Some 15 years ago M. Kontsevich and A. Rosenberg proposed a heuristic principle
according to which the family of schemes {Repn (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 Repn (A)
for all n. In recent years, many interesting structures in noncommutative geometry have …

After surveying relevant literature (on representation schemes, homotopical algebra, and
non-commutative algebraic geometry), we provide a simple algebraic construction of relative
derived representation schemes and prove that it constitutes a derived functor in the sense
of Quillen. Using this construction, we introduce a derived Kontsevich-Rosenberg principle.
In particular, we construct a (non-abelian) derived functor of a functor introduced by Van den
Bergh that offers one (particularly significant) realization of the principle. We also prove a …