Topics in Homological Algebra

Abstract

Homological algebra is the study of homology in an algebraic setting. In this project we explore various standard tools of homological algebra such as Ext groups, Tor groups, Long exact sequence of cohomology etc, and the concepts required to realize those tools such as projective modules and resolutions, cochain complexes, etc. We also present a few topics from category theory initially to better understand these tools. We then focus on the specific case of group cohomology and related results which we apply to profinite groups. Finally we apply a few results from cohomology of profinite group to arrive at the Golod-Shafarevich inequality for finite p-groups.