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Microdifferential systems in the complex domain

Author: Pierre Schapira
Publisher: Berlin ; New York : Springer-Verlag, 1985.
Series: Grundlehren der mathematischen Wissenschaften, 269.
Edition/Format:   Print book : EnglishView all editions and formats
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Additional Physical Format: Online version:
Schapira, Pierre, 1943-
Microdifferential systems in the complex domain.
Berlin ; New York : Springer-Verlag, 1985
(OCoLC)624472738
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Pierre Schapira
ISBN: 038713672X 9780387136721 354013672X 9783540136729
OCLC Number: 10995833
Description: x, 214 pages : illustrations ; 24 cm.
Contents: I. Microdifferential Operators.- Summary.- 1. Construction of the Ring ?x.- 1.1. Differential Operators.- 1.2. Formal Microdifferential Operators.- 1.3. Microdifferential Operators.- Exercises.- 2. Division Theorems.- 2.1. A Banach Algebra of Operators.- 2.2. The Spaath and Weierstrass Theorems.- Exercises.- 3. Refined Microdifferential Cauchy-Kowalewski Theorem.- 3.1. Statement of the Theorem.- 3.2. The Abstract Cauchy-Kowalewski Theorem in Scales of Banach Spaces.- 3.3. Proof of Theorem 3.1.1.- Exercises.- 4. Microdifferential Modules Associated to a Submanifold.- 4.1. The Sheaf CZ|X.- The Case when Z is a Hypersurface.- 4.3. The Sheaf ?Y?X.- Exercises.- 5. Quantized Contact Transformations.- 5.1. Division by an Ideal.- 5.2. Adjoint.- 5.3. Quantized Contact Transformations.- 5.4. Examples.- Exercises.- 6. Systems with Simple Characteristics.- 6.1. Equivalence of Operators.- 6.2. The Regular Involutive Case.- 6.3. Holonomic Systems with Simple Characteristics.- Exercises.- Notes.- II. ?X-modules.- Summary.- 1. Filtered Rings and Modules.- 1.1. Noetherian and Zariskian Filtrations.- 1.2. Homological Properties.- 1.3. Characteristic Ideal.- 1.4. Sheaves of Filtered Modules.- 1.5. Examples.- Exercises.- 2. Structure of the Ring ?X.- 2.1. The Ring ?X(0).- 2.2. Main Properties of ?X.- 2.3. Characteristic Cycle.- 2.4. Holonomic Modules.- 2.5. Adjunction of a Dummy Variable.- 2.6. DX-Modules.- Exercises.- 3. Operations on ?X-modules.- 3.1. Definitions.- 3.2. Operations on BS|X.- 3.3. CS|X.- 3.4. Operations on ?X-modules.- 3.5. Complement on Inverse Images.- Exercises.- Notes.- III. Cauchy Problem and Propagation.- Summary.- 1. Microcharacteristic Varieties.- 1.1. Normal Cones.- 1.2. 1-microcharacteristic Variety.- 1.3. Characteristic Varieties Associated to a Submersion.- 1.4 Characteristic Varieties Associated to an Embedding.- 1.5. Characteristic Variety of the Systems Induced on a Submanifold.- 1.6. Coherency of the Systems Induced on a Submanifold.- Exercises.- 2. The Cauchy Problem.- 2.1. The Cauchy Problem for a System.- 2.2. Application: The Cauchy Problem with Data Ramified along Hypersurfaces.- Exercises.- 3. Propagation.- 3.1. Propagation at the Boundary for one Operator.- 3.2. Propagation for Systems.- Exercises.- 4. Constructibility.- 4.1. Real and Complex Stratifications.- 4.2. Micro-support of Sheaves.- 4.3. Micro-supports and Microcharacteristic Varieties.- Exercises.- Notes.- Appendices.- A. Symplectic Geometry.- A.1. Symplectic Vector Spaces.- A.2. Symplectic Manifolds.- A.3. Homogeneous Symplectic Structures.- A.4. Contact Transformations.- B. Homological Algebra.- B.1. Categories and Derived Functors.- B.2. Rings and Modules.- B.3. Graded Rings and Modules.- B.4 Koszul Complexes.- B.5. The Mittag-Leffler Condition.- C. Sheaves.- C.1. Presheaves and Sheaves.- C.2. Cohomology of Sheaves.- C.3. ?ech Cohomology.- C.4. An Extension Theorem.- C.5. Coherent Sheaves.- D.1. Support and Multiplicities.- D.2. Homological Dimension.- List of Notations and Conversions.
Series Title: Grundlehren der mathematischen Wissenschaften, 269.
Responsibility: Pierre Schapira.

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