Abstract: The goal of the dissertation is to formulate Levine's Coniveau Tower in the setting of Derived Algebraic Geometry and in a self-contained way. The first section includes prerequisites: after a brief introduction to Animated Algebraic Geometry to ease referencing, a presentation of foundations of Motivic Homotopy Theory from an ``animated" standpoint follows. The latter is an elaboration on lectures by Hoyois and has the goal to motivate notions, terminology, and axioms that will appear in the main body. Moreover, a - yet unsatisfactory - digression on (co)dimension theory in animated algebraic geometry is included. In the second section, the main part is developed. Levine's machinery is reformulated in the language of infinity categories to achieve a formal construction for the Coniveau Tower provided a - yet conjectural - ``nice enough" notion of (co)dimension. Finally, the dissertation is complemented by an appendix on spectra, t-structures, and spectral sequences. We purposely omit Levine's versions of Chow's Moving Lemma, as their generalization to the derived setting is yet unclear; for instance, a suitable derived analogous of generic points has not been discovered yet.

Abstract: After a quick review of classical blow-ups and regular closed immersions, the first part of the thesis aims at providing a self-contained introduction to both Animated Higher Algebra and Animated Algebraic Geometry by merging the approach of Lurie's HA, SAG with that illustrated in the introduction to Khan's PhD thesis. At every stage, a model-independent formulation is attempted. Thereafter, a review of the recent work by Khan-Rydh on derived blow-ups of quasi-smooth closed subschemes follows. Apart from familiarity with general infinity-category theory and standard algebraic geometry, no other prerequisite is assumed. Thus, the thesis includes appendices reviewing free sifted completion of infinity-categories, symmetric monoidal infinity-categories, and the theory of infinity-topoi.

Abstract: The goal is to review recent developments on applications of methods from Set-theoretical Homological Algebra to Representation Theory of modules. Starting from the study of (Flat) Mittag-Leffler modules, we investigated deconstructibility issues via left and right approximations of modules. This led us to review the recent contributions by Saroch and Angeleri-Hügel-Saroch-Trlifaj to the solution of the Countable Telescope Conjecture for Module Categories, as well as its applications to the Enochs’ Conjecture. Finally, we considered both Saroch's advancements in the introduction to his Habilitation Thesis and the parallel and very promising contramodule approach, applied by Positselski and Trlifaj to investigate the properties of small precovering classes of modules. The final outcome of the project is an attempt to present the aforementioned results in a self-contained unifying manner.

Abstract: Rewriting some notes of Jan Trlifaj, we investigated the theory of approximation of modules via envelopes and covers and by means of set-theoretical constructions in homological algebra. In this framework, the FCC states that each module admits an `optimal left approximation' via a flat cover, thus leading to the existence of minimal flat resolutions of modules.