Leseprobe
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Homological Algebra (PMS-19), Volume 19
Samuel Eilenberg, Henri Cartan
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Explicit formulae (L-function), Spectral sequence, Commutative diagram, Exact sequence, Linear differential equation, Endomorphism, Epimorphism, Cohomology, Commutative ring, Subcategory, Variable (mathematics), Direct sum, Functor, Algebraic topology, Associative algebra, Monoidal category, Inverse limit, Duality (mathematics), Universal coefficient theorem, Homology (mathematics), Subalgebra, Module (mathematics), Diagram (category theory), Axiom, Product topology, Topological space, Galois theory, Mathematical induction, Ring (mathematics), Derived functor, Unification (computer science), Free abelian group, Group algebra, Lie algebra, Theorem, Direct product, Differential operator, Quotient module, Abelian group, Cokernel, Inclusion map, Projective module, Homological algebra, Homomorphism, Hyperhomology, Zero object (algebra), Tensor product, Right inverse, Special case, Topology, Finitely generated module, Commutative property, Induced homomorphism, Algebra homomorphism, Exact category, Global dimension, Hochschild homology, Set (mathematics), Injective module, Pontryagin duality, Hereditary ring, Equivalence class, Integral domain, Noetherian ring, Category of modules, Quotient algebra, Ideal (ring theory), Noetherian, Fundamental group, Homotopy