This volume of the Encyclopaedia presents a modern approach to homological algebra, which is based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geomtry and algebraic topology.

Algebra V: Homological Algebra (Encyclopaedia of Mathematical Sciences)

Science & Math

Igor R. Shafarevich

Aleksei Ivanovich Kostrikin

Yuri Manin

S.I. Gel'fand

Mathematics

Springer, Verlag

This EMS volume consists of two parts. The first part is devoted to cohomology of algebraic varieties. The second part deals with algebraic surfaces. The authors, who are well-known experts in the field, have taken pains to present the material rigorously and coherently. The book contains numerous examples and insights on various topics.This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetical algebraic geometry, complex analysis and related fields.

Algebraic Geometry II: Cohomology of Algebraic Varieties: Algebraic Surfaces (Encyclopaedia of Mathematical Sciences)

This book is a revised and expanded new edition of the first four chapters of Shafarevich’s well-known introductory book on algebraic geometry. Besides correcting misprints and inaccuracies, the author has added plenty of new material, mostly concrete geometrical material such as Grassmannian varieties, plane cubic curves, the cubic surface, degenerations of quadrics and elliptic curves, the Bertini theorems, and normal surface singularities.

Basic Algebraic Geometry 1: Varieties in Projective Space

Springer

Wholeheartedly recommended to every student and user of mathematics, this is an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields studied in every university maths course, through Lie groups to cohomology and category theory, the author shows how the origins of each concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Required reading for mathematicians, from beginners to experts.

Basic Notions of Algebra

Basic Algebraic Geometry 2: Schemes and Complex Manifolds