Cohomological methods for determining numerical invariants of algebras and modules

This thesis is dedicated to the study of two seemingly unrelated problems. In the first half, we use the notion of cohomological degree to study the HomAB problem: the problem of finding a uniform bound on the number of generators of the module HomR(A, B) in terms of the numerical invariants of A and B.; In the second half of this thesis we introduce and study tracking numbers, invariants defined for graded modules over standard graded algebras. Several estimations are obtained which are used to bound the length of chains of algebras occurring in the construction of integral closure of a graded domain.; The original motivation for both of these problems is the study of the complexity of algorithms to compute integral closure of an algebra. Most algorithms achieve this by constructing an algebra between A and its integral closure. The answer to the HomAB problem will provide us with estimates on how many generators these algebras have. The study of tracking numbers will lead us to bounds on how many such algebras we have to construct.