Some 15 years ago M. Kontsevich and A. Rosenberg [KR] proposed a heuristicprinciple according to which the family of schemes ${Rep_n(A)}$ parametrizingthe finite-dimensional represen- tations of a noncommutative algebra A shouldbe thought of as a substitute or "approximation" for Spec(A). The idea is thatevery property or noncommutative geometric structure on A should induce acorresponding geometric property or structure on $Rep_n(A)$ for all n. Inrecent years, many interesting structures in noncommutative geometry haveoriginated from this idea. In practice, however, if an associative algebra Apossesses a property of geometric nature (e.g., A is a NC completeintersection, Cohen-Macaulay, Calabi-Yau, etc.), it often happens that, forsome n, the scheme $Rep_n(A)$ fails to have the corresponding property in theusual algebro-geometric sense. The reason for this seems to be that therepresentation functor $Rep_n$ is not "exact" and should be replaced by itsderived functor $DRep_n$ (in the sense of non-abelian homological algebra). Thehigher homology of $DRep_n(A)$, which we call representation homology,obstructs $Rep_n(A)$ from having the desired property and thus measures thefailure of the Kontsevich-Rosenberg "approximation." In this paper, which ismostly a survey, we prove several results confirming this intuition. We alsogive a number of examples and explicit computations illustrating the theorydeveloped in [BKR] and [BR].
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