Author	Schapira, Pierre. author
Title	Microdifferential Systems in the Complex Domain [electronic resource] / by Pierre Schapira
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 1985
Connect to	http://dx.doi.org/10.1007/978-3-642-61665-5
Descript	X, 216 p. online resource

The words "microdifferential systems in the complex domain" refer to seveยญ ral branches of mathematics: micro local analysis, linear partial differential equations, algebra, and complex analysis. The microlocal point of view first appeared in the study of propagation of singularities of differential equations, and is spreading now to other fields of mathematics such as algebraic geometry or algebraic topology. Howยญ ever it seems that many analysts neglect very elementary tools of algebra, which forces them to confine themselves to the study of a single equation or particular square matrices, or to carryon heavy and non-intrinsic formulaยญ tions when studying more general systems. On the other hand, many algeยญ braists ignore everything about partial differential equations, such as for example the "Cauchy problem", although it is a very natural and geometriยญ cal setting of "inverse image". Our aim will be to present to the analyst the algebraic methods which naturally appear in such problems, and to make available to the algebraist some topics from the theory of partial differential equations stressing its geometrical aspects. Keeping this goal in mind, one can only remain at an elementary level

I. Microdifferential Operators -- Summary -- ยง 1. Construction of the Ring ?x -- Exercises -- ยง 2. Division Theorems -- Exercises -- ยง 3. Refined Microdifferential Cauchy-Kowalewski Theorem -- Exercises -- ยง 4. Microdifferential Modules Associated to a Submanifold -- Exercises -- ยง 5. Quantized Contact Transformations -- Exercises -- ยง 6. Systems with Simple Characteristics -- Exercises -- Notes -- II. ?X-modules -- Summary -- ยง 1. Filtered Rings and Modules -- Exercises -- ยง 2. Structure of the Ring ?X -- Exercises -- ยง 3. Operations on ?X-modules -- Exercises -- Notes -- III. Cauchy Problem and Propagation -- Summary -- ยง 1. Microcharacteristic Varieties -- Exercises -- ยง 2. The Cauchy Problem -- ยง 3. Propagation -- Exercises -- ยง 4. Constructibility -- 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