1st Edition

Introduction to Abelian Model Structures and Gorenstein Homological Dimensions

By Marco A. P. Bullones Copyright 2016
    370 Pages 36 B/W Illustrations
    by Chapman & Hall

    370 Pages 36 B/W Illustrations
    by Chapman & Hall

    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.

    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

    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.

    "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