Rolf Farnsteiner

Homology of Associative Algebras

Homological algebra originated in algebraic topology, where certain groups, the so-called homology groups, provide information on the homotopy type of topological spaces. Various problems occurring within algebraic topology motivated the parallel development of the homology theory of groups (Eilenberg-MacLane, 1943), associative algebras (Hochschild, 1945), and Lie algebras (Chevalley-Eilenberg, 1948). 

The initial flavor of the subject matter is reflected in Hochschild's introductory remarks to one of his early papers (On the cohomology theory for associative algebras, Ann. of Math. 47 (1946), 568-579): "The cohomology theory for associative algebras may be regarded as a superstructure built upon the classical representation theory....There is good evidence that it will provide the appropriate methods for investigating a variety of structural characteristics which seem to lie outside the domain of the classical representation theory."

In the early 1950's the aforementioned approaches were unified by Cartan and Eilenberg into the theory of derived functors. Since then homological algebra has become an independent algebraic discipline that continues to find new applications. Aside from providing a unifying concept for classical work by Maschke and Schur on group representations, by Weyl on representations of Lie algebras, and Wedderburn-Malcev on the structure of associative algebras, homological algebra has also been applied in commutative algebra (for instance, Serre's homological characterization of regularity) and modern representation theory (representation type, Auslander-Reiten quivers, module varieties).

 

Publications

  1.  On the cohomology of associative algebras and Lie algebras.
        Proc. Amer. Math. Soc. 99 (1987), 414-420

   2.  On the vanishing of homology  and cohomology groups of associative algebras.
        Trans. Amer. Math. Soc. 306 (1988), 651-665

   3.  Cohomology groups of infinite dimensional algebras.
        Math. Z. 199 (1988), 407-423

   4.  On the cohomology of ring extensions.
        Adv. in Math. 87 (1991), 42-70