De Gruyter Expositions in Mathematics – serie

The category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules.The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules.In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules.In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems.The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

No detailed description available for "Subgroup Lattices of Groups".

No detailed description available for "The Stefan Problem".

No detailed description available for "Compact Projective Planes".

No detailed description available for "Combinatorial Homotopy and 4-Dimensional Complexes".

No detailed description available for "The Analytical and Topological Theory of Semigroups".

No detailed description available for "Compositions of Quadratic Forms".

No detailed description available for "Blow-Up in Quasilinear Parabolic Equations".

No detailed description available for "Finite Soluble Groups".

No detailed description available for "Infinite Dimensional Lie Superalgebras".

No detailed description available for "The Riemann Zeta-Function".

No detailed description available for "Contact Geometry and Linear Differential Equations".

No detailed description available for "Elliptic Problems in Domains with Piecewise Smooth Boundaries".

No detailed description available for "Nilpotent Groups and their Automorphisms".

No detailed description available for "The Restricted 3-Body Problem: Plane Periodic Orbits".

No detailed description available for "Orthogonal Decompositions and Integral Lattices".

No detailed description available for "Moduli Spaces of Abelian Surfaces".

No detailed description available for "Unitary Representation Theory of Exponential Lie Groups".

No detailed description available for "The Link Invariants of the Chern-Simons Field Theory".

No detailed description available for "Semigroups in Algebra, Geometry and Analysis".

No detailed description available for "An Introduction to Lorentz Surfaces".

No detailed description available for "The Adjunction Theory of Complex Projective Varieties".

No detailed description available for "Conformal Geometry of Discrete Groups and Manifolds".

No detailed description available for "Evolution Equations and Lagrangian Coordinates".

No detailed description available for "Lectures in Real Geometry".

No detailed description available for "Monoids, Acts and Categories".

No detailed description available for "Nonlinear Wave Equations Perturbed by Viscous Terms".

No detailed description available for "Algebra in the Stone-Cech Compactification".

No detailed description available for "Character Theory of Finite Groups".