OPERATIONS WITH REGULAR HOLONOMIC D-MODULES WITH SUPPORT A NORMAL CROSSING

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1 OPERATIONS WITH REGULAR HOLONOMIC D-MODULES WITH SUPPORT A NORMAL CROSSING JOSEP ÀLVAREZ MONTANER Abstract. The aim of this wor is to describe some operations in the category of regular holonomic D-modules with support a normal crossing and variation zero introduced in [2]. These operations will allow us to compute the characteristic cycle of the local cohomology supported on homogeneous prime ideals of these modules. In particular, we will be able to describe their Bass and dual Bass numbers.. Introduction Let X = C n, O X the sheaf of holomorphic functions in C n,andd X the sheaf of differential operators in C n with holomorphic coefficients. Galligo, Granger and Maisonobe [8], described in terms of linear algebra the category Mod(D X ) T hr of regular holonomic D X -modules such that their solution complex RHom DX (M, O X ) are perverse sheaves relatively to the stratification given by the union T of the coordinate hyperplanes in C n. In Section 2 we recall the definition and the basic properties of the category Dv= T of modules with variation zero introduced in [2] (see also [3]). Moreover, we define the category of modules with unipotent monodromy that is also a full abelian subcategory of Mod(D X ) T hr. This category is closed under extensions and includes D T v=. In Section 3 we describe some operations in the category Dv=. T We have to point out that we will consider the case of R = [x,...,x n ] being the polynomial ring in n independent variables over any field of characteristic zero and D being the ring of differential operators over R. We can consider this case due to the good behavior of this category with respect to flat base change (see [3]). First, we describe the restriction to an homogeneous prime ideal of a module with variation zero. However, the main result of this section is a description of the ernel, the coernel and the image of the homomorphism λ i : M M[ x i ]of localization of a module with variation zero M by the variable x i. In Section 4, by using the results of the previous section and Brodmann s exact sequence, we give an algorithm that allows us to compute the characteristic cycle of the local cohomology modules Hp (M), where R is an homogeneous prime ideal. In particular, we give a different approach to the computation of the Bass numbers µ p (,M):=dim (pα)ext p R pα (( ),M pα ) given by K. Yanagawa in []. 2 Mathematics Subject Classification. Primary 32C38, Secondary 6D4. Key words and phrases. D-modules, Bass numbers.

2 2 J. ÀLVAREZ MONTANER Finally, in Section 5 we define Matlis duality in the category Dv=. T This duality theory is nothing but the duality in the lattice {, } n. The Matlis dual of an injective Dv=-module T is projective so, by using the results of the previous section, we describe projective resolutions in Dv=. T In the sequel we will denote = ε + + ε n where ε,...,ε n is the natural basis of Z n. For all α {, } n, X α will be the linear subvariety of X defined by the homogeneous prime ideal =<x i α i =>. For unexplained terminology on the theory of algebraic D-modules we refer to [4], [6]. 2. Preliminaries Let C n be the category whose objects are families {M α } α {,} n of finitely dimensional complex vector spaces, endowed with linear maps M α u i v Mα+εi, M i α Mα+εi for each α {, } n such that α i =. These maps are called canonical (resp., variation) maps, and they are required to satisfy the conditions: u i u j = u j u i, v i v j = v j v i, u i v j = v j u i and v i u i + id is invertible. Such an object will be called an n-hypercube. A morphism between two n-hypercubes {M α } α and {N α } α is a set of linear maps {f α : M α N α } α, commuting with the canonical and variation maps. In [9], an equivalence of categories between Mod(D X ) T hr and C n is described. The functor Mod(D X ) T hr C n is a contravariant exact functor. The construction of the n-hypercube corresponding to an object M of Mod(D X ) T hr is given in [9]. From the construction one can describe the composition v i u i in terms of the partial monodromy around the hyperplane x i =. We also want to point out that the n-hypercube describes the characteristic cycle of the corresponding D X -module M. Namely, let CC(M) = m α TX α C n be the characteristic cycle of M. Then, one has the equality dim C M α = m α. The papers [2] and [3] study objects in the category Mod(D X ) T hr following property: having the Definition 2.. We say that an object M of Mod(D X ) T hr has variation zero if the morphisms v i are zero for all i n and all α {, } n with α i =. Modules with variation zero form a full abelian subcategory of Mod(D X ) T hr that will be denoted Dv=. T The simple objects of Dv= T are of the form: D X D X ({x i α i =}, { j α j =}). This module is isomorphic to the local cohomology module H α X α (O X ). Every holonomic module has finite length, so if M Dv= T then, there exists a finite increasing filtration {F j } j of M by objects of Dv= T such that for all j one has D X -module isomorphisms F j /F j H α X α (O X ), α {, } n.

3 OPERATIONS 3 The category Dv=, T regarded as a subcategory of Mod(D X ) T hr, is not closed under extensions. However, its objects can be characterized by the following particular filtration: Proposition 2.2. ([2]) An object M of Mod(D X ) T hr has variation zero if and only if there is a increasing filtration {F j } j n of M by objects of Mod(D X ) T hr and there are integers m α for α {, } n such that for all j n one has D-module isomorphisms F j /F j (H α X α =j α (O X )) mα. 2.. Closing the category D T v= for extensions. In this section we will find the minimal full abelian subcategory of Mod(D X ) T hr containing DT v= that is closed under extensions. Definition 2.3. We say that an object M of Mod(D X ) T hr has m-trivial monodromy if the composition of morphisms (v i u i ) m is zero for all i n and all α {, } n with α i =. Modules with m-trivial monodromy form a full abelian subcategory of Mod(D X ) T hr that will be denoted D(vu) T m =. Notice that we have D T v= D T vu= D T (vu) 2 = DT (vu) m = In particular, we get an increasing filtration of the following category: Definition 2.4. We say that an object M of Mod(D X ) T hr has unipotent monodromy if the composition of morphisms v i u i is nilpotent for all i n and all α {, } n with α i =. Modules with unipotent monodromy form a full abelian subcategory of Mod(D X ) T hr that will be denoted Duni T. It is easy to see that the simple objects of the category Duni T are isomorphic to the local cohomology modules H α X α (O X ), for α {, } n. The categories D(vu) T m =, regarded as a subcategories of Mod(D X ) T hr, are not closed under extensions for all m, but Duni T is. Proposition 2.5. The category D T uni, regarded as a subcategory of Mod(D X) T hr,is closed under extensions. Proof: Let M i π M M be an exact sequence in Mod(D X ) T hr such that M, M Duni T. Consider, for all α {, }n such that

4 4 J. ÀLVAREZ MONTANER α i =, the commutative diagram in the category C n of n-hypercubes: u i.. u i u i. M α v i π α Mα v i i α M α v i M α+εi u i π α+εi M α+εi u i i α+εi M α+εi u i M α π α Mα i α M α where (v i u i )m =and(v i u i )m =. It is easy to chec that for m max{m,m },(v i u i ) m =som Duni T. Since every holonomic module has finite length, it follows that the objects of Duni T are characterized as follows: Proposition 2.6. An object M of Dhr T has unipotent monodromy if and only if there is a finite increasing filtration {F j } j of M by objects of Dhr T such that for all j one has D-module isomorphisms F j /F j H α X α (O X ), α {, } n. However, notice that the modules M Duni T are not characterized by a filtration given by the height (as in Proposition 2.2), unless they are modules with variation zero, i.e. we can not give a filtration {F j } j of M where the submodules F j correspond to the n-hypercubes: M β if β j (F j ) β = otherwise, the canonical and variation maps being either zero or equal to those in M. 3. Operations in the category D T v= By using flat base change, we can define the category D T v= of modules with variation zero, as well the corresponding category of n-hypercubes, for the case of D being the ring of differential operators over R, where R is the polynomial or the formal power series ring in n independent variables, x,...,x n, over any field of characteristic zero (see [3, Remar 4.]). From now on, we will only consider the case R = [x,...,x n ] in order to tae advantage of the Z n -graded structure of these modules given by the equivalence of categories (see [2]), between the category of modules with variation zero and the category of straight modules introduced by K. Yanagawa []. 3.. Restriction to a face ideal. Let Z α Z n be the coordinate space spanned by {ε i α i =}, α {, } n. The restriction of R to the homogeneous prime ideal R is the Z α -graded -subalgebra of R R [pα] := [x i α i =].

5 OPERATIONS 5 The restriction to of a Z n -graded module M is the R [pα]-module M [pα] := β Z α Restriction gives us a functor that plays in some cases the role of the localization functor. For details on the description of the morphisms and further considerations we refer to []. Let I = I α I αm be the minimal primary decomposition of a squarefree monomial ideal I R. Then, the restriction of I to the face ideal is the squarefree monomial ideal I [pα] R [pα] whose face ideals in the minimal primary decomposition are those face ideals I αj contained in. Namely I [pα] = I αj. α j α Notice that the restriction to of the local cohomology module H β p β (R) supported on a face ideal p β R vanishes if and only if p β. The restriction to of a module M Dv= T is again a module with variation zero. This can be seen considering the restriction of a fixed increasing filtration of M, =F F F n = M given by Proposition 2.2. Proposition 3.. i) The vertices of the α -hypercube corresponding to M [pα] are the vector spaces (M [pα]) γ = M γ for γ α. ii) The map u j :(M [pα]) γ (M [pα]) γ+εj is the same map as u j : M γ M γ+εj. Proof: Let CC(M) = m γ TX γ X be the characteristic cycle of M. Then, the characteristic cycle of M [pα] is M β CC(M [pα]) = γ α m γ T X γ X so we get the vertices of the n-hypercube. In order to get the linear maps u i s we only have to point out that the filtration of the module M [pα] is determined by the filtration of M, in particular they have the same extension problems. Then we are done by [2, Section 3] and [3, Theorem 4.] Localization by a variable. Let M be a module in D T v= and x i R a variable. We will describe the n-hypercube of the localization M[ x i ]. First of all, it is worthwhile to point out that localization by a variable is an exact functor in the category of modules with variation zero due to the fact that R[ x i ] is a flat module in D T v=. It also follows that localization commutes with restriction to homogeneous prime ideals. Proposition 3.2. i) The vertices of the n-hypercube corresponding to M[ x i ] are the vector spaces M[ x i ] γ = M γ if γ i =. In this case we also have M[ x i ] γ+εi = M γ.

6 6 J. ÀLVAREZ MONTANER ii) The map u j : M[ x i ] γ M[ x i ] γ+εj,whereγ i =,is: Id if j = i u j : M γ M γ+εj if j i. In this case, this map is also equal to u j : M[ x i ] γ+εi M[ x i ] γ+εi+ε j. Proof: The vertices of the n-hypercube corresponding to M[ x i ] can be easily described by means of a formula given in [5]. Namely, let CC(M) = m α TX α X be the characteristic cycle of M. Then, the characteristic cycle of M[ x i ]is CC(M[ x i ]) = α i= m α (T X α X + T X α+εi X) In order to get the linear maps u i s we will use induction on the length l of M. Let l =, i.e. M = H α (R) is a local cohomology module. If α i then M[ x i ]=. Ifα i =, we just have to prove that the map u i : M[ x i ] α M[ x i ] α+εi is the identity. For simplicity we will use the restriction to +εi. Then localizing by x i the minimal graded injective resolution: H α (R) E(R/ )() E(R/+εi )() we get an isomorphism H α (R)[ x i ] = E(R/ )(), so we get the desired result by the description of injective modules given in the proof of [2, Theorem 4.3]. Notice that the same argument can be used to describe the n-hypercube of the localization H α (R)[ ] of a local cohomology module by any squarefree monomial x β x β := x β xβn n, β {, } n. The case l = 2 is proved as well since a module of length two has to be, for convenient α, β {, } n and j {,...,n}, isomorphic to one of the following modules H α (R) H α (R), H α (R) H β p β (R), H α (R)[ ] x j that correspond to the n-hypercubes with non vanishing part 2,, If l>2 we consider the submodule M M whose corresponding n-hypercube only has the vertices and linear map we want to study. Namely, u j : M γ M γ+εj, where γ i =. Ifthen-hypercube of M has another vertex different from zero then length M < length M so we are done by induction. If the n-hypercube of M only has the vertices M γ and M γ+εj, i.e. M = M, we can give a precise description of this module in terms of the ran of u j. Namely, let m =r(u j ), m γ =dim M γ and m γ+εj =dim M γ+εj. Then, M is the direct sum of m copies of H γ p γ (R)[ x j ], m γ m copies of H γ p γ (R),

7 OPERATIONS 7 m γ+εj m copies of H γ+εj p γ+εj (R). So we are done since we can reduce to the cases of length l =andl = Image, ernel and coernel of the localization by a variable. Let M be a module in D T v= and x i R a variable. We will describe the n-hypercubes of the image, ernel and coernel of the morphism λ i : M M[ x i ] of localization of M by the variable x i. Proposition 3.3. i) The vertices of the n-hypercube corresponding to Im λ i are computed from the characteristic cycle CC(Im λ i )= α i=(m α T X α X +r(u i ) T X α+εi X). ii) The map u j :(Imλ i ) γ (Imλ i ) γ+εj are the same as the corresponding for the module M. Proof: We will use induction on the length l of M. Letl =, i.e. M = H α (R) is a local cohomology module. If α i then Im λ i =. Ifα i = we have the exact sequence: H (x i) (M) M M[ x i ] H (x i) (M). Notice that H(x (M) i) =andh (x i) (M) =H α+εi +εi (R) so we are done. The case l = 2 is easy to compute using the description of a module of length two. If l>2, we consider the submodule M M whose corresponding n-hypercube only has the vertices and linear map we want to study. Namely, u j : M γ M γ+εj, where γ i =. It follows as in the proof of Proposition 3.2. Once the n-hypercube for Im λ i is determined we can easily compute the n- hypercube for Ker λ i and Coer λ i. Proposition 3.4. i) The vertices of the n-hypercube corresponding to Ker λ i are computed from the characteristic cycle CC(Ker λ i )=CC(M) CC(Im λ i ). ii) The map u j :(Kerλ i ) γ (Kerλ i ) γ+εj are the same as the corresponding for the module M. Proposition 3.5. i) The vertices of the n-hypercube corresponding to Coer λ i are computed from the characteristic cycle CC(Coer λ i )=CC(M[ x i ]) CC(Im λ i ). are the same as the correspond- ii) The map u j :(Coerλ i ) γ (Coerλ i ) γ+εj ing for the module M[ x i ]. Example: Let R = [x,x 2,x 3 ]. Given the 3-hypercube of a module M D T v= (see below), consider the morphism λ 3 : M M[ x 3 ]. Then we have the following exact sequences of 3-hypercubes: Kerλ 3 M Imλ 3

8 8 J. ÀLVAREZ MONTANER Imλ 3 M[ x 3 ] Coerλ 3 It is not difficult to chec out that: M = H(x (R)[ ) x 2 ] H(x 2 (R) x 2,x 3) M[ x 3 ] = H(x (R)[ ) x 2x 3 ] = E(R/(x ))() Kerλ 3 = H 2 (R) (xx 2,x 3) Im λ 3 = H (x) (R)[ x 2 ] Coerλ 3 = H 2 (x,x 3) (R)[ x 2 ] = E(R/(x,x 3 ))() 4. Bass numbers of modules with variation zero It is easy to prove that the injective objects of Dv= T are of the form: D D({x i α i =}, {x j j + α j =}), α {, }n. These modules are isomorphic to the shifted graded injective hulls E(R/ )() of the quotients R/. In particular, the minimal injective resolution of a module with variation zero M is in the form: I (M) : I d I d dm I m dm, where the j-th term is I j = E(R/ )() µ j(,m). α {,} n

9 OPERATIONS 9 The Bass numbers of M with respect to the face ideal R are the invariants defined by µ j (,M). In general, the Bass numbers µ p (p,m):=dim (p) Ext p R p ((p),m p ) with respect to any prime ideal p R can be described as the multiplicities of the characteristic cycle of Hp(M). p Namely, by using the same arguments as in [, Proposition 2.] we have: Proposition 4.. Let p R be a prime ideal and CC(H p p(m)) = λ p,p,α T X α X be the characteristic cycle of the local cohomology module H p p(m). Then, the Bass numbers with respect to p of M are µ p (p,m)=λ p,p,αp, where X αp is the subvariety of X =Spec(R) defined by p. Our aim in this section is to compute the characteristic cycle of the local cohomology modules Hp (M), where R is a face ideal. In particular, we give a different approach to the computation of the Bass numbers of these modules given by K. Yanagawa in []. We have to point out that, by means of [7, Theorem.2.3], one may compute the Bass numbers of M with respect to any prime ideal. To this purpose we will use the Brodmann sequence H p +εi (M) H p (M) H p (M)[ x i ] H p+ +εi (M) in an iterated way. By using the additivity of the characteristic cycle with respect to short exact sequences it will be enough to compute the characteristic cycle of the ernel and coernel of the localization morphism. This will be done by using the description given in Propositions 3.4 and 3.5. We present the following: Algorithm: INPUT: A module with variation zero M Dv= T and the face ideal R. Denote := (x i α i =, i ). OUTPUT: The characteristic cycle of Hp (M) p. For i =,...,n, while α i =: Consider the Brodmann exact sequence: H p i (M) H p i (M) λi,p H p i (M)[ x i ] H p+ i (M) Compute the characteristic cycle of Ker λ i,p and Coer λ i,p, by using Propositions 3.4 and 3.5. p CC(H p i (M)) = CC(Ker λ i,p )+CC(Coer λ i,p ), p.

10 J. ÀLVAREZ MONTANER Example: Let R = [x,x 2,x 3 ]. Consider the 3-hypercube of the module M Dv= T studied in the previous example. Then: For i =, 2, 3, consider the Brodmann sequences H (x i) (M) M λ i, M[ x i ] H (x i) (M). By using Propositions 3.4 and 3.5 we get: CC(H(x (M)) = T ) X (,,) X + TX (,,) X + TX (,,) CC(H(x (M)) = T 2) X (,,) CC(H(x (M)) = T 3) X (,,) X + TX (,,) X + TX (,,) CC(H(x (M)) = T 3) X (,,) X + TX (,,) For i<j 3, consider the Brodmann sequences H (x i,x j) (M) H (x i) (M) λj, H (x i) (M)[ x j ] H (x i,x j) (M). By using Propositions 3.4 and 3.5 we get: CC(H(x (M)) = T,x 2) X (,,) CC(H(x (M)) = T,x 3) X (,,) CC(H(x (M)) = T,x 3) X (,,) X + TX (,,) CC(H(x (M)) = T 2,x 3) X (,,) Consider the Brodmann sequence H m (M) H (x,x 2) (M) λ3, H (x,x 2) (M)[ x 3 ] H m (M). By using Propositions 3.4 and 3.5 we get: CC(H(x (M)) = T,x 2,x 3) X (,,)

11 OPERATIONS In particular, the Bass numbers of M are: p γ µ µ µ 2 (x ) - - (x 2 ) (x 3 ) (x,x 2 ) (x,x 3 ) - (x 2,x 3 ) - - (x,x 2,x 3 ) Dual Bass numbers of modules with variation zero It is easy to prove that the projective objects of Dv= T are of the form: D D({x i i + α i =}, { j α j =}), α {, }n. This modules are isomorphic to the localizations R[ x ]. In particular, the minimal α projective resolution of a module with variation zero M is in the form: P (M) : P m dm d P d P, where the j-th term is P j = R[ πj(pα,m) x α ]. α {,} n The dual Bass numbers of M with respect to the face ideal R are the invariants defined by π j (,M). d m 5.. Matlis duality. Recall that for Z n -graded modules, the Matlis dual of a Z n - graded R-module M is defined as: M := Hom R (M, E(R/m)). Notice that the Matlis dual defines a duality of the type α α among its graded pieces. In order to find a duality theory for the n-hypercubes we should loo for the duality in the lattice {, } n. More precisely, a duality of the type α α. In the category of module with variation zero we can mimic the definition of Matlis duality because of the equivalence of categories given in [2, Theorem 4.3]. Definition 5.. Let M D T v= be a module with variation zero. We define the Matlis dual of M as: M := Hom D (M, D D(x,...,x n ) ) Nevertheless we can prove that this definition gives in fact the notion of duality in the lattice {, } n. Proposition 5.2. i) The vertices of the n-hypercube corresponding to the module with variation zero M are the vector spaces M α = M α

12 2 J. ÀLVAREZ MONTANER ii) The map u i : M γ M γ+ε i is the dual of u i : M γ εi M γ. Proof: We only have to notice that, by using [2, Lemma 4.4], we have: D Hom D (M, D(x )=Hom,...,x n) D(M,E )= Hom R (M, E(R/m)()) = = Hom R (M, E(R/m))() =M (), where the last module is the Matlis dual of the Z n -graded module M shifted by. The Matlis dual functor is exact contravariant and it is easy to prove that the Matlis dual of an injective Dv=-module T is projective, more precisely we have ( E(R/ )()) = R[ x α ] and the Matlis dual of a simple Dv=-module T is simple, namely we have (H α (R)) = H α (R) Dual Bass numbers. By using Matlis duality we can compute the dual Bass numbers of a projective resolution of a module with variation zero M D T v=. Namely we have: Proposition 5.3. Let M be the Matlis dual of a module with variation zero M D T v=. Then, we have: π j (,M):=µ j (p α,m ). Example: Let R = [x,x 2,x 3 ]. Consider the 3-hypercubes of the module with variation zero M Dv= T studied in the previous examples and its Matlis dual M : Then, the dual Bass numbers of M are: p α p γ π π π 2 () - - (x ) - - (x 2 ) - (x 3 ) (x,x 2 ) (x,x 3 ) (x 2,x 3 ) - - (x,x 2,x 3 ) - - -

13 OPERATIONS 3 References [] J. Àlvarez Montaner, Some Numerical Invariants of Local Rings, Proc. Amer. Math. Soc. 32, (24), [2] J. Àlvarez Montaner, R. García López and S. Zarzuela, Local cohomology, arrangements of subspaces and monomial ideals, Adv. Math. 74 (23), no., [3] J. Àlvarez Montaner and S. Zarzuela, Linearization of local cohomology modules, in Commutative Algebra: Interactions with Algebraic Geometry, Eds.: L. L. Avramov, M. Chardin, M. Morales, C. Polini, Contem. Math. 33,, Amer. Math. Soc. (23). [4] J. E. Björ, Rings of differential operators, North Holland Mathematics Library, Amsterdam, 979. [5] J. Briançon, P. Maisonobe and M. Merle, Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom, Invent. Math. 7 (994), [6] S. C. Coutinho, A primer of algebraic D-modules, London Mathematical Society Student Texts, Cambridge University Press, 995. [7] S. Goto and K. Watanabe, On Graded Rings, II (Z n - graded rings), Toyo J. Math. (2) (978), [8] A. Galligo, M. Granger and Ph. Maisonobe, D-modules et faisceaux pervers dont le support singulier est un croisement normal, Ann. Inst. Fourier 35 (985), 48. [9] A. Galligo, M. Granger and Ph. Maisonobe, D-modules et faisceaux pervers dont le support singulier est un croisement normal. II, in Differential systems and singularities (Luminy, 983) Astérisque 3 (985), [] E. Miller, The Alexander duality functor and local duality with monomial support, J. Algebra 23 (2), [] K. Yanagawa, Bass numbers of local cohomology modules with supports in monomial ideals, Math. Proc. Cambridge Philos. Soc. 3 (2), Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Avinguda Diagonal 647, Barcelona 828, SPAIN address: Josep.Alvarez@upc.es