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Introduction to Abelian Model Structures and Gorenstein Homological Dimensions




ISBN 9781498725347
Published August 17, 2016 by Chapman and Hall/CRC
370 Pages 36 B/W Illustrations

 
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Book 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.

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

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Author(s)

Biography

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.

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