Homological Algebra

Homological algebra is a branch of mathematics that studies algebraic structures using techniques from topology and geometry. It focuses on understanding the properties of objects like groups, rings, and modules through sequences of these structures, known as complexes. By examining how these objects relate to one another and how they change under various operations, mathematicians can gain insights into their underlying relationships and characteristics. Key concepts include exact sequences, which help identify when certain algebraic processes yield meaningful information, and derived functors, which quantify how well certain operations behave.