Introduction to Abelian Model Structures and Gorenstein Homological Dimensions: 1st Edition (Hardback) book cover

Introduction to Abelian Model Structures and Gorenstein Homological Dimensions

1st Edition

By Marco A. P. Bullones

Chapman and Hall/CRC

344 pages | 36 B/W Illus.

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pub: 2016-08-17
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Description

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.

Reviews

"The main goal of this book is to provide a multitude of model category structures for categories such as categories of chain complexes, categories of modules, and more specifically, Gorenstein categories (a Grothendieck category with extra properties). The author provides these model category structures by making use of the Hovey Correspondence, which allows one to associate a model category structure to a complete cotorsion pair in an abelian category […]

This text is based on the thesis of the author and the majority of original results presented here are related to the model category structures coming from homological dimensions. The intended audience includes graduate students pursuing a degree in the field and researchers interested in the development of model category structures associated to Gorenstein homological dimensions. It is well written and is well suited for the target audience."

- Bruce R. Corrigan-Salter, Mathematical Reviews, August 2017

Table of Contents

Categorical and algebraic preliminaries

Universal constructions

Introduction

The opposite category and duality

Limits and colimits

Abelian categories

Introduction

Additive categories

Abelian categories and exact sequences

Grothendieck categories

Extension functors

Introduction

Projective and injective objects

Projective and injective dimensions

Extension groups via cohomology

Extension groups via Baer description

Applications of Baer extensions: adjunction properties for sphere and disk chain complexes

Torsion functors

Introduction

Monoidal categories

Closed monoidal categories on chain complexes over a ring

Derived functors of -⊗- and Hom(–,–)

Flat chain complexes

Torsion functors and flat dimensions

Interactions between homological algebra and homotopy theory

Model categories

Introduction

Weak factorization systems

Model categories

The homotopy category of a model category

Monoidal model categories

Cotorsion pairs

Introduction

Complete and hereditary cotorsion pairs

Eklof and Trlifaj theorem

Compatible cotorsion pairs

Induced cotorsion pairs of chain complexes

Hovey Correspondence

Introduction

Hovey Correspondence

Abelian factorization systems

Proof of the Hovey Correspondence

Abelian model structures on monoidal categories

Further reading

Classical homological dimensions and abelian model structures on chain complexes

Injective dimensions and model structures

Introduction

n-Injective model structures on chain complexes

Degreewise n-injective model structures on chain complexes

Projective dimensions and model structures

Introduction

Projective dimensions and cotorsion pairs of R-modules

The category Mod(R) of modules over a ringoid

Projective dimensions of modules over ringoids and special precovers

n-projective model structures

Degreewise n-projective model structures

Flat dimensions and model structures

Introduction

The n-flat modules and cotorsion pairs

The n-flat cotorsion pair of chain complexes and model structures

The homotopy category of differential graded model structures

Degreewise n-flat model structures

Further reading

Gorenstein homological dimensions and abelian model structures

Gorenstein-projective and Gorenstein-injective objects

Introduction

Properties of Gorenstein-projective and Gorenstein-injective objects

Gorenstein-projective and Gorenstein-injective dimensions

Gorenstein-injective dimensions and model structures

Introduction

Gorenstein categories

Cotorsion pairs from Gorenstein homological dimensions

Hovey’s model structures on Gorenstein categories

The Gorenstein n-injective model structure

Homotopy categories in Gorenstein homological algebra

Gorenstein-homological dimensions of chain complexes

Further reading

Gorenstein-projective dimensions and model structures

Introduction

G-projective dimensions and cotorsion pairs

Model structures from Gorenstein-projective dimensions

Gorenstein-flat dimensions and model structures

Introduction

Gorenstein-flat modules

Gorenstein-flat dimensions

Gorenstein-flat dimensions of chain complexes

Gorenstein-homological dimensions of graded modules

Further reading

Bibliography

Index

About the Author

Dr. Marco A. Pérez is a postdoctoral fellow at the Mathematics Institute of the Universidad Nacional Autónoma de México, where he works on Auslander–Buchweitz approximation theory and cotorsion pairs. He was previously a postdoctoral associate at the Massachusetts Institute of Technology, working on category theory applied to communications and linguistics. Dr. Pérez’s research interests cover topics in both category theory and homological algebra, such as model category theory, ontologies, homological dimensions, Gorenstein homological algebra, finitely presented modules, modules over rings with many objects, and cotorsion theories. He received his PhD in mathematics from the Université du Québec à Montréal in the spring of 2014.

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT012000
MATHEMATICS / Geometry / General
MAT022000
MATHEMATICS / Number Theory