## Induced $${\mathcal D}$$-modules and differential complexes.(English)Zbl 0705.32005

Let $$X\to Y$$ be a proper morphism of complex manifolds or of smooth algebraic varieties. It is known that one can define for any bounded complex $$M^*$$ of $${\mathcal D}_ X$$-modules with coherent cohomologies a duality isomorphism $$f_*{\mathbb{D}}M^*\to {\mathbb{D}}f_*M^*.$$ In this formula $$f_*$$ is the direct image of $${\mathcal D}$$-modules and $${\mathbb{D}}$$ is the dual functor $${\mathbb{R}} Hom_{{\mathcal D}_ X}(M^*,{\mathcal D}_ X(d_ x))$$ (case of right $${\mathcal D}_ X$$-modules).
The aim of this paper is, in particular, to give a new proof of this result, based on the notion of induced $${\mathcal D}_ X$$-module introduced by the author. These are the $${\mathcal D}_ X$$-modules of the type $$L\otimes_{{\mathcal O}_ X}{\mathcal D}_ X$$, for some $${\mathcal O}_ X$$- module L. Applying the functor $$\otimes_{{\mathcal D}_ X}{\mathcal O}_ X$$ one gets the notion of differential morphisms $$L\to L'$$ which are $${\mathcal D}_ X$$ linear map induced by $${\mathcal D}_ X$$-linear morphisms $$L\otimes {\mathcal D}_ X\to L'\otimes {\mathcal D}_ X$$. The author introduces also the notion of differential morphisms of finite order p which correspond to maps $$L\to L'\otimes F_ p{\mathcal D}_ X$$. In § 1, he proves some equivalences of categories quite useful for the next sections:
The category $$D^ b({\mathcal O}_ X,Diff)$$ (resp. $$D^ b({\mathcal O}_ X,Diff)^ f)$$ of complex of $${\mathcal O}_ X$$-modules (resp. with finite order differential), is equivalent to $$D^ b({\mathcal O}_ X)$$. In the first derived category are inverted the D-quasi-isomorphism, i.e. those morphism of complexes which via $$\otimes_{{\mathcal O}_ X}{\mathcal D}_ X$$ induce quasi-isomorphisms in $$C^ b({\mathcal D}_ X)$$. Similarly the author defines $$D^ b({\mathcal O}_ X,Diff)_{coh}$$ via this equivalence. In § 2 and § 3, he obtains then his main results: First, definition of dual functors $$in$$ both categories and then proof of the compatibility of $${\mathbb{D}}$$, with the De Rham functor in the holonomic case, and with $$f^*$$ in general. Finally in the last § 4, the author introduces the concept of diagonal pairing and gives two applications of it, to a definition of the dual functor $${\mathbb{D}}$$ first and then to a simplification in the proof of the Riemann Hilbert correspondence.
Reviewer: J.M.Granger

### MSC:

 32C37 Duality theorems for analytic spaces 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules 14F40 de Rham cohomology and algebraic geometry
Full Text:

### References:

 [1] BOREL (A.) . - Algebraic D-Modules . - Academic Press, 1987 . MR 89g:32014 | Zbl 0642.32001 · Zbl 0642.32001 [2] BERNSTEIN (J.) . - Algebraic theory of D-Modules . - Preprint, 1983 . [3] BERTHELOT (P.) and OGUS (A.) . - Notes on Crystalline cohomology . Princeton University Press, 1978 . MR 58 #10908 | Zbl 0383.14010 · Zbl 0383.14010 [4] GROTHENDIECK (A.) . - Éléments de géométrie algébrique III . Publ. Math. IHES, t. 11, 1961 . Numdam · Zbl 0122.16102 [5] HARTSHORNE (R.) . - Residues and duality . - Lecture Notes in Math., t. 20, 1966 . MR 36 #5145 | Zbl 0212.26101 · Zbl 0212.26101 [6] KASHIWARA (M.) . - On the maximally overdetermined system of linear differential equations I , Publ. RIMS, Kyoto Univ., t. 10, 1975 , p. 563-579. Article | MR 51 #6891 | Zbl 0313.58019 · Zbl 0313.58019 [7] KASHIWARA (M.) . - B-functions and holonomic systems , Invent. Math., t. 38, 1976 , p. 33-53. MR 55 #3309 | Zbl 0354.35082 · Zbl 0354.35082 [8] KASHIWARA (M.) . - The Riemann-Hilbert Problem for Holonomic Systems , Publ. RIMS, Kyoto Univ., t. 20, 1984 , p. 319-365. Article | MR 86j:58142 | Zbl 0566.32023 · Zbl 0566.32023 [9] KASHIWARA (M.) and KAWAI (T.) . - On holonomic system of microdifferential equations III , Publ. RIMS, Kyoto Univ., t. 17, 1981 , p. 813-979. Article | MR 83e:58085 | Zbl 0505.58033 · Zbl 0505.58033 [10] MEBKHOUT (Z.) . - Théorème de bidualité pour les Dx-Modules holonomes , Ark. Mat., t. 20, 1982 , p. 111-124. MR 84a:58075 | Zbl 0525.32025 · Zbl 0525.32025 [11] MEBKHOUT (Z.) . - Une autre équivalence de catégorie , Compositio Math., t. 51, 1984 , p. 63-88. Numdam | MR 85k:58073 | Zbl 0566.32021 · Zbl 0566.32021 [12] RAMIS (J.-P.) , RUGET (G.) et VERDIER (J.-L.) . - Dualité relative en géométrie analytique complexe , Invent. Math., t. 13, 1971 , p. 261-283. MR 46 #7553 | Zbl 0218.14010 · Zbl 0218.14010 [13] SAITO (M.) . - Modules de Hodge polarisables , Publ. RIMS, Kyoto Univ., t. 24, 1988 , p. 849-995. Article | MR 90k:32038 | Zbl 0691.14007 · Zbl 0691.14007 [14] SAITO (M.) . - Mixed Hodge Modules , Preprint RIMS-585. · Zbl 0635.14008 [15] SAITO (M.) . - Introduction to mixed Hodge Modules , Preprint RIMS-605 (to appear in Astérisque). · Zbl 0753.32004 [16] SCHNEIDERS (J.-P.) . - Un théorème de dualité relative pour les modules différentiels , C.R. Acad. Sc. Paris, t. 303, 1986 , p. 235-238. MR 87k:32018 | Zbl 0605.14016 · Zbl 0605.14016 [17] SCHNEIDERS (J.-P.) . - Dualité pour les modules différentiels, dissertation . - Univ. Liège, 1986 - 1987 . [18] VERDIER (J.L.) . - Catégories dérivées , SGA 4 1/2, Lecture Notes in Math., t. 569, 1977 , p. 262-311. MR 57 #3132 | Zbl 0407.18008 · Zbl 0407.18008 [19] VERDIER (J.L.) . - Dualité dans la cohomologie des espaces localement compacts , Séminaire Bourbaki, n^\circ 300, 1966 , Benjamin. Numdam | Zbl 0268.55006 · Zbl 0268.55006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.