Derived Representation Schemes And Non-Commutative Geometry

Abstract

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 theorem allowing one to finitely present derived representation schemes of an associative algebra whenever one has an explicit finite presentation for an almost free resolution of that algebra; using this theorem, we calculate several examples (including some computer calculations of homology).