Hochschild Cohomology Of Finite-dimensional Algebras

The Hochschild cohomology groups of finite-dimensional algebra is intro-duced by the Hochschild 1945 and developed by Cartan Eilenberg. It plays an important role in a number of branches of mathematics such as ring theory, the representation theory of algebras, algebraic topology and algebraic geom-etry, etc. The calculation on various Hochschild cohomology algebra proves significance in algebra as well as its representation theory. In recent years, remarkable results have been achieved in Hochschild cohomology of a few im-portant algebras. In view of this, the thesis describes the most basic sort of Hochschild cohomology theory in a detailed way and aims at reviewing the development concerned in recent years.For a start, the article summarizes the background and development of Hochschild cohomology theory. What follows is an introduction of the rele-vant concepts on quiver, complex and Hochschild cohomology. The third part introduces minimal projective resolution constructed by Happel and discusses the minimal projective resolution of some algebras in the form of theorems. Finally there is the introduction of Hochschild cohomology of some important algebras along with some important results achieved in recent years, during which process, the proof of some theorems is specified and a few concrete examples of the calculation of Hochschild cohomology are presented.