Algebra II: Textbook for Students of Mathematics

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Springer, Feb 12, 2017 - Mathematics - 370 pages

This book is the second volume of an intensive “Russian-style” two-year undergraduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them.

The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses.

This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study.


 

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Contents

1 Tensor Products
1
2 Tensor Algebras
20
3 Symmetric Functions
57
4 Calculus of Arrays Tableaux and Diagrams
75
5 Basic Notions of Representation Theory
99
6 Representations of Finite Groups in Greater Detail
130
7 Representations of Symmetric Groups
151
8 sl2Modules
173
10 Extensions of Commutative Rings
227
11 Affine Algebraic Geometry
241
12 Algebraic Manifolds
265
13 Algebraic Field Extensions
295
14 Examples of Galois Groups
315
Hints to Some Exercises
335
References
355
Index
356

9 Categories and Functors
187

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About the author (2017)

A.L. Gorodentsev is professor at the Independent University of Moscow and at the Faculty of Mathematics at the National Research University „Higher School of Economics“.

He is working in the field of algebraic and symplectic geometry, homological algebra and representation theory connected with geometry of algebraic and symplectic varieties.

He is one of the first developers of the “Helix Theory” and semiorthogonal decomposition technique for studying the derived categories of coherent sheaves.

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