Homological Algebra and Its Application: A Descriptive Study

Abstract

Algebra has been used to define and answer issues in almost every field of mathematics, science, and engineering. Homological algebra depends largely on computable algebraic invariants to categorise diverse mathematical structures, such as topological, geometrical, arithmetical, and algebraic (up to certain equivalences). String theory and quantum theory, in particular, have shown it to be of crucial importance in addressing difficult physics questions. Geometric, topological and algebraic algebraic techniques to the study of homology are to be introduced in this research. Homology theory in abelian categories and a category theory are covered. the n-fold extension functors EXTn (-,-) , the torsion functors TORn (-,-), Algebraic geometry, derived functor theory, simplicial and singular homology theory, group co-homology theory, the sheaf theory, the sheaf co-homology, and the l-adic co-homology, as well as a demonstration of its applicability in representation theory.