Relative homological algebra and exact model structures

Abstract

This thesis is concerned with relative homological algebra and exact model categories. There are two examples of such relative homological theories which are of interest in this thesis. One example comes from commutative algebra in the study of maximal Cohen--Macaulay approximations and its generalizations in Gorenstein homological algebra. Another example comes from the theory of purity in locally finitely presented additive categories. Finally, in the last part of the thesis, induced abelian model categories in categories of quiver representations are considered, with special emphasis in the case of model structures associated to the context of Ding projective and Ding injective representations. The thesis consists of an expository text, which consists of an introduction and three chapters, and three papers (two of them are already published and the third one is submitted), A. Quillen equivalences for stable categories (joint with S.Estrada and H.Holm). Journal of Algebra 501, 130-149 (2018) B. A note on homotopy categories of FP-Injectives}. Homology, Homotopy and Applications 21(1) 95-105, (2019) C. Abelian model structures on categories of quiver representations.