## Homoogical Algebra

#### A brief history of the subject:

Homological algebra, as we understand it today, is the study of chain complexes in an abelian category .The subject has its roots in topology and algebra specifically in the Riemann-Poincaré definition of the homology of a space and in Hilbert`s theorem of Syzygies. Before 1950, mathematicians had already difined homology and cohomology theories for various kinds of geometric and algebraic structures, without any clear link among them. The discovery of derived functors by Cartan and Eilenberg around 1951 and their subsequent book on the subject showed that all of these homology theories are instances of derived functors and can be studied with similar methods. This was the birth of homological algebra . it was soon realized that the right set up for the study of derived functors is the derived category of an abelian category (Grothendieck-Verdier) . This led to the notion of triangulated category . The next major developments include homotopical algebra and the theory of nonabelian derived functors(Quillen, Dold-Pupe) , K-theory, Cyclic cohomology (Connes).

There is hardly any field of modern mathematics that is not touched, influenced, or helped by homological algebra. While it can be studied perse, homological algebra is mostly studied because it is such a useful tool in other areas of mathematics. That is the point of view we adopt in this course. I will cover all the necessary background material as they may be needed.

#### Course Outline:

• Chain complexes
• Derived functors
• Tor and Ext
• Group homology and cohomology
• Lie algebra homology and cohomology
• Homological algebra in analysis and physics (BRST complex)
• Hochschild and cyclic homology
• The derived category, triangulated categories

Marking Policy:  Based on assignments and small research projects.

#### Textbook: An Introduction to Homological Algebra, by C. Weibel, published by Cambridge University Press.