Homological algebra is a branch of mathematics that studies homology in a general algebraic setting, providing a framework for analyzing and solving problems in algebraic topology, algebraic geometry, and beyond. It uses chain complexes and exact sequences to explore the relationships between different algebraic structures, revealing deep insights into their properties and interactions.
The Tor functor is a fundamental tool in homological algebra, used to measure the failure of a functor to be exact. It plays a crucial role in understanding the structure of modules over a ring by providing a way to compute the torsion and extension groups between them.