2 Coherent D-Modules. 2.1 Good filtrations

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1 2 Coherent D-Modules As described in the introduction, any system o linear partial dierential equations can be considered as a coherent D-module. In this chapter we ocus our attention on coherent D-modules and study their basic properties. Among other things, or a coherent D X -module M we deine its characteristic variety as a subvariety o the cotangent bundle T X o X. This plays an important role or the geometric (or microlocal) study o M. 2.1 Good iltrations Recall that the ring D X has the order iltration {F i D X } i Z such that the associated graded ring gr F D X = i=0 F i D X /F i 1 D X is naturally isomorphic to the shea π O T X o commutative rings consisting o symbols o dierential operators, where π : T X X denotes the cotangent bundle (see Section 1.1). By the aid o the commutative approximation gr F D o the non-commutative ring D, we will deduce various results on D using techniques rom commutative algebra (algebraic geometry). We note that some o the results in this chapter can be ormulated or more general iltered rings, in which cases they are presented and proved in Appendix D. Hence readers should occasionally consult Appendix D according to reerences to it in this chapter. Our irst task is to give a commutative approximation o modules over D. Let M be a D X -module quasi-coherent over O X. We consider a iltration o M by quasicoherent O X -submodules F i M (i Z) satisying the conditions: F i M F i+1 M, F i M = 0 (i 0), M = i Z F im, (F j D X )(F i M) F i+j M. In this case, we call (M, F ) a iltered D X -module; the module

2 58 2 Coherent D-Modules gr F M := i Z F i M/F i 1 M obtained by F is a graded module over gr F D X = π O T X. This module is clearly quasi-coherent over O X. Proposition Let (M, F ) be a iltered D X -module. Then the ollowing conditions are equivalent to each other: (i) gr F M is coherent over π O T X. (ii) F i M is coherent over O X or each i, and there exists i 0 0 satisying (F j D X )(F i M) = F j+i M (j 0, i i 0 ). (iii) There exist locally a surjective D X -linear morphism : DX m M and integers n j (j = 1, 2,..., m) such that (F i n1 D X F i n2 D X F i nm D X ) = F i M (i Z). Proo. By Proposition D.1.1 the conditions (i) and (iii) are equivalent. It is easily checked that (iii) holds i and only i F i M is coherent over O X or each i and one can ind i 0 as in (ii) locally on X. Then the global existence o i 0 ollows rom this since X is quasi-compact. Deinition Let (M, F ) be a iltered D X -module. We say that F is a good iltration o M i the equivalent conditions in Proposition are satisied. Theorem (i) Any coherent D X -module admits a (globally deined) good iltration. Conversely, a D X -module endowed with a good iltration is coherent. (ii) Let F, F be two iltrations o a D X -module M and assume that F is good. Then there exists i 0 0 such that F i M F i+i 0 M (i Z). I, moreover, F is also a good iltration, there exists i 0 0 such that F i i 0 M F i M F i+i 0 M (i Z). Proo. (i) By Corollary D.1.2 an object o Mod qc (D X ) is coherent i and only i it admits a good iltration locally on X. Hence it is suicient to show that any coherent D X -module M admits a global good iltration. By Corollary (i), M is generated by a globally deined coherent O X -submodule M 0. I we set F i M = (F i D X )M 0 (i N), then this is a global good iltration o M. The statement (ii) ollows rom Proposition D.1.3.

3 2.2 Characteristic varieties (singular supports) Characteristic varieties (singular supports) Let M be a coherent D X -module and choose a good iltration F on it (Theorem 2.1.3). Let π : T X X be the cotangent bundle o X. Since we have gr F D X π O T X, the graded module gr F M o M obtained by F is a coherent module over π O T X by Proposition We call the support o the coherent O T X-module gr F M := O T X π 1 π O T X π 1 (gr F M) the characteristic variety o M and denote it by Ch(M) (it is sometimes called the singular support o M and denoted by SS(M)). As we see below Ch(M) does not depend on the choice o a good iltration F on M. Since gr F M is a graded module over the graded ring O T X, Ch(M) is a closed conic (i.e., stable by the scalar multiplication o complex numbers on the ibers) algebraic subset in T X. Let U be an aine open subset o X. Then T U is an aine open subset o T X with coordinate algebra gr F D U (U), and Ch(M) T U coincides with the support o the coherent O T U -module associated to the initely generated gr F D U (U)-module gr F M(U). Hence in the notation o Section D.3 we have Ch(M) T U ={p T U (p) = 0 ( J M(U) )}, and its decomposition into irreducible components is given by Ch(M) T U = {p T U (p) = 0 ( p)}. p SS 0 (M(U)) By Lemmas D.3.1 and D.3.3 we have the ollowing. Theorem (i) Let M be a coherent D X -module. Then the set Ch(M) does not depend on the choice o a good iltration F. (ii) For a short exact sequence o coherent D X -modules, we have 0 M N L 0 Ch(N) = Ch(M) Ch(L). By the above theorem, the characteristic variety is a geometric invariant o a coherent D-module. From now on we introduce the notion o the characteristic cycle, which is a iner invariant o a coherent D-module (obtained by taking the multiplicities into account). Let V be a smooth algebraic variety and assume that we are given a coherent O V -module G. Then we can deine an algebraic cycle Cyc G associated to G as ollows. Denote by I(supp G) the set o the irreducible components o the support o G. Let C I(supp G). Take an aine open subset U o V such that C U = C,

4 60 2 Coherent D-Modules and denote the deining ideal o C U by p C O U (U). Then we obtain a local ring O U (U) pc with maximal ideal p C O U (U) pc and an O U (U) pc -module G(U) pc. Note that O U (U) pc and G(U) pc do not depend on the choice o U (in scheme-theoretical language they are the stalks o O V and G at the generic point o C). By a standard act in commutative algebra G(U) pc is an artinian O U (U) pc -module, and its length m C (G) is deined. We call it the multiplicity o G along C. For an irreducible subvariety C o V with C supp G we set m C (G) = 0. We call the ormal sum Cyc G := m C (G)C C I(supp G) the associated cycle o G. Let M be a coherent D X -module. By choosing a good iltration F o M we can consider a coherent O T X-module gr F M. By Lemma D.3.1 the cycle Cyc( gr F M) does not depend on the choice o a good iltration F. Deinition For a coherent D X -module M we deine the characteristic cycle o M by CC(M) := Cyc( gr F M) = m C ( gr F M)C, C I(Ch(M)) where F is a good iltration o M. For d N we denote its degree d part by CC d (M) := C I(Ch(M)) dim C=d By Lemma D.3.3 we have the ollowing. Theorem Let m C ( gr F M)C. 0 M N L 0 be an exact sequence o coherent D X -modules. Then or any irreducible subvariety C o T X such that C I(Ch(N)) we have m C ( gr F N) = m C ( gr F M) + m C ( gr F L). In particular, or d = dim Ch(N) we have CC d (N) = CC d (M) + CC d (L). Example Let M be an integrable connection o rank r > 0. Set F i M = 0 (i < 0), F i M = M (i 0). Then F deines a good iltration on M and gr F M M O r X holds locally. Moreover, since X Ann π O T X (gr F M), we get Ch(M) = T X X = s(x) X (s : x (x, 0), the zero-section o T X) and CC(M) = rt X X. Conversely, integrable connections are characterized by their characteristic varieties as ollows.

5 2.2 Characteristic varieties (singular supports) 61 Proposition For a non-zero coherent D X -module M the ollowing three conditions are equivalent: (i) M is an integrable connection. (ii) M is coherent over O X. (iii) Ch(M) = T X X X (the zero-section o T X). Proo. Since the equivalence (i) (ii) is already proved in Theorem and (i) (iii) is explained in Example 2.2.4, it remains to prove the part (iii) (ii). Since the problem is local, we may assume that X is an aine algebraic variety with a local coordinate system {x i, i } 1 i n. Then we have T X = X C n. Assume that Ch(M) = TX X. This means that or a good iltration F o M we have Ann OX [ξ 1,...,ξ n ](gr F M) = n O X [ξ] ξ i. Here we denote by ξ i the principal symbol o i, and we identiy π O T X with O X [ξ 1,...,ξ n ]. Now let us set I = n i=1 O X [ξ] ξ i. Since the ideal I is noetherian, we have I m 0 Ann OX [ξ 1,...,ξ n ](gr F M) or m 0 0. Since the set {ξ α α =m 0 } generates the ideal I m 0, we have i=1 α F j M F j+m0 1M ( α =m 0 ). On the other hand, since F is a good iltration, we have F i D X F j M = F i+j M (j 0). It ollows that F m0 +j M = (F m0 D X )(F j M) = α m 0 O X α F j M F j+m0 1M (j 0). This means F j+1 M = F j M = M (j 0). Since each F j M is coherent over O X, M is also O X -coherent. Exercise For a coherent D X -module M = D X u D X /I (I = Ann DX u) consider the good iltration F i M = (F i D X )u. I we deine a iltration on I by F i I := F i D X I, we have gr F M gr F D X / gr F I. In this case, the graded ideal gr F I := i 0 F ii/f i 1 I is generated by the principal symbols σ (P ) o P I. Thereore, or an arbitrary chosen set {σ (P i ) 1 i m } o generators o gr F I, we have I = m i=1 D X P i and Ch(M) ={(x, ξ) T X σ (P i )(x, ξ) = 0, 1 i m }. However, or a set {Q i } o generators o I, the equality

6 62 2 Coherent D-Modules Ch(M) ={(x, ξ) σ (Q i )(x, ξ) = 0, 1 i m} does not always hold. In general, we have only the inclusion Ch(M) { (x, ξ) σ (Q i )(x, ξ) = 0, 1 i m}. Find an example so that this inclusion is strict. Remark In general, it is not easy to compute the characteristic variety o a given coherent D-module as seen rom Exercise However, thanks to recent advances in the theory o computational algebraic analysis we now have an eective algorithm to compute their characteristic varieties. Moreover, we can now compute most o the operations o D-modules by computer programs. For example, we reer to [Oa1], [Oa2], [SST], [Ta]. It is also an interesting problem to determine various invariants o special holonomic D-modules introduced in the theory o hypergeometric unctions o several variables (see [AK], [GKZ]). 2.3 Dimensions o characteristic varieties One o the most undamental results in the theory o D-modules is the ollowing result about the characteristic varieties o coherent D-modules. Theorem The characteristic variety o any coherent D X -module is involutive with respect to the symplectic structure o the cotangent bundle T X. This result was irst established by Sato Kawai Kashiwara [SKK] by an analytic method. Dierent proos were also given by Malgrange [Ma5], Gabber [Ga], and Kashiwara Schapira [KS2]. Here, we only note that in view o Lemma E.2.3 it is a consequence o Gabber s theorem (Theorem D.3.4), which is a deep result on a certain class o iltered rings (the proo o Theorem D.3.4 is not given in this book). An important consequence o Theorem is the ollowing result. Corollary Let M be a coherent D X -module. Then or any irreducible component o Ch(M) we have dim dim X. In particular, we have dim Ch(M) dim X i M = 0. Remark Note that Corollary is weaker than Theorem 2.3.1; however, the weaker statement Corollary is almost suicient or arguments in this book. In act, we will need the stronger statement Theorem (or rather its analytic counterpart Theorem below) only in the proo o Kashiwara s constructibility theorem or solutions o analytic holonomic D-modules (Theorem below). Since we will also present a proo o the corresponding act or algebraic holonomic D-modules due to Beilinson Bernstein without using Theorem (see Theorem below), the readers who are only interested in algebraic D-modules can skip Section 4.6. In the rest o this section we will give a direct proo o Corollary ollowing Kashiwara [Kas16]. We irst establish the ollowing result.

7 2.3 Dimensions o characteristic varieties 63 Theorem For any coherent D X -module M there exists a canonical iltration 0 = C 2 dim X+1 M C 2 dim X M C 1 M C 0 M = M o M by coherent D X -modules such that any irreducible component o is s-codimensional in T X. Ch(C s M/C s+1 M) Proo. Let U be an aine open subset o X. We apply the result in Section D.5 to A = D X (U). Then by Lemma D.5.1 and Theorem D.5.3, together with Proposition , we obtain a iltration 0 = C 2 dim X+1 (M U ) C 2 dim X (M U ) C 0 (M U ) = M U o M U by coherent D U -modules such that any irreducible component o Ch(C s (M U )/C s+1 (M U )) is s-codimensional in T U. We see by the cohomological description o the iltration given in Proposition D.5.2 that it is canonical and globally deined on X. Lemma Let S be a smooth closed subvariety o X and let M be a coherent D S -module. Set N = 0 i M, where i : S X denotes the embedding. Let ρ i : S X T X T S and let ϖ i : S X T X T X be natural morphisms induced by i. Then we have Ch(N) = ϖ i ρi 1 (Ch(M)). Proo. Note that the problem is local on S. By induction on the codimension o S one can reduce the problem to the case where S is a hypersurace o X (see Proposition and Lemma below). Assume that S is a hypersurace o X deined by x = 0. Take a local coordinate {x i, i } 1 i n o X such that x = x 1 and set = 1. Then we obtain a local identiication N C[ ] C i M (see Section 1.5). Take a good iltration G o M such that G 1 M = 0, and deine a iltration F o N by F j N = j C k i G j l (M). l=0 k l Then F is a good iltration o N satisying Hence we have F j N/F j 1 N = j C l i (G j l M/G j l 1 M). l=0 gr F N C[ξ] C gr G M (C[x, ξ]/c[x, ξ]x) C gr G M, where ξ is the principal symbol o. From this we easily see that Ch(N) = supp gr F N = ϖ i ρ 1 i The proo is complete. (supp gr G M = ϖ i ρ 1 (Ch(M)). i

8 64 2 Coherent D-Modules Proo o Corollary By Theorem we have only to show that dim Ch(M) dim X or any non-zero coherent D X -module M. We prove it by induction on dim X. It is trivial in the case dim X = 0. Assume that dim X>0. I supp M = X, then we have Ch(M) TX X, and hence dim Ch(M) dim T X X = dim X. Thereore, we may assume rom the beginning that supp M is a proper closed subset o X. By replacing X with a suitable open subset (i necessary) we may urther assume that supp M is contained in a smooth hypersurace S in X. Let i : S X be the embedding. By Kashiwara s equivalence there exists a non-zero coherent D S -module L satisying M = 0 i L. Then by Lemma we have Ch(M) = ϖ i ρi 1 (Ch(L)) and hence dim Ch(M) = dim Ch(L) + 1. On the other hand, the hypothesis o induction implies dim Ch(L) dim S = dim X 1. It ollows that dim Ch(M) dim S + 1 = dim X. Deinition A coherent D X -module M is called a holonomic D X -module (or a holonomic system, or a maximally overdetermined system) i it satisies dim Ch(M) dim X. By Theorem characteristic varieties o holonomic D-modules are C - invariant Lagrangian subset o T X. Holonomic D X -modules are the coherent D X -modules whose characteristic variety has minimal possible dimension dim X. Assume that the dimension o the characteristic variety Ch(M) is small. This means that the ideal deining the corresponding system o dierential equations is large, and hence the space o the solutions should be small. In act, we will see later that the holonomicity is related to the inite dimensionality o the solution space. Example Integrable connections are holonomic by Proposition Example The D X -module B Y X or a closed smooth subvariety Y o X is holonomic (see Example 1.6.4). In this case the characteristic variety Ch(B Y X ) is the conormal bundle TY X o Y in X. 2.4 Inverse images in the non-characteristic case We have shown in Proposition that the inverse image o a coherent D-module with respect to a smooth morphism is again coherent; however, the inverse images with respect to non-smooth morphisms do not necessarily preserve coherency as we saw in Example In this section we will give a suicient condition on a coherent D-module M so that its inverse image is again coherent. For a morphism : X Y o smooth algebraic varieties there are associated natural morphisms T X ρ X Y T Y ϖ T Y.

9 2.4 Inverse images in the non-characteristic case 65 Note that i is a closed embedding (resp. smooth), then ρ is smooth (resp. a closed embedding) and ϖ is a closed embedding (resp. smooth). We set TX Y := ρ 1 (T X X) X Y T Y. When is a closed embedding, TX Y is the conormal bundle o X in Y. The ollowing is easily checked. Lemma Let : X Y and g : Y Z be morphisms o smooth algebraic varieties. Then we have the natural commutative diagram T X ρ X Y T Y ϖ ϕ X Z T Z ψ T Y ρ g Y Z T Z ϖ g T Z such that ρ ϕ = ρ g, ϖ g ψ = ϖ g, and the square in the right upper corner is cartesian. Deinition Let : X Y be a morphism o smooth algebraic varieties and let M be a coherent D Y -module. We say that is non-characteristic with respect to M i the condition ϖ 1 (Ch(M)) T X Y X Y TY Y is satisied. Remark We can easily show that i a closed embedding : X Y is non-characteristic with respect to a coherent D Y -module M, then ρ ϖ 1 ϖ 1 (Ch(M)) T X is a inite morphism. (Ch(M)) : This deinition is motivated by the theory o linear partial dierential equations, as we see below. Example Consider the case where : X Y is the embedding o a hypersurace. Then the conormal bundle T X Y is a line bundle on X. Let P D Y be a dierential operator o order m 0 and set M = D Y /D Y P. In this case Ch(M) is exactly the zero set o the principal symbol σ m (P ), and hence is non-characteristic with respect to the coherent D Y -module M i and only i (σ m (P ))(ξ) = 0 ( ξ T X Y \ (the zero-section o T X Y )). Take a local coordinate {z i, i } 1 i n o Y such that z 1 is the deining equation o X, and let (z 1,...,z n ; ζ 1,...,ζ n ) be the corresponding coordinate o T Y. Then the condition can be written as

10 66 2 Coherent D-Modules or equivalently as σ m (P ) ( 0,z 2,...,z n ; 1, 0,...,0) ) = 0 ( (z 2,...,z n )), m ζ m 1 σ m (P ) ( 0,z 2,...,z n ; 0,...,0 ) = 0 ( (z 2,...,z n )). In the classical analysis, i this is the case, we say that Y is a non-characteristic hypersurace o X with respect to the dierential operator P. Let us show that H 0 (L M)is a locally ree D X -module o rank m. By deinition we have H 0 (L M) = (D Y /z 1 D Y ) DY (D Y /D Y P) D Y /(z 1 D Y + D Y P ). Set D = (j 2,...,j n ) O Y j j n n D Y. By the above consideration we may assume that P is o the orm We will show that m 1 P = 1 m + P i 1 i (P i D ). i=0 DX m D Y /(z 1 D Y + D Y P) (Q 0,Q 1,...,Q m 1 ) m 1 Q j j 1 is an isomorphism o D X -modules. For this we have only to show that or any R D Y there exist uniquely Q D Y and R 0,...,R m 1 D satisying m 1 R = QP + R j j 1. j=0 j=0 Note that D Y = j=0 D j 1. Hence we can write uniquely that R = p S j j 1 (S j D ). j=0 I p m, then R S p p m 1 P p 1 j=0 D j 1. Hence we obtain the existence o Q and R 0,...,R m 1 as above by induction on p. In order to show uniqueness it is suicient to show that D Y P ( m 1 j=0 D j 1 ) = 0. Assume that or Q D Y we have QP m 1 j=0 D j 1. I Q = 0, we can write Q = p j=0 T j j i (T j D ) with T p = 0. Then we have QP T p m+p 1 + m+p 1 j=0 D j 1. This is a contradiction. Hence we have Q = 0.

11 2.4 Inverse images in the non-characteristic case 67 Example A smooth morphism : X Y is non-characteristic with respect to any coherent D Y -module. The aim o this section is to prove the ollowing. Theorem Let : X Y be a morphism o smooth algebraic varieties and let M be a coherent D Y -module. Assume that is non-characteristic with respect to M. (i) H j (L M) = 0 or j = 0. (ii) H 0 (L M) is a coherent D X -module. (iii) Ch(H 0 (L M)) ρ ϖ 1 (Ch M). For the proo we need the ollowing. Lemma Let : X Y be an embedding o a hypersurace and let M be a coherent D Y -module. Assume that is non-characteristic with respect to M. Then or any u M there exists locally a dierential operator P D Y such that Pu = 0 and is non-characteristic with respect to D Y /D Y P. In particular, there exists locally an exact sequence r D Y /D Y P i M 0, i=1 where is non-characteristic with respect to D Y /D Y P i or any i. Proo. It ollows rom Ch(D Y u) Ch(M) that is also non-characteristic with respect to the D Y -submodule D Y u o M. Note that Ch(D Y u) is the zero-set o gr F I or I ={Q D Y Qu = 0}. Since TX Y is a line bundle on X, there exists locally P I such that is non-characteristic with respect to D Y /D Y P. Proo o Theorem (Step 1) We irst consider the case when X is a hypersurace {z 1 = 0} o Y. Let us show (i). Since L M D b (D X ) is represented by the complex 1 M z 1 1 M concentrated in degrees 1 and 0, it suices to show that 1 M z 1 1 M is injective. Assume that u M satisies z 1 ( 1 u) = 0. By Lemma there exists P D Y such that Pu = 0 and is non-characteristic with respect to D Y /D Y P. Then P D Y is a dierential operator o the orm in Example Let m 0 be the order o P and set ad z1 (P ) =[z 1,P]=z 1 P Pz 1 D Y. Then ad m z 1 (P ) D Y is a multiplication by an invertible unction. Hence rom ad m z 1 (P )u = 0 we obtain u = 0. The assertion (i) is proved. Let us show (ii) and (iii). Take a good iltration F o M. Then gr F M is a coherent gr F D Y -module such that the support o the associated coherent O T Y -module gr F M := O T Y π 1 Y gr F D Y πy 1 (grf M)

12 68 2 Coherent D-Modules is Ch(M), where π Y : T Y Y denotes the projection. We set N = M (= H 0 (L M)) and deine a iltration F o N by F i N = Im( F i M M). It is suicient to show that gr F N is a coherent gr F D X -module such that the support o the associated coherent O T X-module gr F N := O T X π 1 X grf D X πx 1 (grf N) is contained in ρ ϖ 1 (Ch(M)). Note that we have a canonical epimorphism gr F M gr F N. Set gr F M := O T X π 1 X grf D X πx 1 ( gr F M). Since is non-characteristic with respect to M, we have that the restriction ϖ 1 (supp gr F M) T X o ρ to ϖ 1 (supp gr F M)is a inite morphism. Hence it ollows rom a standard act in algebraic geometry that gr F M = (ρ ) ϖ gr F M. In particular, gr F M is a coherent O T X-module whose support is contained in ρ ϖ 1 (Ch(M)). The coherence o gr F M over O T X implies the coherence o gr F M over gr F D X. It remains to show that gr F N is a coherent gr F D X -module. Since gr F M is a coherent gr F D X -module, it is suicient to show that F i N is coherent over O X or each i (see Proposition 2.1.1). This ollows rom the deinition o F i N since F i M is coherent and M is quasi-coherent over O X. (Step 2) We treat the case when : X Y is a general closed embedding. We can prove the assertion by induction on the codimension o X using Lemma as ollows (details are let to the readers). The case when codim Y X = 1 was treated in Step 1. In the general case, we can locally actorize : X Y as a composite o X g Z h Y where g and h are closed embeddings o smooth varieties with codim Z X, codim Y Z<codim Y X. Lemma and our assumption on M implies that there exists an open neighborhood U o X in Z satisying ϖh 1 (Ch(M)) T U Y U Y TY Y. Hence we may assume that Z is non-characteristic with respect to M rom the beginning. Then by our hypothesis o induction we have H i (Lh M) = 0 or i = 0 and L = H 0 (Lh M)is a coherent D Z -module with Ch(L) ρ h ϖh 1 (Ch(M)). We easily see by Lemma that g is non-characteristic with respect to L. Hence by our hypothesis o induction we have H i (L M) = H i (Lg L) = 0 or i = 0 and H 0 (L M) = H 0 (Lg L) is a coherent D X -module satisying Ch(H 0 (L M)) = Ch(H 0 (Lg L)) ρ g ϖg 1 (Ch(L)) ρ g ϖg 1 ρ hϖ 1 (Ch(M)) = ρ ϖ 1 (Ch(M)). h

13 2.5 Proper direct images 69 (Step 3) I : X = Y Z Y is the irst projection, then the assertions ollows easily rom the isomorphism L M M O Z. (Step 4) To handle the case o a general morphism : Y X, we may actorize as Y g Y X p X, where g is the graph embedding deined by y (y, (y)) and p is the second projection. Then the result ollows rom Step 2 and Step 3 by using Lemma and the arguments similar to those in Step 2. Remark Under the assumption o Theorem it is known that we have actually Ch(H 0 (L M)) = ρ ϖ 1 (Ch(M)) (see [Kas8] and [Kas18]). 2.5 Proper direct images In this section we show the ollowing. Theorem Let : X Y be a proper morphism. Then or an object M in D b c (D X) the direct image M belongs to Db c (D Y ). Proo. Since we assumed that X and Y are quasi-projective, is a projective morphism. Namely, is actorized as X i Y P n p Y by a closed embedding i (i(x) = ( (x), j (x)), j : X P n ) and a projection p = pr Y to Y. Hence it is enough to prove our theorem or each case. (i) The case o closed embeddings i : X Y : The problem being local on Y, we may take a ree resolution F M o M Dc b(d X) such that each term F j is isomorphic to D n j X. Using the exactness o the unctor i we have only to prove the coherence o i D X over D Y. We see by i D X = i (D Y X DX D X ) = i (i 1 (D Y OY 1 Y ) i 1 O Y X ) = D Y OY ( 1 Y OY i X ) that i D X is locally isomorphic to D Y /D Y I X where I X O Y is the deining ideal o X. In particular, it is coherent. (ii) The case o projections p : X = Y P n Y : Since the problem is local on Y, we may assume that Y is an aine variety. By Theorem and Proposition there exists a resolution F M o M in Dc b(d X), where F is a bounded complex o D X -modules such that each term F j o F is a direct summand o a ree D X -module o inite rank. Then it is suicient to show p F j Dc b(d Y ) or any j. Assume that F j is a direct summand o DX n. Then p F j Dqc b (D Y ) is a direct summand o p Dn X. Hence it is enough to show p D X Dc b(d Y ). By

14 70 2 Coherent D-Modules D Y X = Y P n OY P n p ( D Y OY 1 ) Y DY P n, we have D X = Rp (D Y X L D X D X ) Rp (D Y P n) D Y C RƔ(P n, P n). p Now we recall that the only non-vanishing cohomology group o RƔ(P n, P n) is H n (P n, P n) C. Thereore, we get that RƔ(P n, P n) C[ n] and p D X D Y [ n]. Remark Under the assumption o Theorem it is known that we have ( Ch M ) ϖ ρ 1 (Ch(M )). As we saw in Lemma the equality holds in the case where is a closed embedding. The proo or the general case is more involved. 2.6 Duality unctors We irst try to ind heuristically a candidate or the dual o a let D-module. Let M be a let D X -module. Then Hom DX (M, D X ) is a right D X -module by right multiplication o D X on D X. By the side-changing unctor OX 1 X we obtain a let D X -module Hom DX (M, D X ) OX 1 X. Since the unctor Hom D X (,D X ) is not exact, it is more natural to consider the complex RHom DX (M, D X ) OX 1 X o let D X -modules. In order to judge which cohomology group o this complex deserves to be called dual, let us consider the ollowing example. Let X = C (or an open subset o C) and M = D X /D X P (P = 0). By applying the unctor Hom DX (,D X ) to the exact sequence 0 D X P D X M 0 o let D X -modules we get an exact sequence 0 Hom DX (M, D X ) D X P D X (note Hom DX (D X,D X ) D X ). Hence in this case, we have Ext 0 D X (M, D X ) = Hom DX (M, D X ) = Ker(P : D X D X ) = 0 and the only non-vanishing cohomology group is the irst one Ext 1 D X (M, D X ) D X /P D X. The let D X -module obtained by the side changing 1 X is isomorphic to

15 Ext 1 D X (M, D X ) OX 1 X D X /D X P, 2.6 Duality unctors 71 where P is the ormal adjoint o P. From this calculation, we see that Ext 1 is more suited than Ext 0 to be called dual o M. More generally, i n = dim X and M is a holonomic D X -module, then we can (and will) prove that only the term ExtD n X (M, D X ) survives and the resulting let D X -module ExtD n X (M, D X ) OX is also holonomic. Hence the correct deinition o the dual DM o a holonomic 1 X D X -module M is given by DM = ExtD n X (M, D X ) OX 1 X. For a non-holonomic D X -module one may have other non-vanishing cohomology groups, and hence the duality unctor should be deined as ollows or the derived categories. Deinition We deine the duality unctor D = D X : D (D X ) D + (D X ) op by DM := RHom DX (M, D X ) OX 1 X [dim X] = RHom DX (M, D X OX 1 X [dim X]) (M D (D X )). We use the ollowing notation since shits o complexes by dimensions o varieties will oten appear in the subsequent parts. Notation For an algebraic variety X we denote its dimension dim X by d X. Example We have H k (DD X ) = { D X OX 1 X (k = d X ), 0 (k = d X ). Lemma Let M be a coherent D X -module. Then or any aine open subset U o X we have (Ext i D X (M, D X ))(U) = Ext i D X (U) (M(U), D X(U)). Proo. Take a resolution P M U o M U by ree D U -modules o inite rank. Since U is aine, P (U) M(U) gives a resolution o M(U) by ree D X (U)-modules o inite rank. By deinition we have (Ext i D X (M, D X ))(U) = (H i (Hom DU (P,D U )))(U). Set L = Hom DU (P,D U ). Since U is aine and L is a complex o coherent right D U -modules, we have H i (L )(U) = H i (L (U)) (see Remark (ii)). Moreover, we have L (U) = Hom DU (P,D U ) = Hom DX (U)(P (U), D X (U)). Here, the irst equality is obvious, and the second equality ollows easily rom the act that P is a complex o ree D U -modules (or one can use the D-ainity o U). Thereore, we obtain (Ext i D X (M, D X ))(U) = H i (Hom DX (U)(P (U), D X (U))) = Ext i D X (U) (M(U), D X(U)). The proo is complete.

16 72 2 Coherent D-Modules Proposition (i) The unctor D sends D b c (D X) to D b c (D X) op. (ii) D 2 Id on D b c (D X). Proo. (i) We may assume that M = M Mod c (D X ). Then we see rom (the proo o) Lemma that H i (DM) Mod c (D X ) or any i. The boundedness o DM also ollows rom Lemma and Proposition (ii). (ii) We irst construct a canonical morphism M D 2 M or M D b (D X ). First note that D 2 M RHom D op X (RHom D X (M,D X ), D X ), where RHom DX (M,D X ) and D X are regarded as objects o D b (D op X ) (complexes o right D X -modules) by the right multiplication o D X on D X, and the let D X - action on the right-hand side is induced rom the let multiplication o D X on D X. Set H = RHom DX (M,D X ). By applying H 0 (RƔ(X, )) to RHom DX C D op X (M C H,D X ) RHom DX (M,RHom D op X (H,D X )), we obtain Hom DX C D op X (M C H,D X ) Hom DX (M,RHom D op X (H,D X )). Hence the canonical morphism M C H (= M C RHom DX (M,D X )) D X in D b (D X C D op X ) gives rise to a canonical morphism M RHom D op X (H,D X )(= D 2 M) in D b (D X ). It remains to show that M D 2 M is an isomorphism or M Dc b(d X). Since the question is local, we may assume that X is aine. Then we can replace M with D X by Proposition (see the proo o Theorem 2.5.1). In this case the assertion is clear. Corollary D is ully aithul on D b c (D X). The ollowing theorem gives an estimate or the dimensions o the characteristic varieties Ch(H i (DM)) or M Mod c (D X ). Theorem Let X be a smooth algebraic variety and M a coherent D X -module. (i) codim T X Ch(ExtD i X (M, D X ) OX 1 X ) i. (ii) ExtD i X (M, D X ) = 0 (i < codim T X Ch(M)). This theorem is a consequence o Theorem D.4.3 and Lemma Corollary Let M be a coherent D X -module. (i) H i (DM) = 0 unless (d X codim T X Ch(M)) i 0.

17 (ii) codim T X Ch(H i (DM)) d X + i. (iii) M is holonomic i and only i H i (DM) = 0 (i = 0). (iv) I M is holonomic, then DM H 0 (DM) is also holonomic. 2.6 Duality unctors 73 Proo. The statements (i) and (ii) are just restatements o Theorem The statement (iv) and the only i part o (iii) ollows rom (i), (ii) and Corollary Let us show the i part o (iii). Assume that H i (DM) = 0(i = 0), i.e., DM H 0 (DM). Set M = H 0 (DM). Then we have DM = D 2 M M and H 0 (DM ) M by the preceding result D 2 = Id. On the other hand by (ii) we have codim Ch(H 0 (DM )) d X, and hence DM M is a holonomic D X -module. Example Let X = C and Y ={0}. Then we have B Y X D X /D X x, where x is the coordinate o X. Hence by the irst part o this section we have DB Y X D X /D X x B Y X. More generally, we have DB Y X B Y X or any smooth closed subvariety Y o a smooth variety X. This ollows rom Example , Theorem below and B Y X = i O Y, where i : Y X is the embedding. Example Let M be an integrable connection. Then by Proposition 1.2.9, Hom OX (M, O X ) is a let D X -module (an integrable connection). Let us show that First consider the locally ree resolution DM Hom OX (M, O X ). 0 D X OX d X X D X OX X D X O X 0 o O X given in Lemma Since M is locally ree over O X, D X OX X OX M is a locally ree resolution o M. Using this resolution we can calculate Ext d X D X (M, D X ) by the complex ( d X 1 ( d X ) Hom D D O O M, D) Hom D D O O M, D 0 (d X 1 (d X ) Hom O O M, D) Hom O O M, D Hom O ( M, d X 1 O D ) δ Hom O ( M, d X O D ). On the other hand since M is locally ree over O X, we have an exact sequence Hom O (M, d X 1 O D) δ Hom O (M, d X O D) Hom O (M, d X ) 0 o right D X -modules. Hence as a right D X -module we have

18 74 2 Coherent D-Modules Ext d X D (M, D) Hom O(M, d X ). Passing to a let D X -module by the side-changing unctor, we inally obtain DM Hom O (M, d X ) O ( d X ) 1 Hom O (M, O). Theorem (i) The rings D X (U) and D X,x, where U is an aine open subset o U and x is a point o X, have let and right global dimensions d X. (ii) Any M Mod qc (D X ) admits a resolution 0 P dx P 1 P 0 M 0 o length d X by locally projective D X -modules. I M Mod c (D X ), we can take all P i s to be o inite rank. Proo. (i) Since the category o right D-modules is equivalent to that o let D- modules we only need to show the statement or let global dimensions. Since D X (U) is a let noetherian ring with inite let global dimension, its let global dimension coincides with the largest integer m such that there exists a initely generated D X (U)-module M satisying Ext m D X (U) (M, D X(U)) = 0. By Theorem we have Ext i D X (U) (M, D X(U)) = 0 or any initely generated D X (U)-module M and i > d X. Moreover, by Example Ext d X DX (U) (O X(U), D X (U)) = X (U) = 0. Hence the let global dimension o D X (U) is exactly d X. The statement or D X,x ollows rom this. (ii) ollows rom (i) and the proo o Corollary (ii). We note the ollowing basic result, which is a consequence o Proposition D.4.2. Proposition Let X be a smooth algebraic variety and M a coherent D X - module. Then we have Ch(M) = Ch(ExtD i X (M, D X ) OX 1 X ). 0 i d X In particular, i M is holonomic, then the characteristic varieties o M and its dual DM are the same. In the rest o this section we give a description o RHom DX (M,N ) or M D b c (D X), N D b (D X ) in terms o the duality unctor. Lemma For M D b c (D X) and N D b (D X ), we have RHom DX (M,N ) RHom DX (M,D X ) L D X N. Proo. Note that there exists a canonical morphism RHom DX (M,D X ) L D X N RHom DX (M,N ). Hence we may assume that M = D X. In this case the assertion is obvious since both sides are isomorphic to N.

19 2.6 Duality unctors 75 Proposition For M D b c (D X), N D b (D X ) we have isomorphisms RHom DX (M, N ) ( X L O X D X M ) L D X N [ d X ] in D b (C X ). In particular, we have or N D b (D X ). X L D X (D X M L O X N )[ d X ] RHom DX (O X, D X M L O X N ) (2.6.1) RHom DX (O X,N ) X L D X N [ d X ] (2.6.2) Proo. We irst show (2.6.2). By Lemma we may assume that N = D X. In this case we have RHom DX (O X,D X ) Hom DX D X OX 0 d X X,D X Hom DX D X OX X,D X 0 d X Hom OX X,D X Hom OX X,D X 0 d X 1 X OX D X 1 X OX D X X [ d X ] by Lemma The isomorphism (2.6.2) is proved. Let us show (2.6.1). We have RHom DX (M, N ) RHom DX (M, D X ) L D X N ( X L O X D X M ) L D X N [ d X ] by Lemma The second and the third isomorphisms ollow rom Proposition and (2.6.2), respectively. Applying RƔ(X, ) to (2.6.1), we obtain the ollowing. Corollary Let p : X pt be the projection to a point. Then or M D b c (D X) and N D b (D X ) we have isomorphisms R Hom DX (M, N ) p (D X M L O X N ) [ d X ] R Hom DX (O X, D X M L O X N ).

20 76 2 Coherent D-Modules 2.7 Relations among unctors Duality unctors and inverse images The main result in this subsection is the ollowing. Theorem Let : X Y be a morphism o smooth algebraic varieties, and let M be a coherent D Y -module. (i) Assume L M D b c (D X). Then there exists a canonical morphism D X (L M) L (D Y M). (ii) Assume that is non-characteristic with respect to M (hence L M = M and M is coherent by Theorem 2.4.6). Then we have D X (L M) L (D Y M). Proo. (i) By Proposition and Proposition (ii) we have a sequence Hom D b (D Y )(M, M) Hom D b (D Y ) (O Y, D Y M L O Y M) Hom D b (D X ) (L O Y, L (D Y M) L O X L M) Hom D b (D X ) (O X, L M L O X L (D Y M)) Hom D b (D X ) (D X(L M), L (D Y M)) o morphisms, and hence we obtain a canonical morphism D X (L M) L (D Y M) as the image o id M. (ii) By using the decomposition o into a composite o the graph embedding X X Y and the projection X Y Y we may assume that is either a closed embedding or a projection. Assume that : X = T Y Y is the projection. Since the question is local on Y, we may assume that Y is aine. In this case we may urther assume that M = D Y. Then we have D X (L D Y ) D X (O T D Y ) O T (D Y OY 1 Y )[d Y ] L (D Y D X ). Assume that : X Y is a closed embedding. In this case we may assume that is an embedding o a hypersurace (see the proo o Theorem 2.4.6). By Lemma we may urther assume that M = D Y /D Y P. Choose a local coordinate {z i, i } 1 i n as in Example Then we have D Y M D Y /D Y P [d Y 1] = D Y /D Y P [d X ], where P is the ormal adjoint o P with respect to the chosen coordinate (see Section 2.6). Denote by m the order o the dierential operator P. By Example we have D X (L M) D X (DX m ) D m X [d X], L (D Y M) DX m [d X]. The proo that the canonical morphism D X (L D X ) L (D Y M) is actually an isomorphism is let to the readers.

21 2.7 Relations among unctors Duality unctors and direct images In this section we will prove the commutativity o duality unctors with proper direct images. Let : X Y be a proper morphism o smooth algebraic varieties. We irst construct a morphism Tr : O X [d X ] O Y [d Y ] in Dc b(d Y ) (d X = dim X, d Y = dim Y ), which is called the trace map o. In the case o analytic D-modules, this morphism can be constructed using resolutions by currents (Schwartz distributions) (Morihiko Saito, Kashiwara, Schneiders [Sch] or see [Bj2, p. 120]). In our situation dealing with algebraic D-modules we decompose into a composite o a closed embedding and a projection and construct the trace map in each case. First, assume that i : X Y is a closed embedding. By applying the canonical O Y in Dc b(d Y ). By morphism i i Id to O Y we get a morphism i i O Y i O Y = i O Y [d X d Y ]=O X [d X d Y ] it gives i O X[d X d Y ] O Y. We obtain the required morphism Tr i ater taking the shit [d Y ]. Next consider the case o a projection X = P n Y Y. By O X = O P n O Y the problem is reduced to the case where Y consists o a single point. So let us only consider the case p : P n pt, where pt denotes the algebraic variety consisting o a single point. In this case p O Pn is given by Hence there exist isomorphisms ( ]) RƔ P n, [O P n 1 P n n P n. H 0( ) O P n[n] τ 0( ) O P n[n] H n( P n ), P n p p (use the Hodge spectral sequence). Using the canonical isomorphism H n( P n, P n) C given by the standard trace morphism in algebraic geometry, we obtain the desired morphism O P n[n] τ 0( ) O P n[n] C = O pt. p p Let : X Y be a general proper morphism o smooth algebraic varieties. We can decompose into a composite o a closed embedding i : X P n Y and the projection p : P n Y Y. Then the trace morphism Tr : O X [d X ] O Y [d Y ]

22 78 2 Coherent D-Modules is deined as the composite o O X [d X ]= O X [d X ] p i p O P n Y [d Y + n] O Y [d Y ]. One can show that the trace morphism Tr does not depend on the choice o the decomposition = p i and that it is unctorial in the sense that or two proper morphisms : X Y and g : Y Z we have Tr g = Tr g g Tr. We omit the details. The main result in this section is the ollowing. Theorem Let : X Y be a proper morphism. Then we have a canonical isomorphism DY : Dc b (D X) Dc b (D Y ) o unctors. D X Proo. We irst construct a canonical morphism D X D Y o unctors. Let M Dc b(d X). By D X M = R (RHom DX (M,D X ) L D X D X Y ) L O Y 1 Y [d X ] = R (RHom DX (M,D X Y )) L O Y 1 Y [d X ], ( ) D Y M = RHom DY M,D Y L O Y 1 Y [d Y ], it is suicient to construct a canonical morphism ( ) (M ) : R (RHom DX (M,D X Y [d X ])) RHom DY M,D Y [d Y ] in D b c (Dop Y ). By the projection ormula (Corollary 1.7.5) we have D X Y [d X ]= L D Y [d X ] O X [d X ] L O Y D Y and hence the trace morphism Tr induces a canonical morphism D X Y [d X ] D Y [d Y ]. Using this (M ) is deined as the composite o R (RHom DX (M,D X Y [d X ])) R RHom 1 D Y (D Y X L D X M,D Y X L D X D X Y [d X ])) RHom DY (R (D Y X L D X M ), R (D Y X L D X D X Y )[d X ])

23 2.7 Relations among unctors 79 ( ) = RHom DY M, D X Y [d X ] ( ) RHom DY M,D Y [d Y ]. It remains to prove that (M ) is an isomorphism or any M. By decomposing into a composite o a closed embedding and a projection we may assume rom the beginning that is either a closed embedding i : X Y or a projection p : X = P n Y Y. In each case there exists locally on Y a resolution F M o M in D b c (D X), where F is a bounded complex o D X -modules such that each term F j o F is a direct summand o a ree D X -module o inite rank. This is obvious in the case o a closed embedding. In the case o a projection this is a consequence o Theorem and Proposition Thereore, we may assume rom the beginning that M = D X (see the proo o Theorem 2.5.1). Let i : X Y be a closed embedding. In this case (D X ) is given by the composite o i (RHom DX (D X,i D Y )[d X ] ( RHom DY D X, i D Y )[d X ] i i ( = RHom DY D X, i D Y )[d Y ] i ( RHom DY D X,D Y )[d Y ], i where the irst isomorphism is a consequence o Kashiwara s equivalence. Hence it is suicient to show that RHom DY ( i D X, i i D Y ) RHom DY ( i D X,D Y ) is an isomorphism. Set U = Y \ X and let j : U X be the embedding. By the distinguished triangle i i D Y D Y j i j D Y +1 we have only to show that RHom DY ( i D X, j j D Y ) = 0. By Propositions and (ii), we obtain ( ) ( ) RHom DY D X, j D Y i RHom DX D X,i! j D Y i j = i i! j j D Y = 0. Let p : X = P n Y Y be the projection. By D X = D P n D Y the problem is easily reduced to the case when Y consists o a single point and we can only consider the case p : X = P n pt, where pt is the algebraic variety consisting o a single point. In this case we have D P n pt = O P n,d pt P n = P n and hence j

24 80 2 Coherent D-Modules Rp (RHom DX (D X,D X Y [d X ])) = R Hom DP n (D P n, O P n)[n] =RƔ(P n, O P n)[n] C[n], ( ) RHom DY D X,D Y [d Y ] p = R Hom C (RƔ(P n, n P ), C) Hom C(C[ n], C) = C[n]. Thereore, it is suicient to show that (D P n) is non-trivial. Note that (D P n)[ n] is given by the composite o R Hom DP n (D P n, O P n) R Hom C ( P n, P n L D P n O P n) R Hom C (RƔ(P n, P n), RƔ(P n, P n L D P n O P n)) R Hom C (RƔ(P n, P n), τ n RƔ(P n, P n L D P n O P n)) R Hom C (RƔ(P n, P n), C[ n]) R Hom C (RƔ(P n, P n), RƔ(P n, P n)). We easily see that the morphism R Hom DP n (D P n, O P n) R Hom C (RƔ(P n, P n), RƔ(P n, P n)) is induced by the canonical morphism O P n Hom OP n ( P n, P n) and it is nontrivial. Corollary (Adjunction ormula). Let : X Y be a proper morphism. Then we have an isomorphism ( RHom DY M, N ) R RHom DX (M, N ) or M D b c (D X) and N D b (D Y ). Proo. We have R RHom DX (M, N ) R (( X L O X D X M ) L D X L N )[ d Y ] R (( X L O X D X M ) L D X D X Y L 1 D Y 1 N )[ d Y ] R (( X L O X D X M ) L D X D X Y ) L D Y N [ d Y ] D X M L D Y N [ d Y ] D Y M L D Y N [ d Y ] ( RHom DY M, N ) by Proposition and Theorem