is coherent for all k and is 0 for k < codim(char(m)), (ii) codim(char(ext k (M, X))) k, (iii) char(ext k (M, X)) char(m), (iv) Ext k (M, X) = 0 for k > d X. Corollary 1.5.2. Let M be a coherent X -module. Ext d X (M, X) has pure dimension d X. Then the support of Proof. First we construct by induction a finite free filtered resolution of Fl M, that is, a filtered exact sequence of Fl X -modules Fl L 1 Fl L 0 Fl M 0 where the Fl L j s are filtered finite free. We denote by d j the differential. Set: Fl L := Fl L 1 Fl L 0 0, gr L := gr L 1 gr L 0 0.

22 CHAPTER 1. THE RING X Then gr L 1 gr L 0 gr M 0 is exact. Put L j = Hom (L j, X ), L = Hom (L, X ) = 0 L 0 L 1 One defines a filtration Fl L j on L j by setting Fl m L j = {ϕ Hom (L j, X ); ϕ(fl k L j ) Fl k+m X for all k}. Clearly, this filtration on L j gr L j. In other words, is good and moreover Hom gr (gr L j, gr ) Hom gr (gr L, gr ) gr L. Put Z k = Ker(L k d k L k+1 ), I k = Im(L k 1 L k ) H k (L ) = Z k /I k. We endow Z k with the induced filtration and H k (L ) with the filtration image of Fl Z k. Since Ext k (M, X) H k (L ), we get a filtration Fl Ext k (M, X) on this module. Moreover Ext k (gr M, gr gr X)) H k (gr L ). In order to complete the proof, we need a lemma. Lemma 1.5.3. gr H k (L ) is a subquotient of H k (gr L ). Proof of Lemma 1.5.3. On the other-hand, H k (gr m L ) = Fl m(l k ) (dk ) 1 Fl m 1 L k+1 Fl m 1 (L k ) + dk 1 Fl m L k 1 Fl m (Z k ). Fl m 1 (Z k ) + d k 1 Fl m L k 1 gr m H k (L ) = The result then follows from Fl m (Z k ) Fl m 1 (Z k ) + I k Fl m (Z k ). Fl m 1 (Z k ) + d k 1 Fl m L k 1 Fl m 1 (Z k ) + I k Fl m (Z k ).

1.5. HOMOLOGICAL PROPERTIES OF X 23 End of proof of Theorem 1.5.1. It follows that (1.22) char(ext k (M, X)) supp(ext k gr (gr M, gr X))). (i) By Theorem 3.5.6, Ext k O ( gr M, O T X)) = 0 for k < codim(char(m)). By (1.22), we get that Ext k (M, X) = 0 for k < codim(char(m)). (ii) By Theorem 3.5.6, codim(supp(ext k gr (gr M, gr X))) k. By (1.22), we get that codim(char(ext k (M, X))) k. (iii) follows from the inclusion supp(ext k gr (gr M, gr X)) supp(gr M). (iv) follows from (ii) and the involutivity of the characteristic variety of Ext k (M, X). Example 1.5.4. Let d X = 1. Then any coherent ideal I of X is projective since Ext j ( X/I, X ) = 0 for j > 1. Let t denote a local holomorphic coordinate. The left ideal of X generated by t 2 and t t 1 is projective. By Theorem 1.3.4, its characteristic is T X. Since it is contained in X, its multiplicity on T X is 1. This module does not admits a single generator, and it follows that it is not free. Free resolutions Theorem 1.5.5. Let M be a coherent X -module. Then, locally on X, M admits a finite free resolution of length d X. In other words, there locally exists an exact sequence 0 L d X L 0 M 0, where the L i s are free of finite rank over X and n d X. Proof. Set n = d X. Since we argue locally, we may endow M with a good filtration Fl M. We may locally find a finite free filtered resolution Fl L n Fl L 0 Fl M 0. On the other-hand, we know that Ext j gr (gr M, gr X) = 0 for j > n. Set K n = Ker(L n 1 L n 2 ) and let us endow K n with the induced filtration. Then the sequence 0 gr K n gr L n 1 gr L 0 gr M 0

24 CHAPTER 1. THE RING X is exact and it follows that gr K n is projective. Since projective modules over gr X are stably free, there exists a finite free X module L such that gr K n gr L is free and this implies that K n L is a free X -module. The sequence 0 K n L L n 1 L L 0 M 0 is a finite free resolution of M. Homological dimension Let R be a ring. Recall that the global homological dimension of R, gld(r), is the biggest d N { } such that there exist left R-modules M and N with Ext d (M, N) 0. R For a sheaf of rings R on a topological space X, the global homological dimension of R, gld(r), is the biggest d N { } such that there exist sheaves of R-modules M and N with Ext d (N, M) 0. R The weak global homological dimension of R, wgld(r), also called the Tor-dimension of R, is the biggest d N { } such that there exists a right R-module N and a left R-module M with T ord R (N, M) 0. For a sheaf of rings R, wgld(r) is the maximum of wgld(r x ), for x X. Lemma 1.5.6. (i) The O X -module X is flat. (ii) If a X -module I is injective in the category Mod( X ), then it is injective in the category Mod(O X ). Proof. (i) Locally, X is isomorphic to O X (N). (ii) follows from (i). Indeed, if N is a X -module, then Hom O (N, I) Hom ( X O N, I). Recall that if M and N are two left X -modules, Hom O (M, N ) has a natural structure of a left X -modules. By Lemma 1.5.6 we get that the natural forgetful functor b ( X ) b (O X ) commutes with RHom O. Lemma 1.5.7. Let M, N Mod( X ). Then RHom (M, N ) RHom (O X, RHom O (M, N )).

1.5. HOMOLOGICAL PROPERTIES OF X 25 Proof. Since this formula is true when replacing RHom with Hom, it is enough to show that if N is an injective X -module, then H j (RHom (O X, Hom O (M, N ))) = 0 for j > 0. Choose a finite free X -resolution L of O X (for example, take L = SP X ( X )). Notice that L O M M is a quasi-isomorphism of left X -modules. Using the fact that N is O X and X -injective, we get: RHom (O X, Hom O (M, N )) RHom (O X, RHom O (M, N )) RHom (L, Hom O (M, N )) RHom (L O M, N ) RHom (M, N ) Hom (M, N ). Theorem 1.5.8. Let x X. The global homological dimension gld( X,x ) is d X. In other words, the conditions (i) (ii) below are satisfied: (i) let M and N be two X,x -modules. Then Ext j X,x (M, N) = 0 for j > d X, (ii) there exist two X,x -modules M and N such that Ext j X,x (M, N) 0, with j = d X. Proof. (i) By classical results (see [We94, Th. 4.1.2]), it is enough to prove the result when assuming that M is finitely generated. Since X,x is noetherian, there exists a coherent X module M defined in a neighborhood of x such that M = M x. Then the result follows from Theorem 1.5.5 in this case. (ii) Choose M = O X,x and N = X,x. Theorem 1.5.9. The weak global dimension wgld( X,x ) of X is equal to d X. In other words, the conditions (i) (ii) below are satisfied: (i) for any left (resp. right) X -module M (resp. N ), one has T or j (N, M) = 0 for j > d X, (ii) there exist a left X -module M and a right X -module N, such that T or d X (N, M) 0. Proof. (i) It is well known that if R is a ring, wgld(r) is less or equal to gld(r) (see [We94, Ch. 4]). Therefore, wgld( X ) is bounded by gld( X,x ), that is, by d X. (ii) Choose N = Ω X and M = O X.

26 CHAPTER 1. THE RING X Theorem 1.5.10. The global dimension of X is 2d X + 1. In other words, the conditions (i) (ii) below are satisfied: (i) let M and N be two X -modules. Then Ext j (M, N ) = 0 for j > 2d X + 1, (ii) there exist two X -modules M and N such that Ext 2d X+1 (M, N ) 0. Proof. Let n = dim X. (i) By Lemma 1.5.7 one has RHom (M, N ) RHom (O X, RHom O (M, N )). Let SP X ( X ) be the Spencer complex of X. This complex has length n, is locally free and is qis to O X. On the other hand, consider a resolution in the category Mod( X ): 0 N n+1 N n N 0 N 0 such that N 0,..., N n are X -injective. Then these modules will be O X - injective and it follows from Theorem 3.5.7 that N n+1 is O X -injective. Set L i = Hom OX (M, N i ). This is a left X -module, and a flabby sheaf. Consider the complex L := 0 L 0 L n+1 0. Then RHom (M, N ) is represented by the complex Hom (SP X ( X ), L ). This complex has length 2n+1 and its components are flabby sheaves. Therefore RHom (M, N ) RΓ(X; Hom (SP X ( X ), L )) is concentrated in degree [0, 2n + 1]. (ii) Let x X. One has Ext j (O X,x, (N) X ) 0 for j = 2n + 1. Indeed, RHom (O X,x, X ) Ω X [ n], we get Ext j+n (O X,x, ( X ) (N) ) H j (RΓ {x} (X; Ω (N) X )). Then the result follows from Proposition 3.5.8.

1.6. ERIVE CATEGORY AN UALITY 27 1.6 erived category and duality Recall that Mod( X ) is a Grothendieck category (see for example [KS06, Th. 18.1.6]) and thus has enough injectives. One denotes by Mod coh ( X ) the thick abelian subcategory of Mod( X ) consisting of coherent modules and by b coh ( X) the full triangulated category of the bounded derived category b ( X ) consisting of objects with coherent cohomology. If M b coh ( X), we set (1.23) char(m) = j char(h j (M)). Internal operations We denote by RHom O the right derived functor of Hom O and by the left derived functor of O acting on -modules. Hence, we get the functors : b ( X ) b ( X ) b ( X ), : b ( op X ) b ( X ) b ( op X ), RHom O (, ) : b ( X ) op b ( X ) b ( X ), RHom O (, ) : b ( op X )op b ( op X ) b ( X ). The tensor product is commutative and associative, that is, for L, M, N in b ( X ) there are natural isomorphisms M N N M and (M N ) L M (N L). Moreover O X M M. There are also natural functors RHom (, ) : b ( X ) op b ( X ) b (C X ), L : b ( op X ) b ( X ) b (C X ). These functors are related by the formulas (1.24) and (1.25) below. Proposition 1.6.1. For L, M, N in b ( X ) and K in b ( op X ) there are natural isomorphisms (1.24) (1.25) K L (M N ) (K M) L N, RHom (L, RHom O (M, N )) RHom (L M, N ).

28 CHAPTER 1. THE RING X uality We define the duality functors on b ( X ) or b ( op X ), all denoted by and, by setting (1.26) (M) := RHom (M, X ) (M b ( X ) or M b ( op X )), (1.27) (M) := RHom (M, X O Ω 1[d X]) (M b coh ( X)), (1.28) (M) := RHom (M, Ω X [d X ] O X ) (M b coh (op X )). Proposition 1.6.2. For M, N in b ( X ), we have a natural morphism X (1.29) RHom (O X, M N ) RHom (M, N ) and if M of N belongs to b coh ( X), this morphism is an isomorphism. Proof. We have the isomorphism RHom (O X, M N ) RHom (O X, X ) L ( M N ) Ω X L ( M N ) [ d X ] (Ω X M) L N [ d X ] M L N RHom (M, N ). Cleary, if M of N belongs to b coh ( X), the last morphism is an isomorphism. Proposition 1.6.3. (i) The functor : b coh ( X) op b coh (op X ) is welldefined and satisfies id and similarly with. (ii) If M b coh ( X), then char( (M)) = char(m). Proof. (i) There is a natural morphism id. To prove it is an isomorphism, we argue by induction on the amplitude of M and reduce to the case where M is a coherent X -module. More precisely, assume H j (M) = 0 for j / [j 0, j 1 ] and the result has been proved for modules with amplitude j 1 j 0 1. Consider the distinguished triangle (d.t. for short) (1.30) H j 0 (M)[ j 0 ] M τ >j 0 (M) +1 and apply the functor. We get a new d.t. with two objects isomorphic to two objects of the d.t. (1.30). hence the third objects of these d.t. will be isomorphic.

1.6. ERIVE CATEGORY AN UALITY 29 Hence, we are reduced to treat the case of M Mod coh ( X ). We may argue locally and replace M with a bounded complex of finite free X - modules. It reduces to the case where M = X. (ii) It is enough to prove the inclusion char( (M)) char(m). We argue by induction on the amplitude of M. Assume H j (M) = 0 for j / [j 0, j 1 ]. Consider the distinguished triangle (1.30) Applying the functor we find the d.t. (τ >j 0 M) M (H j 0 (M))[j 0 ] +1 Since char(m) = char(h j 0 (M)) char(τ >j 0 (M)), the induction proceeds, and we are reduced to the case where M is a coherent X -module. Then the result follows from Theorem 1.5.1 (iii).

30 CHAPTER 1. THE RING X

Chapter 2 Operations on -modules 2.1 External product Let X and Y be two manifolds. For a X -module M and a Y -module N, we define their external product, denoted M N, by M N := X Y X Y (M N ). Note that the functor M M N is exact. Theorem 2.1.1. Let M b coh ( X) and N b coh ( Y ). Then M N b coh ( X Y ) and char(m N ) = char(m) char(n ). Proof. (i) By dévissage, one reduces to the case where M Mod coh ( X ) and N Mod coh ( Y ). (ii) Let us show that M N is coherent. Consider finite free presentations of M and N : M 1 X P M 0 X M 0, N 1 Y Q N 0 Y N 0. Then ( X Y ) N 1+M 1 ( ) P 0 0 Q ( X Y ) N 0+M 0 M N 0 is a finite free presentation of M N over X Y. To conclude, apply the exact functor X Y X Y to this sequence. 31

32 CHAPTER 2. OPERATIONS ON -MOULES (iii) Let us endow M and N with good filtrations Fl M and Fl N. Set Fl k (M N ) = Fl i (M) Fl j (N ). i+j=k Then {Fl k (M N )} k is a good filtration on M N and the result follows from gr (M N ) gr (M) gr gr (N ) where gr is defined similarly as. 2.2 Transfert bimodule Let f : X Y be a morphism of complex manifolds. Recall (see (3.14)) that to f are associated the maps (2.1) T X f X Y T Y fτ T Y. We shall construct a ( X, f 1 Y )-bimodule denoted X Y which shall allow one to pass from left Y -modules to left X -modules and from right X -modules to right Y -modules. Set X Y = O X f 1 O Y f 1 Y. This sheaf on X is naturally endowed with a structure of an (O X, f 1 Y )- bimodule. We shall endow it of a structure of a left X -module by defining the action Θ X and verifying that this action satisfies the hypothesis of Corollary 1.1.3. Let v Θ X. Then f v O X f 1 O Y f 1 Θ Y. Hence f v = j a j w j, with a j O X and w j f 1 Θ Y. efine the action of v on a P O X f 1 O Y f 1 Y by setting (2.2) v(a P ) = v(a) P + j aa j w j P.

2.2. TRANSFERT BIMOULE 33 If one chooses a local coordinate system (y 1,..., y m ) on Y and writes f = (f 1,..., f m ), then v(f ϕ) = m j=1 v(f j ) ϕ y j, which implies f v = m v(f j ) yj. j=1 A section P of X Y may formally be written as P = α a α(x) α y. By composing the monomorphism Y Hom CY (O Y, O Y ) with X Y = O X f 1 O Y f 1 Y we get the monomorphisms X Y O X f 1 O Y f 1 Hom CY (O Y, O Y ) Hom CX (f 1 O Y, O X ) and the section 1 X Y := 1 1 X Y corresponds to the canonical morphism Note that Y being flat over O Y, f 1 O Y O X ϕ ϕ f. X Y O X L f 1 O Y f 1 Y. One also introduces the (f 1 Y, X )-bimodule Y X by setting Y X = Ω X OX X Y f 1 O Y f 1 Ω 1 Y. Proposition 2.2.1. Let f : X Y, g : Y Z be morphisms of manifolds and set h = g f : X Z. Then there is an isomorphism of ( X, h 1 Z )- bimodules (2.3) L X Y f 1 Y f 1 Y Z X Z. In particular, the left hand side is concentrated in degree zero.

34 CHAPTER 2. OPERATIONS ON -MOULES Proof. One has the isomorphisms of (O X, h 1 Z )-bimodules: L X Y f 1 Y f 1 L Y Z = (O X f 1 O Y f 1 Y ) L f 1 Y f 1 L (O Y g 1 O Z g 1 Z ) O X L f 1 O Y (f 1 Y L f 1 Y f 1 O Y L h 1 O Z h 1 Z ) O X L h 1 O Z h 1 Z O X h 1 O Z h 1 Z. (Recall that Z is flat over O Z.) Then, one checks that these isomorphisms extend as isomorphisms of ( X, h 1 Z )-bimodules. Proposition 2.2.2. (i) Assume f is submersive. Then X Y is X - coherent and f 1 Y -flat. (ii) Assume f is a closed embedding. Then X Y is Y -coherent and X -flat. Proof. (i) Since the problem is local on X, we may assume that X = Z Y and f is the second projection. In this case, X Y O Z Y. Note that if x = (t, y) is a local coordinate system on Z Y with t = (t 1,..., t m ), then X Y X / X t where X t denotes the left ideal generated by ( t1,..., tm ). (ii) For a local coordinate system y = (t, x) on Y such that X = {t = 0}, we have X Y Y /t Y where t Y denotes the right ideal generated by (t 1,..., t m ). If f is submersive, one has X Y X / X Θ f where X Θ f denotes the left ideal generated by the vector fields tangent to the leaves of f. If f is a closed embedding, one has X Y Y /I X Y where I X Y denotes the right ideal generated sections of O Y vanishing on X. Notice that any morphism f : X Y may be decomposed as f : X X Y Y where the first map is the graph (closed) embedding and the second map is the projection. Example 2.2.3. One has X pt O X and pt X Ω X.

2.3. INVERSE IMAGES 35 Inverse and direct images of -modules efinition 2.2.4. Let f : X Y be a morphism of complex manifolds. (i) One defines the inverse image functor f 1 : b ( Y ) b ( X ) by setting for N b ( Y ): f 1 L N := X Y f 1 Y f 1 N. (ii) One defines the direct image functors f, f! : b ( X ) b ( Y ) by setting for M b ( op X ): f M := Rf (M L X Y ), f! M := Rf! (M L X Y ). Using the bimodule Y X, one defines similarly the inverse image of a right Y -module or the direct images of a left X -module. Note that, if g : Y Z is another morphism of complex manifolds, we have (2.4) (g f) 1 f 1 g 1, (2.5) (g f) g f, (2.6) (g f)! g! f!. 2.3 Inverse images efinition 2.3.1. Let N be a coherent Y -module. One says that f is non characteristic for N (or N is non characteristic for f) if f is non characteristic for char(n ). (See efinition 3.1.10.) Example 2.3.2. (i) Since char(o Y ) = TY Y, the Y -module O Y is non characteristic for any morphism f : X Y. Note that f 1 O Y O X. (ii) See Exercise 2.2. Example 2.3.3. Assume to be given a coordinate system (y) = (x 1,..., x n, t) = (x, t) on Y such that X = {t = 0}. Let P be a differential operator of order m. Then X is non-characteristic with respect to P (i.e., for the Y -module Y / Y P ) in a neighborhood of (x 0, 0) X if and only if P is written as (2.7) P (x, t; x, t ) = a j (x, t, x ) j t 0 j m where a j (x, t, x ) is a differential operator not depending on t of order m j and a m (x, t) (which is a holomorphic function on Y ) satisfies: a m (x 0, 0) 0.

36 CHAPTER 2. OPERATIONS ON -MOULES Lemma 2.3.4. Let X, Y and P be as in Example 2.3.3. Let N = Y / Y P. Then X Y Y N m X. Proof. Notice that X Y N Y /(t Y + Y P ). By the Weierstrass preparation theorem, any Q(x, t, x, t ) Y written uniquely as may be Q(x, t, x, t ) = S(x, t, x, t ) P (x, t, x, t ) + m 1 j=0 R j (x, t, x ) j t. Hence, Q(x, t, x, t ) Y may be written uniquely as Q(x, t, x, t ) = S(x, t, x, t ) P (x, t, x, t ) + t T (x, t, x ) + Proposition 2.3.5.. For M, N b ( X ), one has m 1 j=0 P j (x, x ) j t. M N δ 1 (M N ), where δ : X X X is the diagonal embedding. Proof. Let us identify X with, the diagonal of X X. One has the chain of isomorphisms δ 1 (M N ) O L O X X L (M N ) O L O (M N ) M N. Corollary 2.3.6. Let f : X Y be a morphism of complex manifolds. For N 1, N 2 b ( Y ), one has f 1 (N 1 N 2 ) f 1 N 1 f 1 N 2.

2.3. INVERSE IMAGES 37 Proof. enote by δ X the diagonal embedding X X X and similarly with δ Y, and denote by f : X X Y Y the map associated with f. One has the chain of isomorphisms f 1 (N 1 N 2 ) f 1 δ Y 1 (N 1 1 1 δ X (f N 1 N 2 ) δ X 1 f 1 1 f N 2) f 1 (N 1 N 2 ) N 1 f 1 N 2. Theorem 2.3.7. Let N Mod coh ( Y ) and assume that f is non characteristic for N. Then (a) f 1 N is concentrated in degree 0, (b) f 1 N is X-coherent, (c) char(f 1 N ) f dfπ 1 char(n ). Remark 2.3.8. In fact, there is a better result, namely char(f 1 N ) = f d fπ 1 char(n ) and the characteristic cycle of f 1 N is the image by f dfπ 1 of the characteristic cycle of N (see [Ka83]). Proof. The map f : X Y decomposes as X h X Y p Y where h is the graph embedding and p is the projection. Using (2.4) and Lemma 3.1.13, it is enough to prove the result for p and for h. Hence, we shall treat separately the case where f is submersive and the case where f is a closed embedding. (i) Assume f : X Y is submersive. The problem is local on X. Hence, we may assume X = Y Z and f is the projection. In this case, f 1 ( ) O X. Hence, this functor is exact and the result follows from Theorem 2.1.1. (ii) Assume f : X Y is a closed embedding. Let d denote the codimension of X in Y. Since our problem is local, we may assume that there are submanifolds X = X 0 X 1 X d = Y. Using (2.4) and Lemma 3.1.13 again, we are reduced to treat the case d = 1. Since the problem is local we may assume to be given a local coordinate system in a neighborhood of x 0 X, (y) = (x 1,..., x n, t) = (x, t) on Y such that X = {t = 0}. Let (x, t; ξ, τ) denote the associated coordinate system on T Y. Set Λ = char(n ). By

38 CHAPTER 2. OPERATIONS ON -MOULES the hypothesis, (x 0, 0; 0, 1) / Λ. By Corollary 1.3.6, for each section u of N defined in a neighborhood of (x 0, 0), there exists a differential operator P, say of order m, such that (2.8) P u = 0, σ m (P )(x 0, 0; 0, 1) 0. (iii) Let us prove that f 1 N is concentrated in degree 0. Since X Y Y /t Y, f 1 t N is isomorphic to the complex N N. Hence, we have to show that t acting on N is injective. Let u N with tu = 0. Let P satisfying (2.8). Set Ad(P ) = [P, ]. We obtain Hence, u = 0. Ad m (P )(t)u = m!u = 0. (iv) Let us prove that f 1 N is X-coherent. Let (u 1,..., u N ) be a system of generators of N in a neighborhood of (x 0, 0). For each j, 1 j N, there exists a differential operator P j of order m j, such that P j u j = 0 and σ mj (P j )(x 0, 0; 0, 1) 0. Set M = N j=1 Y / Y P j. It follows from (iii) and Lemma 2.3.4 that f 1 M is concentrated in degree 0 and is X -coherent. enote by v j the canonical generator of Y / Y P j, the image of 1 Y. There is a well-defined Y -linear epimorphism ψ : M N which associates u j to v j. The functor f 1 being right exact, the epimorphism ψ defines the epimorphism f 1 1 1 M f N. Therefore, f N is locally finitely generated. efine the coherent Y -module L by the exact sequence (2.9) 0 L M N 0. It follows from (iii) that the sequence (2.10) 0 f 1 1 1 L f M f N 0 is exact. Since X is non-characteristic for M, it is non-characteristic for its submodule L. Therefore, f 1 1 L is locally finitely generated and f M being coherent, this implies that f 1 N is coherent. (v) Let us prove (c). (v) (a) Let us choose a local coordinate system (x, t) on Y such that X = {(x, t); t = 0}. Then f 1 N N /t N. Set M := f 1 N.

2.3. INVERSE IMAGES 39 Let Fl N = {N j } j Z be a good filtration on N. We define a filtration on Fl M = {M j } j Z by setting (2.11) M j = N j /(t N N j ). (v) (b) Let us show that Fl M is a good filtration. It is enough to check that the M j s are O X -coherent. Since t N N j = k (t N k N j ), and N j is O Y -coherent, this sequence is locally stationary. It follows that M j is O Y -coherent. Being supported by X, M j is O X -coherent. (v) (c) The exact sequence 0 N j 1 N j gr j N 0 gives rise to the exact sequence (2.12) N j 1 /t N j 1 N j /t N j gr j N /t gr j N 0. Note that gr N /t gr N is an O X OY gr X -module, but gr M is simply a gr X -module. We deduce from (2.11) and (2.12) an epimorphism gr j N /t gr j N gr j M, hence, an epimorphism of gr X -modules (2.13) gr N /t gr N gr M. Considering gr N /t gr N as a gr X -module is the same as considering f d (gr N /t gr N ). It follows that the support of gr M in T X is contained in f d (supp(gr N /t gr N ) = f d fπ 1 char(n ). Corollary 2.3.9. Let M, N Mod coh ( X ) and assume that char(m) char(n ) T X X. Then M N is X -coherent and char(m N ) char(m) + char(n ). X Recall that for two conic subsets Λ 1 and Λ 2 of T X, Λ 1 + Λ 2 := {(x; ξ 1 + ξ 2 ); (x; ξ j Λ j, j = 1, 2}. X Proof. Apply Proposition 2.3.5 and Theorem 2.3.7.

40 CHAPTER 2. OPERATIONS ON -MOULES uality and inverse images Let N b ( Y ). Recall that its dual, N b ( Y ) has been constructed in (1.28) Theorem 2.3.10. Let f : X Y be a morphism of complex manifolds and let N b coh ( Y ). Assume that f is non characteristic for N. Then there exists a natural isomorphism : ψ : f 1 N f 1 N. Proof. First, we shall construct the morphism ψ. By Proposition 1.6.2, we have an isomorphism Hom b ( Y ) (N, N ) Hom b ( Y ) (O Y, N N ). It defines the morphism O Y the morphisms N N. Applying the functor f 1 we get f 1 O Y O X f 1 N f 1 N Hence, we have obtained a morphism f 1 N f 1 N. ψ Hom b ( X ) (O X, f 1 N f 1 N ) Hom b ( X ) ( f 1 N, f 1 N ). To prove that ψ is an isomorphism, we proceed as in the proof of Theorem 2.3.7 and reduce to the case where X is a closed hypersurface of Y and N = Y / Y P for a differential operator P of order m. In this case, f 1 N m X and f 1 N m X [d X]. On the other hand, N is represented by the complex 0 P Y Y 0 and it follows that N N [d Y 1]. Therefore, f 1 N m X [d Y 1]. 2.4 Holomorphic solutions of inverse images Let f : X Y be a morphism of complex manifolds and let N 1, N 2 Mod( Y ). There is a natural morphism (2.14) f 1 RHom Y (N 1, N 2 ) RHom X (f 1 N 1, f 1 N 2).

2.4. HOLOMORPHIC SOLUTIONS OF INVERSE IMAGES 41 obtained as the composition f 1 RHom Y (N 1, N 2 ) RHom f 1 Y (f 1 N 1, f 1 N 2 ) Also recall the natural isomorphism (2.15) L RHom X ( X Y f 1 f N 1 1, X Y f 1 f N 1 2 ). f 1 O Y O X. Theorem 2.4.1. (Cauchy-Kowalevski-Kashiwara) Let f : X Y be a morphism of complex manifolds and let N Mod( Y ). Assume that f is non characteristic for N. Then there exists a natural isomorphism : (2.16) f 1 RHom Y (N, O Y ) RHom X (f 1 N, O X). Proof. As in the proof of Theorem 2.3.7, we may check separately the case of a projection and a closed embedding. (a) If f is submersive, the morphism (2.14) is an isomorphism. Indeed, we may reduce to the case where N 1 = N 2 = Y. In such a case, the isomorphism reduces to: f 1 Y RHom X ( X Y, X Y ). We may assume f is the projection X = Y Z Y, and the result is a relative version of the e Rham isomorphism C Z RHom Z (O Z, O Z ). (b) Now assume f is a closed embedding. Again, we reduce to the case where X is a hypersurface. First we treat the case where N = Y / Y P. We may assume that we have a local coordinate system (x, t) such that X = {(x, t); t = 0} and P is a differential operator of order m as in Lemma 2.3.3. P The complex RHom Y (N, O Y ) is represented by the complex 0 O Y X O Y X 0, where O Y X on the left is in degree 0. Since N 1 m X, the complex RHom X (N 1, O X) is represented by the complex OX m in degree 0. The morphism (2.16) reduces to the morphism L 0 O Y P X O Y X 0 γ 0 O m X 0 0 Here, the vertical arrow γ is the morphism which, to f O Y X associates the first m traces of f γ(f) = f X, t f X,..., m 1 t f X.

42 CHAPTER 2. OPERATIONS ON -MOULES Then the theorem asserts that P acting on O Y X is an epimorphism and Ker P acting on this sheaf is isomorphic by γ to OX m. This is the Cauchy- Kovalevski theorem. (c) As in the proof of Theorem 2.3.7, we construct an exact sequence (2.9) 0 L M N 0 where M is a finite direct sum of modules of the type Y / Y P. let us apply the functor RHom Y (, O Y ) to the sequence (2.9) and the functor RHom X (, O X ) to the image by ( 1 ) of the sequence (2.9). Let us set for short Sol Y ( ) := RHom Y (, O Y ) and similarly with Sol X ( ). We find the morphism of distinguished triangles f 1 Sol Y (N ) f 1 Sol Y (M) f 1 Sol Y (L) +1 Sol Y (f 1 N ) Sol Y (f 1 M) Sol Y (f 1 L) +1 Let us apply the cohomology functor H 0 to this morphism of distinguished triangles. We find a morphism of long exact sequences 0 H 0 (A 1 ) H 0 (A 2 ) H 0 (A 3 ) H 1 (A 1 ) u 0 1 u 0 2 0 H 0 (B 1 ) H 0 (B 2 ) H 0 (B 3 ) H 1 (B 1 ). u 0 3 u 1 1 By (b), all morphisms u n 2, n 0 are isomorphisms. It follows that u 0 1 is a monomorphism, and the module M satisfying the non characteristicity hypothesis, the morphism u 0 3 is also a monomorphism. Therefore, u 0 1 is an isomorphism, hence u 0 3 is also an isomorphism. By induction, we get that all u n 1 are isomorphism. 2.5 irect images Good -modules efinition 2.5.1. (i) Let F Mod(O X ). One says that F is good if for any relatively compact open subset U X, there exists a small and filtrant category I, an inductive system {F i } i I of coherent O U -modules and an isomorphism lim F i F U. i