D-Modules and Mixed Hodge Modules

Transcription

1 D-Modules and Mixed Hodge Modules Notes by Takumi Murayama Fall 2016 and Winter 2017 Contents 1 September 19 (Harold Blum) Deinitions [HTT08, 1.1] The correspondence between D-modules and connections [HTT08, 1.2] The relationship between let and right D-modules [HTT08, 1.2] D-modules that are coherent over O [HTT08, Thm ] September 26: The Classical Riemann Hilbert Correspondence (Mircea Mustaţă) 8 3 September 26 (Harold Blum) Good Filtrations [HTT08, 2.1] Characteristic varieties [HTT08, 2.2] October 3: Operations on D-modules (Takumi Murayama) Inverse images [HTT08, 1.3] Inverse images o let D-modules The shea D X Y Direct images [HTT08, 1.3] Direct images o right D-modules Direct images o let D-modules and the shea D Y X The derived category o D-modules [HTT08, ] Derived inverse images and shited inverse images [HTT08, 1.5] October 10 and October 17: Operations on D-modules (Takumi Murayama) Tensor products and box products Derived inverse images and shited inverse images Projections Closed immersions Derived direct images Composition o direct images Projections Closed immersions Properties o general direct images October 17: Kashiwara s Equivalence (Harold Blum) 32 Notes were taken by Takumi Murayama, who is responsible or any and all errors, based on lectures by Harold Blum, Mircea Mustaţă, and the notetaker. Please takumim@umich.edu with any corrections. Compiled on September 7,

2 7 October 24: Holonomicity (Harold Blum) Bernstein s inequality Properties o holonomic D-modules Holonomicity and Functors Finiteness property October 31 (Harold Blum) Holonomic D-modules on C n Fourier transorm Proo o main theorem Excision sequence October 31 and November 7: Duality Functors (Takumi Murayama) Duals and holonomicity Hom in terms o duality unctors Relations with other unctors Inverse images Direct images Adjunction ormulas and six unctor ormalism [HTT08, 3.2.3] November 7 and November 14: Regular holonomic D-modules (Takumi Murayama) Regular holonomic D-modules on curves [Ber82, 4.1] Simple holonomic modules [HTT08, 3.4] Regular holonomic D-modules in general [Ber82, ] D-modules with regular singularities along a divisor November 14: The Riemann Hilbert correspondence (Takumi Murayama) Proo o Riemann Hilbert correspondence assuming Theorem Proo o Theorem 11.5 or direct images November 21: Hodge structures and mixed Hodge structures (Harold Blum) Motivation Hodge structures Mixed Hodge structures November 28 (Harold Blum) Polarized Hodge structures Examples o the weight iltration on varieties Hodge structure on cohomology December 12: Variations o Hodge structure (Takumi Murayama) Geometric variations o Hodge structure Griiths transversality Abstract variations o Hodge structure One application: Hodge loci January 23: V -iltrations and vanishing cycles (Harold Blum) Example Vanishing cycles V -iltrations

3 16 January 30: Pure Hodge modules (Takumi Murayama) A unctorial deinition or vanishing cycles Introduction to Hodge modules Motivation [Sch14a, 4] An example: the canonical bundle and vanishing theorems [Pop16, 5, 9; Sch14a, 5] Pure Hodge modules [Sai89b, 3; Sch14a, Pt. B] Filtered D-modules with Q-structure [Sch14a, 7] Nearby and vanishing cycles or iltered D-modules [Sai88, 3.4, 5.1; Sch14a, 9] A preliminary deinition [Sai89b, n o 3.2] February 13 (Harold Blum) The decomposition theorem Decomposition by strict support Compatibility with iltrations February 20 and March 6: Functors on Hodge modules and an application (Takumi Murayama) Statement o Popa and Schnell s theorem on zeros o one-orms [PS14] Strategy o proo [PS14, n o 10] Technical preliminaries [Sch14a; Pop17, 2] Strictness The duality unctor and polarizations [Sai88, 2.4; Sch14a, 13,29] Direct images o Hodge modules [Sch14a, 27] Inverse images [Sch14a, 30] Proo o Proposition 18.5, assuming Proposition 18.4 [PS14, n o 10] Construction o objects as in Proposition 18.4 [PS14, n o s 11 17] Preliminaries on cyclic covers [EV92, 3] The construction March 27: Mixed Hodge modules (Takumi Murayama) Weakly mixed Hodge modules [Sch14a, 19] Deinition and properties o mixed Hodge modules [Sch14a, 20 21] Algebraic mixed Hodge modules [Sch14a, 22] Derived categories [Sch14a, 23] Weights [Sch14a, 23] April 2: Algebraic mixed Hodge modules (Takumi Murayama) Deinitions and statement o results Open direct images [Sai13, n o 2.3] Passing to the derived category [Beĭ87, 3] April 3 Kodaira Saito vanishing (Harold Blum) Initial reductions Key tools Non-characteristic pullbacks April 10 (Harold Blum) 95 Reerences 96 3

4 1 September 19 (Harold Blum) Today we will discuss basic properties o D-modules. Note X will always be a smooth complex variety. 1.1 Deinitions [HTT08, 1.1] Let Θ X be the shea o derivations on X, that is, Θ X = Der CX (O X ) = {θ End CX θ(g) = θ()g + θ(g) or all, g O X }, where s F or a shea F denotes a local section s o F. Both Θ X and the structure shea O X are subsheaves o End(O X ), where O X corresponds to [O X g g O X ] End CX (O X ). Deinition 1.1. The shea o dierential operators on X is deined as D X := Θ X, O X End(O X ), that is, D X is the sub-c-algebra o End(O X ) generated by Θ X and O X. Remark 1.2. Over singular varieties, you can still study this ring o dierential operators, but it is pretty crazy (e.g., it is not initely generated), and so the notion o a D-module is deined dierently. Let Z be a singular variety, and suppose that it can be embedded into a smooth variety X. Then, we deine D Z -modules as those D X -modules with support on Z. By Kashiwara s theorem, this deinition is independent o the embedding Z X. Even i there isn t such an embedding, you can deine D Z -modules locally by embedding Z locally, and patching together. We can also describe D X locally using coordinates. Let U be an aine open, and let {x i } be a local coordinate system, which we recall is a set {x i, i } where x i O X (U) and i Θ X (U), such that [ i, j ] = 0, i (x j ) = δ ij, Θ U = and which exists by [HTT08, Thm. A.5.1]. Then, we have D U = α N n O U α n O U i, where α = α1 1 αn n. You can check that these α indeed generate D U over O U, and that they are also independent over O U as is done or U = A n in [Cou95, Ch. 1]. Deinition 1.3. We deine the order iltration F on D U locally by F l D U := O U α. You can also deine the iltration globally by α l F l D X (V ) := {P D X (V ) P U F l D(U) or all open aine U X}. Remark 1.4. Since this deinition requires a choice o coordinates, you need to check that using commutator relations does not increase the order, and that the order is independent o coordinate systems. On the other hand, you can also deine the order iltration more intrinsically as in [Cou95, Ch. 3] as ollows: F 0 D X = O X, F l D X = {P End C (O X ) [P, g] F l 1 D X or all g O X }. Coutinho in [Cou95, Thm ] shows that these two deinitions are equivalent, at least or A n. Remark 1.5. Recall the Bernstein iltration deined in [Cou95, Ch. 1]. This iltration doesn t make sense globally, in particular because it doesn t even exhaust everything: not all unctions can be written as polynomials. However, the Bernstein iltration is easier to prove things with than the order iltration when it can be deined, especially when studying holonomicity. 4 i=1

5 Note 1.6. I P F m D X and Q F n D X, then P Q F m+n D X by [Cou95, Prop ]. We can also show [P, Q] F m+n 1 D X by induction on m + n. I m + n = 0, then the claim is clear; or m + n > 0, we use Jacobi s identity to obtain [[P, Q], g] = [Q, [g, P ]] + [P, [Q, g]], and by induction, [g, P ] F m 1 D X and [Q, g] F n 1 D X, so [Q, [g, P ]] and [P, [Q, g]] are in F m+n 2 D X. Deinition 1.7. The graded ring associated to the iltration F on D X is deined as By Note 1.6, we have gr F D X := gr F l (D X ), gr F l (D X ) := F l D X /F l 1 D X. l=0 Key Property 1.8. gr F D X is commutative. Note 1.9. Key Property 1.8 implies that there exists a map Sym Θ X gr F D X o O X -algebras, by the universal property o the symmetric algebra. This map is in act an isomorphism: locally, i U X is an aine open set, with local coordinates {x i, i }, then gr F D U = O U [ξ 1,..., ξ n ], where the ξ i are the images o i in gr F D U. Thus, we have an isomorphism π O T X gr F D X, where π : T X X is the cotangent bundle on X. Deinition A let D-module is a shea M on X such that M(U) is a let D X (U)-module or each open U X. A right D-module is deined similarly. We denote by Mod(D X ) the collection o let D-modules, and Mod(D op X ) the collection o right D-modules. Example Consider s dierential operators P 1,..., P s D A n. Then, consider M := D A n / D A np D A np s Claim [Cou95, Thm ]. Hom DA n (M, O A n) = { O A n P i = 0 or all i}. Proo. I ϕ Hom DA n (M, O A n), then we can look at ϕ(1) =. We know 0 = ϕ(p i 1) = P i ϕ(1) = P i (). In the other direction, you can send 1 to. Alternatively, this is really something general about maps R/I R. 1.2 The correspondence between D-modules and connections [HTT08, 1.2] Proposition Suppose M is an O X -module. Then, giving a (let) D-module structure on M which extends its O X -module structure is equivalent to giving a C-linear map satisying the ollowing properties: (1) θ (s) = θ (s) (2) θ (s) = θ()s + θ (s) (3) [θ1,θ 2](s) = [ θ1, θ2 ](s) where O X, θ, θ 1, θ 2 Θ X, and s M. : Θ End C (M) θ θ There isn t too much to show, since D X is generated by O X and Θ X, and satisies the key relation [θ, ] = θ() rom [HTT08, Exc (4)]. 5

6 Note When M is locally ree, a map satisying (1) and (2) is called a connection, and is called an integrable or lat connection i it satisies (3) as well. Proposition 1.12 also has an adjoint description: a map : Θ End C (M) is equivalent to a map : M Ω M, since in the orward direction, you can deine and in the opposite direction, you can deine : M Ω C M u dx i i (u) : Θ End(M) D [u (D 1) u] The conditions (1) and (2) above then translate to (1 ) The map in act descends to a map M Ω OX M; (2 ) (u) = (u) + d u. To ormulate (3 ), we note that (1 ) and (2 ) imply that there exist unique maps or each p, such that or every ω 1 Ω q and α Ω p q M, we have Now we can ormulate the integrability condition: (3 ) The composition Ω p M Ω p+1 M (1.1) (ω 1 α) = dω 1 α + ( 1) q ω 1 α. M Ω OX M Ω 2 M is zero, and using the map in (1.1) to extend this sequence o maps, the resulting chain is in act a complex. In this way, the adjoint description o Proposition 1.12 gives the de Rham complex or ree. This is oten used by people who study dg-algebras, since the de Rham complex orms a dg-algebra with multiplication given by the map. Since vector bundles with integrable connection have vanishing Chern classes, we have Claim Let X be a smooth projective curve, and L a line bundle on X. Then, L has a D X -module structure i and only i deg L = The relationship between let and right D-modules [HTT08, 1.2] We want to take M Mod(D X ), and associate to it a right D-module M R Mod(D op X ). Locally, this is simple and is done in [Cou95, 16.2]: i {x i, i } are local coordinates, and P = α a α(x) α D X, then t P = ( 1) α α a α (x), which satisies t P Q = t Q t P, that is, t gives a ring anti-automorphism o D X. I M Mod(D X ), we can get a right action by setting m P := t P m, m M, P D X. Since m (P Q) = t Q t P m = (m tp )Q, this deinition gives a right D X -module structure on M. Now we globalize this construction. It turns out we need to use the shea o n-orms to make this work. Claim Ω n X has a natural right D X-module structure, given by ω θ = (Lie θ)ω, (1.2) where Lie is the Lie derivative, deined below. 6

7 Deinition The Lie derivative o a dierential operator θ D X is the map Lie θ : Ω n X Ω n X ω θ(ω(θ 1,..., θ n )) n ω(θ 1,..., [θ, θ i ],..., θ n ) The Lie derivative comes rom dierential geometry, and satisies the ollowing properties: (1) (Lie[θ 1, θ 2 ])ω = [Lie θ 1, Lie θ 2 ]ω; (2) (Lie θ)(ω) = ((Lie θ)ω) + θ()ω; (3) (Lie(θ))ω = (Lie θ)(ω). Properties (1) and (3) turn the deinition in (1.2) into a right D X -action. Property (1) also explains the sign in (1.2): without the negative sign, we would have a let D X -action, instead o a right D X -action. Locally, we have dx 1 dx n P = ( t P )dx 1 dx n ; also, the right D-module structure on Ω n X gives a map D op X End C(Ω n X). Proposition 1.17 (Tensoring D-modules). Let M, N Mod(D X ) and M Mod(D op X ). Then, (i) M OX N Mod(D X ), with the let action given by θ (m n) = (θm) n + m (θn); (ii) M OX N Mod(D op X ), with the right action given by (m n)θ = m θ n m θn. Here, θ Θ X. We give an example o how to prove this, using the connection ormulation 1.12: Proo o (i). Condition (3) says that θ((m n)) = θ()(m n) + (θ(m n)), which we can check: θ((m n)) = θ(m n) = (θm) n + m θn i=1 = (θ)m n + θ(m) n + m θn = θ()m n + θ(m) n + m θ(n) = θ()(m n) + (θ(m n)) Note it s not clear that the ormulas in Proposition 1.17 that the actions are balanced. Proposition We have an equivalence o categories with quasi-inverse given by (Ω n X ) OX. Ω n X OX : Mod(D X ) Mod(D op X ), 1.4 D-modules that are coherent over O [HTT08, Thm ] Proposition I M is a D X -module, and M is coherent over O X, then M is locally ree. Proo. The idea is to work locally; we want to show that or all closed points x X, the module M x is ree. Choose local coordinates {x i, i }. By Nakayama s lemma, there exist sections s 1,..., s m M x that generate M x, such that s 1,..., s m M x /m x M x orm a basis or this vector space. We want to show that s 1,..., s m have no non-trivial relation. Assume i s i = 0, where i O X,x. Deine the order ord x o O X,x as max{l m l }. I the minimal order o the i is zero, then the residue o i s i = 0 in O X,x /m x gives a non-trivial relation on the s i, which is a contradiction. Otherwise, let i 0 be the index such that ord x ( i0 ) is minimal, and choose j such that i / x j 0. Then, we can act on the relation i s i = 0 by j to obtain ( j i )s i = ( i x j ) s i + i j s i = 0. Since ( ) i0 ord x < ord x ( i0 ) ord x ( i ) x j 7

8 or all i by choice o i 0, writing j s i = k hi k s k, we can combine terms to get a non-trivial relation gi s i = 0 where min{ord x (g i )} ord x i ( i0 Thus, repeating this process gives relations i s 0 = 0 with ever-decreasing minimal orders, and so eventually we have a non-trivial relation h i s i = 0 with h i O X,x /m x = C, contradicting our choice o si. 2 September 26: The Classical Riemann Hilbert Correspondence (Mircea Mustaţă) The basic ramework or this theorem is the analytic category. We will explain how to get back to the algebraic case using GAGA, as long as the variety we are working with is complete. Let X be a connected complex maniold. Then, recall the ollowing deinition: Deinition 2.1. A local system on X is a shea L o C-vector spaces on X, such that locally, L C r, where r is called the rank o L. These orm a category Loc, where morphisms are morphisms as sheaves. The basic act about local systems is the ollowing: Fact 2.2. There is an equivalence o categories { } { } Local systems Finite-dimensional on X representations o π 1 (X, x) where a local system L is deined by looking at the monodromy action o an element o π 1 (X, x) on a stalk L x to get a representation π(x, x) GL(L x ). Remark 2.3. A local system is dierent rom a vector bundle, which is locally C r U. Beore stating the Theorem, we deine morphisms in the other category involved: Deinition 2.4. A morphism ϕ: (E, ) (E, ) o vector bundles with integrable connection is a morphism ϕ: E E o sheaves (hence, o vector bundles by Proposition 1.19) such that the diagram ϕ E Ω E 1 ϕ E Ω E commutes. In particular, this implies both cok ϕ and ker ϕ carry integrable connections. Theorem 2.5 (Riemann Hilbert Correspondence). There is an equivalence o categories { } { } Local systems Vector bundles with on X integrable connection where the unctor F rom local systems to vector bundles with integrable connection is given by x j F: L (L C O X, 1 L d) where O X is the shea o holomorphic unctions, and the connection 1 L d is deined to be the composition L C O X and the unctor G in the other direction is given by ). 1 L d (L C O X ) OX Ω X = L C Ω X, G: (E, ) (ker E). 8

9 One part o the proo is clear: Proo that G F id. It s clear that (L C O X, 1 L d) is a vector bundle with integrable connection, and since ker(o X Ω X ) = C because sections o O X with vanishing derivative are constant, we have that (G F)(L) = ker(1 d) = L C ker(o X Ω X ) L. The subtle part is to show that i (E, ) is a vector bundle with integrable connection, then ker is a local system, and that the canonical morphism (F G)(E, ) = (ker C O X ) E is an isomorphism. This is a local assertion, so you can check this locally with a system o coordinates (that is, we can assume X C n is open with coordinates x 1,..., x n ), and assume that E = O n X is moreover trivial, with basis e 1,..., e r. Given these simpliications, a connection on E is described by saying where each e i goes: (e j ) = n i=1 k=1 r Γ k ijdx i e k, where the Γ k ij are Christoel coeicients. Our goal will be to describe the integrability condition into a geometric condition. Deinition 2.6. A lat section is a section s = (s 1,..., s r ) ker Γ(E). The condition that s ker can be written as ( s j e j ) = 0, which is equivalent to s k x i + r Γ k ijs j = 0 or all i n, k r, which is a linear system o partial dierential equations. The condition that is integrable is that ( (e j )) = 0, which means ( ) Γ k ijdx i e k = 0 or all j. i,k j=1 The let-hand side is ( ( ) d Γ k ij dx i ek Γ k ij dx i (e k ) ). The condition is thereore i,k Γ q ij x p Γq pj x i + r k=1 ( ) Γ k ijγ q pk Γk pjγ q ik = 0 or all i, p n, j, q r. Now consider the total space E, which is just Y = X C r π X with coordinates (x 1,..., x n, y 1,..., y r ), and consider the ollowing vector ields on Y : r r v i = xi yk or 1 i n. k=1 j=1 Γ k ijy j These v 1,..., v n span a subbundle F T Y o rank n, since these v i s are independent at every point. We irst note that [v i, v j ] = 0 or all i, j, and so [F, F ] F, since i u 1 = i v i and u 2 = g i v i, then [u 1, u 2 ] is a linear combination o the v i : i v i (g j v j )(h) g j v j ( i v i )(h) = i v i (g j v j (h)) g j v j ( i v i (h)) = i v i (g j )v j (h) + i g j v i (v j (h)) g j v j ( i )v i (h) g j i v j (v i (h)) = ( i v i (g j )v j g j v j ( i )v i ) (h) 9

10 The integrability condition can be translated into the condition that [v i, v j ] = 0 or all i, j. Frobenius theorem asks or an integrable submaniold in T Y such that the tangent bundle is precisely F. More precisely, the integrability condition implied [F, F ] F above, which implies Theorem 2.7 (Frobenius). For all y Y, there exists a submaniold W y Y containing y such that T z W y = F z or all z W y. Moreover, this is unique in a neighborhood o y. Now consider a section s: X X C r o the bundle projection π : Y = X C r X, given by x s (x, s 1 (x),..., s n (x)). The condition or integrability is that T s(p) s(x) F s(p) or all p X. But T x X T x (X C r ) ( T s(p) s(x) = ds(t p X) = im ( n u u, and so the condition T s(p) s(x) F s(p) is equivalent to s i x j = or all i, j. This is exactly the condition (s) = 0. Given a point x X, consider the ollowing map: r k=1 Γ k ijs k j=1 s i x j u j )1 i r ), (ker ) x E x E x /m x E x = C r. ( ) Step 1. The map ( ) is injective. Proo. Suppose s ker 0 C r, i.e., this says that s 1 (x) = = s r (x) = 0. Since s(x) is an integrable submaniold or F T Y, and the same holds or the zero section, this implies that s = 0 in a neighborhood o x by local uniqueness. Step 2. The map ( ) is also surjective, i.e., there are enough lat sections. Proo. Consider α = (α 1,..., α r ) C r = π 1 (x), and let W an integrable submaniold o Y through the point (x, α). Then, W Y By construction, this diagonal map is a local dieomorphism at x. This means that W is isomorphic to X, i.e., X is the image o a section s o E = O r X in a neighborhood U o x such that W = s(u) around (x, α). This implies that (s) = 0, and s(x) = (x, α). X Step 3. Fix x X, We know that (ker ) x C r, and so there exists a neighborhood U o x and V Γ(U, ker ) such that V (ker ) x, and so we have a morphism V C U E, such that at x, it induces V E(x). In particular, ater replacing U by a smaller subset, we may assume that this is an isomorphism o vector bundles on U. I x U, then V (ker ) x E(x ) implies that V (ker ) x. This implies that L U V. 10

11 Corollary 2.8. I (E, ) is a vector bundle with integrable connection, then the de Rham complex is a resolution o ker. 0 E Ω E 2 Ω E n Ω E 0 Proo. By Riemann Hilbert, this is L C the usual de Rham complex on X. A version o the Poincaré lemma (or holomorphic orms) says the usual de Rham complex is a resolution o O X. This is in act an analytic story; you want to say that analytic vector bundles with connection are the same as algebraic vector bundles with connection. But the connection is not a linear object, so you have to be a bit careul, and can t just apply GAGA! Corollary 2.9. I X is a complete complex algebraic variety, then { } { } Algebraic vector bundles Analytic vector bundles with integrable connection with integrable connection Proo. We want to use GAGA, but or this we need to interpret as an O X -linear map, i.e., we need to linearize the integrable connection. Consider : X X X the diagonal embedding, and let I be the ideal deining X. Then, P := O X X /I 2. This is a shea supported on X, with two O X -module structures: one coming rom let, induced by the irst projection p 1 : X X X, and the other coming rom the right, induced by the second projection p 2 : X X X. We have an exact sequence 0 I /I 2 P O X 0 π O and d() = 1 1 I /I 2. d Ω X/C Claim A connection on E is the same as giving an O X -linear map ϕ: E P OX E, such that (π 1) ϕ = id, where you use the right O X -module structure on P or then the let O X -module structure on E to make P OX E into an O X -module. Proo. Such ϕ is given by ϕ(e) = (1 1) e + (e), where (e) I /I 2 OX E. (e) = (e) + ( 1 1 ) e i and only i ϕ(e) (1 1) e = ( ϕ(e) (1 1 e)) + ( 1 1 ) e, which is equivalent to ϕ(e) = ϕ(e) since (1 1) e = (1 ) e, and (1 1 e) = ( 1) e. We thereore have that {algebraic vector bundles with connection} {analytic vector bundles with connection} by GAGA. For integrability: the curvature θ = E Ω E 2 Ω E is always O X -linear, hence it vanishes in the algebraic category i and only i it does in the analytic category. The issue is much more subtle i you are in the quasi-projective case, in which case you naturally get to the world o regular singularities. Typically, i you have a quasi-projective variety, you compactiy so that the complement is snc, and you have a condition o regular singularities, i.e., it has log poles on the boundary. We will come back to this later. We can or example check compatibility will pullbacks: (e j ) = i,k Γ k ijdx i e k, and so ( e j ) = i,j (Γ k ij )d(x k ) e i On the other hand, pushorwards don t make sense except or smooth maps. I you replace categories to have constructible systems and holonomic D-modules you do get compatibility. 11

12 3 September 26 (Harold Blum) 3.1 Good Filtrations [HTT08, 2.1] Recall that X denotes a smooth variety over C, and that D X denotes the ring o dierential operators. We have deined the order iltration F l D X on D X (Deinition 1.3), which locally given by saying that on an open aine subset U X, with a trivialization {x i, i } o the tangent bundle, we have F l D X (U) = O U α, α l where α N n and α = α i. We can look at the graded ring gr F D X := l F l D X /F l 1 D X, which is commutative, and on open aines U o the orm above, gr F D X (U) = O U [ξ 1,..., ξ n ], where ξ i = i. Now what we want to do is to take a D-module M Mod qc (D X ) that is quasi-coherent over O X, and associate to it a graded module over gr F D X. Deinition 3.1. Let M Mod qc (D X ). Then, we say (M, F ) is a iltration o D X -modules i For every integer i, F i M is a quasi-coherent O X -submodule o M; F i M F i+1 M; F i M = 0 or i 0; M = F i M; F i D X F j M F i+j M. Once we have one o these iltrations, we can deine gr F (M) := i Z F i M/F i 1 M. Proposition 3.2. The ollowing conditions are equivalent: (1) gr F M is coherent over gr F D X ; (2) F i M is coherent over O X or all i, and or i 0 0, F j D X F i0 M = F j+i0 M. Proo o (1) (2). Work locally on an open aine U X. Choose generators m 1,..., m s gr F M(U) over gr F D X (U), where m i F ki M \ F ki 1 M. Then, or k max{k i }, the map F k k1 D X F k ks D x F k M mapping e i m i is surjective. Letting l = max{k i }, the map F k l D X F l M F k M is surjective. Deinition 3.3. I (M, F ) is a iltration, where M is a D-module, we say it is a good iltration i one o the equivalent conditions (1) and (2) hold in the previous Proposition. The theorem below describes when good iltrations exist, and how dierent good iltrations are related to each other. Recall that a coherent D X is one whose local sections on every open aine set U are o inite type over D X (U). Theorem 3.4. (1) I M Mod Coh (D X ), then there exists a good iltration. (2) I (M, F ) and (M, F ) are two iltrations on M, and F is good, then there exists i 0 such that F i M F i+i 0 M or all i. In particular, i F, F are both good, then choosing the maximum o the two i 0 values you get, you have F i i 0 M F i M F i+i 0 M or all i, i.e., any two good iltrations are not too ar apart. 12

13 Proo o (1). Choose generators m 1,..., m s locally on an open aine U. Then, let F i M = F i D X (m 1,..., m s ) or i 0, and F i M = 0 or i < 0. The associated graded module gr F M is generated in degree 0. Now using [HTT08, Cor ], these sections on U extend to global sections o some coherent O X -shea F U, which generate M U as a D X -module. The direct sum U F U or a inite cover o X gives a coherent O U module, which globally generates M as a D X -module. Now we can deine the global good iltration in the same way as in the local description above. (2) is rom Property (2) rom Proposition Characteristic varieties [HTT08, 2.2] Let X be a smooth complex variety, and consider again the order iltration F l D X. We can consider the associated graded ring gr F D X := l N F l D X /F l 1 D X π O T X, which is commutative. Now let M Mod qc (D X ). I M has a iltration F, then we get a graded module gr F M := l Z F l M/F l 1 M over gr F D X, where we note that gr F M is Z-graded in general, in contrast to gr F D X which is N-graded. Recall that (M, F ) is a good iltration i gr F M is initely generated over gr F D X ; this is equivalent to saying that each F l M is coherent by Proposition 3.2. We can then deine the ollowing: Deinition 3.5 (Characteristic variety). Let M Mod c (D X ), and let (M, F ) be a good iltration. Then, let gr F M = O T X π 1 (π O T X ) π 1 gr F M, which we note is the associated module o gr F M under relative Spec. Then, we set ch(m) := Supp gr F M. We can make this more explicit as ollows: i we assume X is aine with local coordinates {x i, i }, then and so ch(m) is given by the ideal Ann(gr F M). gr F D X = O X [ξ 1,..., ξ n ], Note 3.6. ch(m) is preserved by scalar multiplication on the ibers o T X X, since the annihilator is a homogeneous ideal in O X [ξ 1,..., ξ n ]. In the sequel, we say that Ann(gr F ) is homogeneous o degree p when F l M F l+p 1 M. Theorem 3.7 (Basic properties o the characteristic variety). (1) ch(m) does not depend on the choice o good iltration; (2) I 0 M N L 0 is a short exact sequence, ch(n) = ch(m) ch(l). The second statement essentially ollows by choosing a iltration on N, which induces iltrations on the others, and gives a relationship between annihilators. We prove the irst statement. Proo o (1). The idea is to use our result rom last time about how good iltrations are related. We work locally. Suppose (M, F ) and (M, F ) are good iltrations, and suppose Ann(gr F M) is homogeneous o degree p, and so m F l M F l+mp m M. Now let i 0 0 such that F i i 0 M F i M F i+i 0 M, 13

14 which exists by Theorem 3.4. Then, m F lm m F l+i0 M F i+i0 M F l+i0+mp mm F l+2i 0+mp mm which is contained in F l+mp 1 as long as 2i 0 1m 1. Thus, m Ann gr F M. Now that the characteristic variety is well-deined, we will talk about a special case. Proposition 3.8. M Mod coh (D X ), coherent over O X. This is equivalent to ch(m) = im{x 0 T X}. Proo. Choose the iltration F i M = 0 or i < 0, and F i M = M or i 0, in which case gr F M = M. For the other direction, assume that ch(m) has a zero section. Suppose that X is aine, with coordinates {x i, i }. Then, (ξ 1,..., ξ n ) m0 Ann(gr F M) or m 0 0. Using properties o the ilration as beore, you see that F i 1 M = F i M = M or i 0. Since the F i M are coherent, this shows that M is coherent. Example 3.9. Let M = D X /D X P. Then, there is a short exact sequence 0 gr(d X P ) gr D X gr(d X /D X P ) 0 Write P = P P r, such that each P i is homogeneous o order i, and P r 0. Then, the let hand side is gr D X P r (x, ξ), and so ch(m) is deined by the radical o the annihilator o P r (x, ξ). Note that P r (x, ξ) is called the symbol o P. This is more subtle when there is more than one dierential operator. This is similar to tangent cone computations, in that taking the symbol o the entire ideal is not the same as taking the symbol o each generator. 4 October 3: Operations on D-modules (Takumi Murayama) We will reely use intuition rom the study o dierential equations on maniolds. Much o the analogies and motivation or constructions have been taken rom [Ber82]; to make them precise, we should probably mention that basic material on distributions, in particular distributions on maniolds, can be ound in the book(s) by Hörmander [Hör03]. To keep track o what is going on, we recall the ollowing motivation that Harold gave: Example Let P 1,..., P s be s dierential operators on A n. Then, letting we had that M := D A n /D A np D A np s, Hom DA n (M, O) { O A n P i = 0 or all i}. So a D-module M keeps track o solutions to a system o dierential equations. 4.1 Inverse images [HTT08, 1.3] Let : X Y be a morphism o smooth complex varieties. The map O Y O X tells us how we can pullback unctions. Since a D-module keeps track o a system o dierential equations, we can ask: Question 4.1. I we pullback a collection o unctions that satisy a system o dierential equations, how does this aect the system that they satisy? This is what the inverse image unctor will do. Deinition 4.2. Let M be a let D Y -module. Then, its inverse image is deined by M := M = O X 1 O Y 1 M, which is the ormula or inverse images o O-modules. What we need to check is that this new shea M has a let D-module structure. Note that we use dierent notation to dierentiate the act that M has a D-module structure, ollowing [Bor+87, IV,4.1]. 14

15 4.1.1 Inverse images o let D-modules Suppose M is a quasi-coherent O Y -module. Let {y i, i } be a local coordinate system on Y. For any ψ 1 s M and ψ O Y, θ Θ Y, we can deine a let action by ψ (ψ s) = ψ ψ s θ(ψ s) = θ(ψ) s + ψ n θ(y i ) i s The second term in θ(ψ s) can be thought o as a kind o chain rule, and allows the deinition to transorm well under change o coordinates. More precisely, we check that the deinition is independent o choice o coordinates, ollowing [Bor+87, VI,4.1]. Let {y j, j } be another local coordinate system. Then, we have n ψ θ(y j ) js = ψ ( y ) ( ) j y k θ(y i ) y j=1 j y k s i,j,k j = ψ [ ] y j y k θ(y i ) ( k s) y i,k j j y j i=1 = ψ i,k = ψ i θ(y i ) ( k s) ik θ(y i ) i s You can also check that [ i, j ] = 0 implies that this deines a lat connection, and so you get a D X -module structure as desired. Instead o checking the deinition transorms well under changes o coordinates, we can also just write down a global description o this action. First, there is a natural map Ω 1 Y Ω1 X, and so by dualizing on X, we get a morphism Θ X Θ Y = O X 1 O Y Θ Y θ θ = j ϕ j θ j and we can deine θ(ψ s) = θ(ψ) s + ψ θ(s) = θ(ψ) s + ψ j ϕ j θ j (s). We note that tracing this description above, you can show that the inverse image is well-behaved under composition, that is, Proposition 4.3. Let X Y g Z be a sequence o morphisms o smooth varieties. Then, (g ) = g. Proo ollowing [Mil99, Thm. 10.3(i)]. This is already true on the level o O X -modules, and so it suices to show the let D X -module structures match locally, or dierential operators o the orm xk : xk (ψ s) = xk (ψ (1 s)) = xk ψ (1 s) + ψ j xk (y j ) yj (1 s) = xk ψ s + ψ j xk (y j ) i yj (z i g) zi s = xk ψ s + ψ i,j = xk ψ s + ψ i xk (y j )( yj (z i g) ) zi s xk (z i g ) zi s. 15

16 Alternatively, it suices to note that the commutative triangle Θ X g Θ Z Θ Y implies that the D X -module structures on (g ) M and (g (M)) are the same The shea D X Y We want an alternative description o the inverse image unctor. Recall rom Proposition 1.18 that the unctor Mod(D X ) Mod(D op X ) was deined by ω X OX. We want a similar description o the inverse image as a tensor product with a suitable shea. Deinition 4.4. We deine D X Y := D Y = O X 1 O Y 1 D Y, which has a (D X, 1 D Y )-bimodule structure: the let D X -module structure comes rom beore, and the right 1 D Y -module structure comes rom acting on the right. By associativity o the tensor product, M D X Y 1 D Y 1 M. Example 4.5. Let i: A r x A n y be the embedding o A r as {y r+1 = y r+2 = = y n = 0}. Then, D X Y = C[x] C[y] C[y, y ] C[x] C[y] C[y, y1,..., yr ] C C[ yr+1,..., yn ] D X C C[ yr+1,..., yn ]. as a let D X -module. This is locally true or an arbitrary closed embedding X Y [HTT08, Ex ]. 4.2 Direct images [HTT08, 1.3] As beore, let : X Y be a morphism o smooth algebraic varieties. There is no way to pushorward regular unctions or systems o dierential equations on X to those on Y, and so we seem stuck. This is where the equivalence o let and right D-modules becomes useul. We irst recall: Proposition We have an equivalence o categories with quasi-inverse given by ω X O X. ω X OX : Mod(D X ) Mod(D op X ), We can think o this in terms o systems o dierential equations as ollows. Solution spaces o unctions are let D-modules, but solution spaces o distributions are right D-modules. And indeed, the unctor ω X or the special case M = O X locally can be described as dx 1 dx n, that is, it transorms a unction into a distribution [HTT08, Lem ]. Thus, the unctor ω X can be thought o as transorming a system o dierential equations or unctions, into one or distributions. Recall that given a distribution (or maybe more correctly, a current) E (say, with compact support) and a map : X Y, we can intergrate E to get a distribution on Y, by the ormula E, ϕ = E, ϕ. This suggests the ollowing: Question 4.6. I we integrate a collection o distributions that satisy a system o dierential equations, how does this aect the system that they satisy? This is what the direct image unctor will do. 16

17 4.2.1 Direct images o right D-modules Let N be a right D X -module. We already have a direct image N o O-modules, but there s no natural right D Y -module structure on N: Example 4.7 [Mil99, p. 46]. Consider the inclusion i o X = {0} into Y = A 1. Then, D X = C and D Y = C[x, ]. Consider the module i D X = C. This is coherent as an O Y -module, but i it had a D Y -module structure, it must also be locally ree (Proposition 1.19), which it is not. Alternatively, you can also compute the characteristic variety. Choose the iltration where gr F (i D X ) = C in degree 0. The annihilator o gr F (i D X ) in gr F D Y C[x, ξ] is some maximal ideal m, and so the characteristic variety Ch(i D X ) would have dimension 0. This contradicts Bernstein s inequality [HTT08, Cor ], since dim(ch(i D X )) = 0 1 = dim A 1. Instead, consider the O X -module N DX D X Y, the right D X -module structure on N and the let D X -module structure on D X Y are eaten up by the tensor product, and so a right 1 D Y -module structure remains. Now i we apply the pushorward, we obtain (N DX D X Y ). which has a right ( 1 D Y )-module structure. Finally, using the canonical map D Y 1 D Y, we can make the ollowing deinition: Deinition 4.8. Let N be a right D X -module. Then, its direct image is deined by which is a right D Y -module. N := (N DX D X Y ), Warning 4.9. This deinition is not compatible with composition! We shouldn t expect it to be because it is the composition o a right-exact unctor with a let-exact unctor. This will hopeully be clearer when we deine the derived versions o and Direct images o let D-modules and the shea D Y X Since we preer let D-modules, we want to rewrite this deinition in terms o let D-modules. The trick is to use Proposition What we want to do is to ind a unctor itting into the commutative diagram below: Mod(D X ) Mod(D Y ) ω X OX Mod(D op X ) ω Y OY Mod(Dop Y ) Since ωy is a quasi-inverse or the vertical unctor on the right, we compose around the square to get a candidate or the direct image o a let D X -module M: ω Y OY (ω X OX M) = ω Y OY ((ω X OX M) DX D X Y ) ω Y OY ((ω X OX D X Y ) DX M) ((ω X OX D X Y 1 O Y 1 ω Y ) DX M) where the second line is an isomorphism o the input o by Lemma [HTT08, Lem ], and the third line is an isomorphism o O Y -modules by the projection ormula. Much like in the case or inverse images, we give the part o this ormula that does not contain M a name, and restate the deinition o the inverse image with this new shea. 17

18 Deinition We deine D Y X := ω X OX D X Y 1 O Y 1 ω Y, which has a ( 1 D Y, D X )-bimodule structure: D X Y rightmost actors switch these. has the opposite structure, and the letmost and Remark D X Y and D Y X are called the transer bimodules or : X Y by [HTT08]. Deinition Let M be a let D X -module. Then, its direct image is deined by M := (D Y X DX M) Lemma We have the ollowing alternate descriptions o D Y X as a ( 1 D Y, D X )-bimodule: D Y X 1 (D Y OY ωy ) 1 O Y ω X D X Y OX ω X/Y, where the bimodule structure is described in [HTT08, Lem ]. Proo. We just use the deinition o D X Y : D Y X = ω X OX D X Y 1 O Y 1 ω Y = ω X OX (O X 1 O Y 1 D Y ) 1 O Y 1 ω Y ω X 1 O Y 1 D Y 1 O Y 1 ω Y ω X 1 O Y 1 (D Y OY ω Y ) ω X 1 O Y 1 (ωy OY D op Y O Y ω Y OY ωy ) ω X 1 O Y 1 (ωy OY D op Y ) 1 (D Y OY ωy ) 1 O Y ω X where the third isomorphism is by [HTT08, Lem ], and the last isomorphism is given by ω η P P η ω. The second description ollows by deinition o the inverse image o O-modules: D Y X 1 (D Y OY ω Y ) 1 O Y ω X (D Y OY ω Y ) OX ω X D Y OX ω X/Y. We return to the example o an embedding o aine spaces: Example Recall the situation o Example 4.5. We have an embedding i: A r x A n y o A r as {y r+1 = y r+2 = = y n = 0}. Then, i 1 ω Y i 1 O Y ω X can be identiied with O X via the section (dy 1 dy n ) 1 (dx 1 dx r ), and so D Y X C[y, y ] C[y] C[x] C[ yr+1,..., yn ] C D X. Again, this is locally true or an arbitrary closed embedding X Y [HTT08, Ex ]. Remark For reasons o symmetry, we can also deine the inverse image unctor or right D-modules, ollowing [Bor+87, p. 244]. Let : X Y be a morphism o smooth complex varieties, and let M be a right D Y -module. Its inverse image is (M) = 1 M 1 D Y D Y X. 18

19 4.3 The derived category o D-modules [HTT08, ] We assume rom now on that all algebraic varieties are quasiprojective. We saw in 3.1 that it is nicest to work with quasi-coherent D-modules. To make the derived unctor machinery work, we will see that this is also essential because we will need locally projective resolutions. Notation We denote Mod qc (D X ) to be the category o D X -modules that are also quasi-coherent O X -modules, and we denote Mod c (D X ) to be the subcategory o Mod qc (D X ) consisting o modules that are coherent as D X -modules. We mainly work with the bounded derived category. Deinition We denote by D b qc(d X ) (resp. D b c(d X )) the ull subcategory o D b (D X ) with cohomology sheaves in Mod qc (D X ) (resp. Mod c (D X )). We can get analogous deinitions or D + (D X ), the category o complexes that are bounded to the let. Proposition 4.18 [HTT08, Props , ; Cor ]. Let X be a quasi-projective variety, and let M Mod qc (D X ). Then, (i) M has a resolution by injective objects in Mod qc (D X ); (ii) M has a resolution by locally ree objects in Mod qc (D X ); (iii) M has a inite resolution by locally projective objects in Mod qc (D X ) o length 2 dim X. Thus, any object in D b qc(d X ) can be represented by a bounded complex o locally projective objects in Mod qc (D X ). I M Mod c (D X ), then all sheaves in the resolutions in (ii) and (iii) can be taken to be o inite rank. This will allow us to deine the derived versions o unctors o D-modules. Remark There are equivalences o categories D b qc(d X ) D b (Mod qc (D X )) and D b c(d X ) D b (Mod c (D X )) [HTT08, Thm ]. Note we will be deining derived unctors or complexes in D b (D X ) which do not have quasicoherent cohomology (at least at irst), and so we need resolutions or these objects as well: Lemma 4.20 [HTT08, Lem ]. Let R be a shea o rings on a topological space X, and let M Mod(R). Then, (i) M has a resolution by injective objects in Mod(R); (ii) M has a resolution by lat objects in Mod(R). 4.4 Derived inverse images and shited inverse images [HTT08, 1.5] Deinition Let : X Y be a morphism o smooth quasi-projective complex varieties. We deine the derived inverse image unctor L : D b (D Y ) D b (D X ) M D X Y L 1 D Y 1 M by using a lat resolution as in Lemma Proposition L descends to a unctor D b qc(d Y ) D b qc(d X ). Proo. A complex M D b qc(d Y ) is quasi-isomorphic to a bounded complex o locally projective objects P. Now as complexes o D X -modules, L M = D X Y L 1 D Y 1 P (O X 1 O Y 1 D Y ) 1 D Y 1 P O X 1 O Y 1 P P, which has quasi-coherent cohomology. 19

20 Warning The unctor L does not necessarily descend to a unctor D b c(d Y ) D b c(d X )! First note L D Y = D X Y L 1 D Y 1 D Y = D X Y. In Example 4.5, we saw that D X Y was locally ree o ininite rank, and so L D Y = D X Y / D b c(d X ). For convenience later on (especially when discussing Kashiwara s equivalence, adjointness, and the Riemann Hilbert correspondence), we will introduce a shit in degree into the derived inverse image unctor. To those o you who know about perverse sheaves, this amounts to preserving perversity. Deinition Let : X Y be a morphism o smooth quasi-projective complex varieties. We deine the shited inverse image unctor : D b (D Y ) D b (D X ) where (M [dim X dim Y ]) i = M i+dim X dim Y. M L M [dim X dim Y ] Proposition Let X Y g Z be a sequence o morphisms o smooth varieties. Then, L(g ) L Lg, (g ) = g. Proo. This ollows by the Grothendieck spectral sequence, since preserves lat complexes [Bor+87, VI, Prop. 4.3], and any complex o modules has a lat resolution by Lemma We also present the proo in [HTT08, Prop ]. First, we have a chain o isomorphisms o (D X, (g ) 1 D Z )-bimodules D X Y L 1 D Y 1 D Y Z = (O X 1 O Y 1 D Y ) L 1 D Y 1 (O Y g 1 O Z g 1 D Z ) (O X 1 O Y 1 D Y ) L 1 D Y ( 1 O Y (g ) 1 O Z (g ) 1 D Z ) = (O X 1 O Y 1 D Y ) L 1 D Y ( 1 O Y L (g ) 1 O Z (g ) 1 D Z ) O X L (g ) 1 O Z (g ) 1 D Z = D X Z where we use repeatedly that D is a locally ree O-module. We thereore have L(g ) (M ) = D X Z L (g ) 1 D Y (g ) 1 M (D X Y L 1 D Y 1 D Y Z ) L 1 g 1 D Y 1 g 1 M D X Y L 1 D Y 1 (D Y Z L g 1 D Y g 1 M) = L (Lg (M )). We now present an example o (shited) inverse images. Example 4.26 (Open embeddings). Let j : U X be an open embedding into a smooth algebraic variety X. Then, j 1 is just restriction to U, and so in particular, D U X = j 1 D X = D U, and j = Lj = j 1. 5 October 10 and October 17: Operations on D-modules (Takumi Murayama) Today, we want to describe the derived inverse image unctor (in more detail) and the derived direct image. We irst give some motivation or the speciic examples we will be studying in both contexts. Let : X Y be any morphism o smooth quasi-projective varieties. We can actor this map as X Γ X Y 20 p2 Y, (5.1)

21 where Γ is a closed immersion since it is the base change o the diagonal : Y Y Y, and X Y Y is a projection map. Thereore, to understand inverse images and direct images o D-modules, it suices to understand closed immersions and projections separately. Along the way, we will point out that because all o our varieties are smooth, we can assume even more about the morphisms Γ and p 2 above. For example, Γ will realize X as local complete intersection in X Y, and p 2 will be smooth. We start irst by talking about tensor products and box products, since these will be useul when we study projections. 5.1 Tensor products and box products First, the biunctor OX : Mod(D X ) Mod(D X ) Mod(D X ) is right-exact in both actors, we can deine its let derived unctor as L O X : D b (D X ) D b (D X ) D b (D X ) by using lat resolutions as D X -modules. Since a lat D X -module is lat over O X, we have a commutative diagram L O X D b (D X ) D b (D X ) D b (D X ) L O X D b (O X ) D b (O X ) D b (O X ) and so the unctor L O X descends to a unctor L O X : D b qc(d X ) D b qc(d X ) D b qc(d X ). Now consider the smooth variety X Y, and consider the two projections X Y p 1 p 2 X Y Let M be a let D X -module, and N a let D Y -module. We want to describe how they pullback to X Y. To do so, we irst make some general remarks about X Y. Since Θ X Y p 1Θ X p 2Θ Y we also expect that D X Y is related to D X and D Y somehow. In act, Lemma 5.1. D X Y O X Y p 1 1 O X C p 1 2 O Y p 1 1 D X C p 1 2 D Y as O X Y -modules. Proo. Choose local coordinates, and notice that p 1 1 D X C p 1 2 D Y (p 1 1 O X C p 1 2 O Y )[ x, y ], where x are the local partials on X, and similarly or Y. Remark 5.2. The statement o this Lemma in [Bor+87, IV,4.5] is a bit wrong: they identiy O X Y and p 1 1 O X C p2 1 O Y and so only the pullbacks o D X and D Y appear. I think their statement is correct i you ignore non-closed points, since the two sheaves O X Y and p 1 1 O X C p 1 2 O Y dier only at non-closed points. We can now deine the box product : 21

22 Deinition 5.3. Let M Mod(D X ) and N Mod(D Y ). Then, we deine M N := O X Y p 1 1 O X C p 1 2 O (p 1 Y 1 M C p 1 2 N) = D X Y p 1 1 D X C p 1 2 D (p 1 Y 1 M C p 1 2 N) (5.2) This description shows M N is exact in both actors (since both projections are lat: they are base changes o the smooth structure morphisms or X and Y ), so it deines unctors Lemma 5.4. This unctor descends to unctors Proo. This is clear by the isomorphisms (5.2). : Mod(D X ) Mod(D Y ) Mod(D X Y ) : D b (D X ) D b (D Y ) D b (D X Y ) : D b qc(d X ) D b qc(d Y ) D b qc(d X Y ) : D b c(d X ) D b c(d Y ) D b c(d X Y ) We now show that tensor products and inverse images are compatible, by irst showing it or box products: Proposition 5.5. (i) Let 1 : X 1 Y 1 and 2 : X 2 Y 2 be morphisms o smooth algebraic varieties. Then or M 1 D b (D Y1 ), M 2 D b (D Y2 ), we have L( 1 2 ) (M 1 M 2 ) L 1 M 1 L 2 M 2. (ii) Let : X Y be a morphism o smooth algebraic varieties. Then, or M, N D b (D Y ), we have L (M L O Y N ) L M L O X L N. Proo. For (i), since is exact, it suices to note that this is true on the level o modules. We want to reduce (ii) to (i) by using the diagonal embedding Y : Y Y Y. First, note Y (M N) = Y (O Y Y p 1 1 O Y C p 1 2 O Y (p 1 1 M C p 1 2 N)) O Y 1 Y O 1 Y Y Y O Y OY C O Y (M C N) M OY N O Y Y 1 Y (p 1 1 O Y C p 1 2 O Y ) (M C N) where tracing the D Y -module structure everywhere, we see that this isomorphism preserves the D Y -module structure. Since preserves lat modules, we have a canonical isomorphism in D b (D X ). Now can prove (ii): M L O Y N L Y (M N ) L (M L O Y N ) L L Y (M N ) L XL( ) (M N ) L X(L M L N ) L M L O X L N. 22