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"D-module theory is essentially the algebraic study of systems of linear partial differential equations. This book, the first devoted specifically to holonomic D-modules, provides a unified treatment of both regular and irregular D-modules. The authors begin by recalling the main results of the theory of indsheaves and subanalytic sheaves, explaining in detail the operations on D-modules and their tempered holomorphic solutions. As an application, they obtain the Riemann–Hilbert correspondence for regular holonomic D-modules. In the second part of the book the authors do the same for the sheaf of enhanced tempered solutions of (not necessarily regular) holonomic D-modules. Originating from a series of lectures given at the Institut des Hautes Études Scientifiques in Paris, this book is addressed to graduate students and researchers familiar with the language of sheaves and D-modules, in the derived sense." [Publisher]

D-modules. --- D-modules, Théorie des. --- Modules (Algebra) --- Modules (algèbre) --- Sheaf theory. --- Faisceaux, Théorie des. --- Geometry, Algebraic. --- Géométrie algébrique.

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Artificial gravity. --- Coriolis effect. --- Manned Mars missions. --- Spacecraft design. --- Spacecraft modules.

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D-module theory is essentially the algebraic study of systems of linear partial differential equations. This book, the first devoted specifically to holonomic D-modules, provides a unified treatment of both regular and irregular D-modules. The authors begin by recalling the main results of the theory of indsheaves and subanalytic sheaves, explaining in detail the operations on D-modules and their tempered holomorphic solutions. As an application, they obtain the Riemann-Hilbert correspondence for regular holonomic D-modules. In the second part of the book the authors do the same for the sheaf of enhanced tempered solutions of (not necessarily regular) holonomic D-modules. Originating from a series of lectures given at the Institut des Hautes Études Scientifiques in Paris, this book is addressed to graduate students and researchers familiar with the language of sheaves and D-modules, in the derived sense.

D-modules. --- Modules (Algebra) --- Sheaf theory. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra)

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The main focus of this monograph is to offer a comprehensive presentation of known and new results on various generalizations of CS-modules and CS-rings. Extending (or CS) modules are generalizations of injective (and also semisimple or uniform) modules. While the theory of CS-modules is well documented in monographs and textbooks, results on generalized forms of the CS property as well as dual notions are far less present in the literature. With their work the authors provide a solid background to module theory, accessible to anyone familiar with basic abstract algebra. The focus of the book is on direct sums of CS-modules and classes of modules related to CS-modules, such as relative (injective) ejective modules, (quasi) continuous modules, and lifting modules. In particular, matrix CS-rings are studied and clear proofs of fundamental decomposition results on CS-modules over commutative domains are given, thus complementing existing monographs in this area. Open problems round out the work and establish the basis for further developments in the field. The main text is complemented by a wealth of examples and exercises.

Mathematics. --- Associative rings. --- Rings (Algebra). --- Associative Rings and Algebras. --- Modules (Algebra) --- Rings (Algebra) --- Algebraic rings --- Ring theory --- Finite number systems --- Modular systems (Algebra) --- Algebraic fields --- Algebra --- Finite groups --- Algebra. --- Mathematics --- Mathematical analysis

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When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.

Homology theory. --- Abelian group. --- Additive group. --- Algebra homomorphism. --- Algebraic topology. --- Anticommutativity. --- Associative algebra. --- Associative property. --- Axiom. --- Betti number. --- C0. --- Category of modules. --- Change of rings. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Commutative property. --- Commutative ring. --- Cyclic group. --- Derived functor. --- Diagram (category theory). --- Differential operator. --- Direct limit. --- Direct product. --- Direct sum of modules. --- Direct sum. --- Duality (mathematics). --- Endomorphism. --- Epimorphism. --- Equivalence class. --- Exact category. --- Exact sequence. --- Existential quantification. --- Explicit formulae (L-function). --- Factorization. --- Field of fractions. --- Finite group. --- Finitely generated module. --- Free abelian group. --- Free monoid. --- Functor. --- Fundamental group. --- G-module. --- Galois theory. --- Global dimension. --- Graded ring. --- Group algebra. --- Hereditary ring. --- Hochschild homology. --- Homological algebra. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Hyperhomology. --- I0. --- Ideal (ring theory). --- Inclusion map. --- Induced homomorphism. --- Injective function. --- Injective module. --- Integral domain. --- Inverse limit. --- Left inverse. --- Lie algebra. --- Linear differential equation. --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monoidal category. --- Natural transformation. --- Noetherian ring. --- Noetherian. --- Permutation. --- Polynomial ring. --- Pontryagin duality. --- Product topology. --- Projective module. --- Quotient algebra. --- Quotient group. --- Quotient module. --- Right inverse. --- Ring (mathematics). --- Ring of integers. --- Separation axiom. --- Set (mathematics). --- Special case. --- Spectral sequence. --- Subalgebra. --- Subcategory. --- Subgroup. --- Subring. --- Summation. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Trivial representation. --- Unification (computer science). --- Universal coefficient theorem. --- Variable (mathematics). --- Zero object (algebra).