A Simple Construction of Derived Representation Schemes

We present a simple algebraic construction of the (non-abelian) derived functors DRep_n(A) of the representation scheme Rep_n(A), parametrizing the n-dimensional representations of an associative algebra A. We construct a related derived version of the representation functor introduced recently by M. Van den Bergh and, as an application, compute the derived tangent spaces TDRep_n(A) to Rep_n(A). We prove that our construction of DRep_n(A) agrees with an earlier construction of derived action spaces, due to I. Ciocan-Fontanine and M. Kapranov; however, our approach, proofs and motivation are quite different. This paper is mainly a research announcement; detailed proofs and applications will appear elsewhere.

💡 Research Summary

The paper introduces a concise algebraic construction of the derived representation scheme DRepₙ(A) associated with an associative algebra A, which parametrises n‑dimensional representations of A. The authors start by recalling the classical representation scheme Repₙ(A) = Spec k