nLab
Abelian categories with applications to rings and modules

  • N. Popescu, Abelian categories with applications to rings and modules, London Mathematical Society Monographs 3, Academic Press 1973, xii + 470 pp

Contents:

Note to the reader

Some terminology, notations and conventions used throughout the book.

Categories and functors

The notion of a category; examples. Special objects and morphisms. Functors. Inductive and projective limits. Adjoint functors; equivalence of categories.

Abelian categories

Preadditive and additive category. The canonical factorization of a morphism; preabelian categories. Abelian categories. Fibred products and fibred sums. Basic lemmas on abelian categories. The isomorphism theorems. Direct sums of subobjects. Inductive limits; the conditions Ab.

Additive functors

Additive functors. Exactness of functors; injective and projective objects. The injective and projective objects in the category Ab. Categories of additive functors; modules. Special objects in abelian categories. Tensor products; a characterization of functor categories. Tensor products of modules. Flat modules. Some remarks on projective modules. Injective envelopes. Semisimple rings.

Localization

Categories of fractions; calculus of fractions. The spectral category of an abelian category. The quotient category of an abelian category relative to a dense subcategory. The section functor. Localization in categories with injective envelopes. Localization in Ab 3 categories. Sheaves over a topological space. Torsion theories. Localizing subcategories in categories of modules. Localization in categories of modules. Left exact functors; the embedding theorem. The study of the localization ring of a ring. The complete ring of quotients. Some remarks on Grothendieck categories. Finiteness conditions on localizing systems. Flat epimorphism of rings. Left quasi-orders of a ring. Rings of fractions. Left orders. Left spectrum of a ring. Bilocalizing subcategories.

The Krull-Remak-Schmidt theorem and decomposition theories

The classical Krull-Remak-Schmidt theorem. The structure of spectral categories. Local coirreducible categories. Decomposition theory. Semi-noetherian categories. Semi-artinian categories. Noetherian and artinian categories. Locally noetherian categories. Noetherian and artinian rings. Primary decomposition theory. Decomposition theories on locally noetherian categories.

Duality

Linearly compact subcategories. Topologically linearly compact rings. The duality theorem for Grothendieck categories. Duality theory for l.n.- and l.f.-categories. Colocalization.

Bibliography

Index

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