A Simple Construction of Derived Representation Schemes

TL;DR

This paper introduces a straightforward algebraic method to construct derived representation schemes for associative algebras, connecting with recent work and computing derived tangent spaces, advancing the understanding of non-abelian derived functors.

Contribution

It provides a new, simpler construction of derived representation schemes and relates it to existing frameworks, with a focus on algebraic and geometric properties.

Findings

Constructed a simple algebraic model for derived representation schemes.

Computed derived tangent spaces of representation schemes.

Established equivalence with previous constructions of derived action spaces.

Abstract

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.