On Simplicial Commutative Algebras with Finite Andre-Quillen Homology

L. Avramov, following D. Quillen, posed a conjecture to the effect that if $R \to A$ is a homomorphism of Noetherian rings then the Andr'e-Quillen homology on the category of A-modules satisfies: $D_s(A\mathrm{\mid }R;-)=0$ for $s\gg 0$ implies $D_s(A\mathrm{\mid }R;-)=0$ for s>2. In an earlier paper, the author posed an extended version of this conjecture which considered A to be a simplicial commutative R-algebra with Noetherian homotopy such that the characteristic of ${\pi }_{0}A$ is non-zero. In addition, a homotopy characterization of such algebras was described. The main goal of this paper is to develop a strategy for establishing this extended conjecture and provide a complete proof when R is Cohen-Macaulay of characteristic 2.