Injective resolutions of BG and derived moduli spaces of local systems

It was suggested on several occasions by Deligne, Drinfeld and Kontsevich t hat all the moduli spaces arising in the classical problems of deformation theory should be extended to natural "derived" moduli spaces which are alwa ys smooth in an appropriate sense and whose tangent spaces involve the enti re cohomology of the sheaf of infinitesimal automorphisms, not just H-I. In this note we give an algebraic construction of such an extension for the s implest class of moduli spaces, namely for moduli of local systems (represe ntations of the fundamental group). (C) 2001 Elsevier Science B.V. All righ ts reserved. MSG: Primary 18G50; secondary 14D22.