Introduction to Abelian Model Structures and Gorenstein Homological Dimensions

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 precoversn-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 structuresThe 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

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.