An Introduction to Hochschild Cohomology 1st Edition by Sarah Witherspoon ISBN 9781470449315 1470449315

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ISBN 10: 1470449315
ISBN 13: 9781470449315
Author: Sarah Witherspoon

This book gives a thorough and self-contained introduction to the theory of Hochschild cohomology for algebras and includes many examples and exercises. The book then explores Hochschild cohomology as a Gerstenhaber algebra in detail, the notions of smoothness and duality, algebraic deformation theory, infinity structures, support varieties, and connections to Hopf algebra cohomology. Useful homological algebra background is provided in an appendix. The book is designed both as an introduction for advanced graduate students and as a resource for mathematicians who use Hochschild cohomology in their work.no cover page[Abstract] This is an advanced graduate level textbook, designed both as an introduction for students to the subject of Hochschild cohomology, and as a resource for mathematicians who use Hochschild cohomology in their work. The text begins with definitions, properties, and many examples. The structure of Hochschild cohomology as a Gerstenhaber algebra is explored in detail. Many other topics of current interest are presented, including smoothness and duality, algebraic deformation theory, infinity structures, support varieties, and connections to Hopf algebra cohomology. Also included is an appendix containing some needed homological background

An Introduction to Hochschild Cohomology 1st Edition Table of contents:

Chapter 1. Historical Definitions and Basic Properties

1.1. Definitions of Hochschild homology and cohomology

1.2. Interpretation in low degrees

1.3. Cup product

1.4. Gerstenhaber bracket

1.5. Cap product and shuffle product

1.6. Harrison cohomology and Hodge decomposition

Chapter 2. Cup Product and Actions

2.1. From cocycles to chain maps

2.2. Yoneda product

2.3. Tensor product of complexes

2.4. Yoneda composition and tensor product of extensions

2.5. Actions of Hochschild cohomology

Chapter 3. Examples

3.1. Tensor product of algebras

3.2. Twisted tensor product of algebras

3.3. Koszul complexes and the HKR Theorem

3.4. Koszul algebras

3.5. Skew group algebras

3.6. Path algebras and monomial algebras

Chapter 4. Smooth Algebras and Van den Bergh Duality

4.1. Dimension and smoothness

4.2. Noncommutative differential forms

4.3. Van den Bergh duality and Calabi-Yau algebras

4.4. Skew group algebras

4.5. Connes differential and Batalin-Vilkovisky structure

Chapter 5. Algebraic Deformation Theory

5.1. Formal deformations

5.2. Infinitesimal deformations and rigidity

5.3. Maurer-Cartan equation and Poisson bracket

5.4. Graded deformations

5.5. Braverman-Gaitsgory theory and the PBW Theorem

Chapter 6. Gerstenhaber Bracket

6.1. Coderivations

6.2. Derivation operators

6.3. Homotopy liftings

6.4. Differential graded coalgebras

6.5. Extensions

Chapter 7. Infinity Algebras

7.1. 𝐴_{∞}-algebras

7.2. Minimal models

7.3. Formality and Koszul algebras

7.4. 𝐴_{∞}-center

7.5. 𝐿_{∞}-algebras

7.6. Formality and algebraic deformations

Chapter 8. Support Varieties for Finite-Dimensional Algebras

8.1. Affine varieties

8.2. Finiteness properties

8.3. Support varieties

8.4. Self-injective algebras and realization

8.5. Self-injective algebras and indecomposable modules

Chapter 9. Hopf Algebras

9.1. Hopf algebras and actions on rings

9.2. Modules for Hopf algebras

9.3. Hopf algebra cohomology and actions

9.4. Bimodules and Hochschild cohomology

9.5. Finite group algebras

9.6. Spectral sequences for Hopf algebras

Appendix A. Homological Algebra Background

A.1. Complexes

A.2. Resolutions and dimensions

A.3. Ext and Tor

A.4. Long exact sequences

A.5. Double complexes

A.6. Categories, functors, derived functors

A.7. Spectral sequences

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Tags: Sarah Witherspoon, Hochschild Cohomology, Introduction