D-modules and finite maps

Abstract

Introduction 1.1. The direct image π + . 1.2. The inverse image π ! 1.3. Connections and simply connected varieties 2. Operations on D-modules over finite maps 2.1. The Jacobian ideal and the étale locus 2.2. Differential operators 2.3. The relative canonical bundle and the isomorphism η 2.4. Direct and inverse images 2.5. Duality and finite maps 2.6. The trace 2.7. A D-module motivation for the isomorphism η 3. Semisimple inverse and direct images 3.1. Descent of D-modules 3.2. Twisted modules and the inertia group 3.3. Decomposition of inverse images 3.4. Decomposition of direct images 3.5. The normal basis theorem 3.6. Proof of Theorem 3.27 3.7. Decomposition for general finite maps 4. More decompositions, modules over liftable differential operators 4.1. About the vanishing trace module 4.2. Explicit decomposition for Galois morphisms 4.3. Composition of direct and inverse images 4.4. Abelian extensions 4.5. Decomposition over liftable differential operators 5. Covering D-modules 5.1. Global sections of differential forms and connections on a projective variety 5.2. D-modules of rank 1 5.3. Étale trivial connections of rank 1 and a residue theorem 5.4. Liouville's theorem 5.5. Representations of finite groups 5.6. The Galois correspondence 5.7. Inverse images of covering modules 6. Simply connected varieties 6.1. Generalities 6.2. The Grothendieck-Lefschetz theorem for connections 6.3. Differential coverings 7. The decomposition of π + (O X ) Date: November 19, 2018. 1 2 ROLF K ÄLLSTR ÖM 7.1. Total ramification 92 7.2. Canonical stratifications and filtrations 94 7.3. Maximal étale and minimal totally ramified factorizations 97 7.4. Étale coverings for étale trivial connections and Galois coverings for finite modules. 99 7.5. Trace characterization of the canonical submodules 103 7.6. Generators and decomposition of the inertia modules 107 8. Complex reflection groups 108 8.1. Generators 108 8.2. Canonical filtrations for symmetric groups 109 8.3. Presentation of exponential modules for complex reflection groups 113 8.4. Simple D-modules for imprimitive complex reflection groups 116 9. Appendix: Minimal extensions 118 9.1. Abstract minimal extensions 118 9.2. Holonomic minimal extensions 121 References 123