Derived Representation Schemes and Noncommutative Geometry

Some 15 years ago M. Kontsevich and A. Rosenberg proposed a heuristic principle according to which the family of schemes {Rep(n)(A)} parametrizing the finite-dimensional representations of a noncommutative algebra A should be thought of as a substitute or 'approximation' for 'Spec(A)'. The idea is that every property or noncommutative geometric structure on A should induce a corresponding geometric property or structure on Rep(n)(A) for all n. In recent years, many interesting structures in noncommutative geometry have originated from this idea. In practice, however, if an associative algebra A possesses a property of geometric nature (e.g., A is a NC complete intersection, Cohen-Macaulay, Calabi-Yau, etc.), it often happens that, for some n, the scheme Rep(n)(A) fails to have the corresponding property in the usual algebro-geometric sense. The reason for this seems to be that the representation functor Rep(n) is not 'exact' and should be replaced by its derived functor DRep(n) (in the sense of non-abelian homological algebra). The higher homology of DRep(n)(A), which we call representation homology, obstructs Rep(n)(A) from having the desired property and thus measures the failure of the Kontsevich-Rosenberg 'approximation.' In this paper, which is mostly a survey, we prove several results confirming this intuition. We also give a number of examples and explicit computations illustrating the theory.