  Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristic
principle according to which the family of schemes ${Rep_n(A)}$ parametrizing
the finite-dimensional represen- tations 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
developed in [BKR] and [BR].
