Set theory of derived functors

This thesis investigates how set theory interacts with the study of derived functors, algebraic tools used to measure how far a functor is from being exact. It focuses on the derived functors of the inverse limit and Ext, both expressible as quotients from certain cohomological complexes. Set theory contributes in two main areas: infinite combinatorics and descriptive set theory. In the combinatorial part, cocycles in derived limits of inverse systems of abelian groups correspond to “coherent” families of functions exhibiting local behavior, while coboundaries express that this behavior holds globally. Thus, derived functors capture a form of incompactness, the gap between local and global properties. The thesis establishes several consistency results about such incompactness for systems indexed by various ideals of sets. The methods include forcing, combinatorial principles of the Constructible Universe (such as Diamond and Square), and properties of cardinal invariants of the continuum. The second line of work connects set theory to descriptive set theory and the theory of Polish modules and modules with a Polish cover. The thesis studies injective and projective objects in the category of pro-Lie Polish abelian groups and analyzes the Borel complexity of the trivial submodule of Ext(C,A), viewed as a module with a Polish cover when A and C are countable flat modules over a countable Dedekind domain. This corresponds to the potential Borel complexity of isomorphism among short exact sequences with fixed end terms. Using a hierarchy of subfunctors indexed by countable ordinals, the thesis defines higher analogues of injective and projective modules and shows that a countable flat module over a countable Dedekind domain that is not injective remains non-injective for all higher analogues.