  This paper is a sequel to [BKR], where we studied the derived affine scheme
DRep_n(A) of the classical representation scheme Rep_n(A) for an associative
k-algebra A. In [BKR], we have constructed canonical trace maps Tr_n(A): HC(A)
-> H[DRep_n(A)]^GL extending the usual characters of representations to higher
cyclic homology. This raises a question whether a well known theorem of Procesi
[P] holds in the derived setting: namely, is the algebra homomorphism
Sym[Tr_n(A)]: Sym[HC(A)] -> H[DRep_n(A)]^GL defined by Tr_n(A) surjective ? In
the present paper, we answer this question for augmented algebras. Given such
an algebra, we construct a canonical dense DG subalgebra DRep_\infty(A)^Tr of
the topological DG algebra DRep_\infty(A)^{GL_\infty}. It turns out that on
passing to the inverse limit (as n -> \infty), the family of maps Sym[Tr_n(A)]
"stabilizes" to an isomorphism Sym[\bar{HC}(A)] = H[DRep_\infty(A)^Tr]. The
derived version of Procesi's theorem does therefore hold in the limit. However,
for a fixed (finite) n, there exist homological obstructions to the
surjectivity of Sym[Tr_n(A)], and we show on simple examples that these
obstructions do not vanish in general. We compare our result with the classical
theorem of Loday-Quillen and Tsygan on stable homology of matrix Lie algebras.
We show that the relative Chevalley-Eilenberg complex C(gl_\infty(A),
gl_\infty(k); k) equipped with the natural coalgebra structure is Koszul dual
to the DG algebra DRep_\infty(A)^Tr. We also extend our main results to
bigraded DG algebras, in which case we show that DRep_{\infty}(A)^Tr =
DRep_{\infty}(A)^GL_{\infty}. As an application, we compute the (bigraded)
Euler characteristics of DRep_\infty(A)^GL_{\infty} and \bar{HC}(A) and derive
some interesting combinatorial identities.
