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-Macau lay, 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 DRepn (in the sense of non-abelian homological algebra). The higher homology of DRepn(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.
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