Introduction to Abelian model structures and Gorenstein homological dimensions

Content: 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 -â  - 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