An Introduction to Homological Algebra

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Author: Charles A. Weibel

Publisher: Cambridge University Press

ISBN: 9780521559874

Category: Mathematics

Page: 450

View: 3042

A portrait of the subject of homological algebra as it exists today.

An Introduction to Homological Algebra

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Author: Joseph J. Rotman

Publisher: Springer Science & Business Media

ISBN: 0387683240

Category: Mathematics

Page: 710

View: 9615

Graduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman’s book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology. In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author has also included material about homotopical algebra, alias K-theory. Learning homological algebra is a two-stage affair. First, one must learn the language of Ext and Tor. Second, one must be able to compute these things with spectral sequences. Here is a work that combines the two.

Introduction to Abelian Model Structures and Gorenstein Homological Dimensions

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Author: Marco A. P. Bullones

Publisher: CRC Press

ISBN: 149872535X

Category: Mathematics

Page: 344

View: 489

Introduction to Abelian Model Structures and Gorenstein Homological Dimensions provides a starting point to study the relationship between homological and homotopical algebra, a very active branch of mathematics. The book shows how to obtain new model structures in homological algebra by constructing a pair of compatible complete cotorsion pairs related to a specific homological dimension and then applying the Hovey Correspondence to generate an abelian model structure. The first part of the book introduces the definitions and notations of the universal constructions most often used in category theory. The next part presents a proof of the Eklof and Trlifaj theorem in Grothedieck categories and covers M. Hovey’s work that connects the theories of cotorsion pairs and model categories. The final two parts study the relationship between model structures and classical and Gorenstein homological dimensions and explore special types of Grothendieck categories known as Gorenstein categories. As self-contained as possible, this book presents new results in relative homological algebra and model category theory. The author also re-proves some established results using different arguments or from a pedagogical point of view. In addition, he proves folklore results that are difficult to locate in the literature.

Leavitt Path Algebras

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Author: Gene Abrams,Pere Ara,Mercedes Siles Molina

Publisher: Springer

ISBN: 1447173449

Category: Mathematics

Page: 289

View: 3882

This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and symbolic dynamics. With its descriptive writing style, this book is highly accessible.

Noncommutative Algebraic Geometry

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Author: Gwyn Bellamy,Daniel Rogalski,Travis Schedler,J. Toby Stafford,Michael Wemyss

Publisher: Cambridge University Press

ISBN: 1107129540

Category: Mathematics

Page: N.A

View: 2266

This book provides a comprehensive introduction to the interactions between noncommutative algebra and classical algebraic geometry.

A Course in Homological Algebra

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Author: Peter J. Hilton,Urs Stammbach

Publisher: Springer Science & Business Media

ISBN: 1441985662

Category: Mathematics

Page: 366

View: 7120

Homological algebra has found a large number of applications in many fields ranging from finite and infinite group theory to representation theory, number theory, algebraic topology and sheaf theory. In the new edition of this broad introduction to the field, the authors address a number of select topics and describe their applications, illustrating the range and depth of their developments. A comprehensive set of exercises is included.

Minimal Resolutions Via Algebraic Discrete Morse Theory

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Author: Michael Jöllenbeck,Volkmar Welker

Publisher: American Mathematical Soc.

ISBN: 0821842579

Category: Mathematics

Page: 74

View: 8525

Forman's discrete Morse theory is studied from an algebraic viewpoint. Analogous to independent work of Emil Skoldberg, the authors show that this theory can be aplied to chain complexes of free modules over a ring and provide four applications of this theory.

An Algebraic Introduction to Complex Projective Geometry

Commutative Algebra

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Author: Christian Peskine,Peskine Christian

Publisher: Cambridge University Press

ISBN: 9780521480727

Category: Mathematics

Page: 244

View: 2542

In this introduction to commutative algebra, the author choses a route that leads the reader through the essential ideas, without getting embroiled in technicalities. He takes the reader quickly to the fundamentals of complex projective geometry, requiring only a basic knowledge of linear and multilinear algebra and some elementary group theory. The author divides the book into three parts. In the first, he develops the general theory of noetherian rings and modules. He includes a certain amount of homological algebra, and he emphasizes rings and modules of fractions as preparation for working with sheaves. In the second part, he discusses polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalization lemma and Hilbert's Nullstellensatz, the author introduces affine complex schemes and their morphisms; he then proves Zariski's main theorem and Chevalley's semi-continuity theorem. Finally, the author's detailed study of Weil and Cartier divisors provides a solid background for modern intersection theory. This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.

Lectures on Algebraic Geometry I

Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces

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Author: Günter Harder

Publisher: Springer Science & Business Media

ISBN: 9783834895011

Category: Mathematics

Page: 300

View: 6443

This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own. In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them.

Introduction to Noncommutative Algebra

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Author: Matej Brešar

Publisher: Springer

ISBN: 3319086936

Category: Mathematics

Page: 199

View: 4770

Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.