Algebra II, 1st ed. 2017
Textbook for Students of Mathematics

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.

§1Tensor Products.- §2 Tensor Algebras.- §3 Symmetric Functions.- §4 Calculus of Arrays, Tableaux, and Diagrams.- §5 Basic Notions of Representation Theory.- §6 Representations of Finite Groups in Greater Detail.- §7 Representations of Symmetric Groups.- §8 sl_2-Modules.- §9 Categories and Functors.- §10 Extensions of Commutative Rings.- §11 Affine Algebraic Geometry.- §12 Algebraic Manifolds.- §13 Algebraic Field Extensions.- §14 Examples of Galois Groups.- References.- Hints to Some Exercises.- Index.

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.

Challenging amount of material thoughtfully organized for deep and fast learning

Large collection of exercises equipped with hints and a lot of problems for independent solution

Simple modern explanation of subjects usually omitted in basic courses, such as representations of the symmetric group, geometry of algebraic varieties, meaty aspects of the category theory etc

Includes supplementary material: sn.pub/extras