Derived Representation Schemes and Noncommutative Geometry

Source abstract

Some 15 years ago M. Kontsevich and A. Rosenberg proposed a heuristic principle according to which the family of schemes { R e p n ( A ) } \{\mathrm {Rep}_n(A)\} parametrizing the finite-dimensional representations of a noncommutative algebra A A should be thought of as a substitute or ‘approximation’ for ‘ S p e c ( A ) \mathrm {Spec}(A) ’. The idea is that every property or noncommutative geometric structure on A A should induce a corresponding geometric property or structure on R e p n ( A ) \mathrm {Rep}_n(A) for all n n . In recent years, many interesting structures in noncommutative geometry have originated from this idea. In practice, however, if an associative algebra A A possesses a property of geometric nature (e.g., A A is a NC complete intersection, Cohen-Macaulay, Calabi-Yau, etc.), it often happens that, for some n n , the scheme R e p n ( A ) \mathrm {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 R e p n \mathrm {Rep}_n is not ‘exact’ and should be replaced by its derived functor DRep n \textrm {DRep}_n (in the sense of non-abelian homological algebra). The higher homology of DRep n ( A ) \textrm {DRep}_n(A) , which we call representation homology, obstructs R e p n ( A ) \mathrm {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.