On simplicial commutative algebras with Noetherian homotopy

In this paper, we introduce a strategy for studying simplicial commutative algebras over general commutative rings R. Given such a simplicial algebra A, this strategy involves replacing A with a connected simplicial commutative k(Weierstrass p)-algebra A(Weierstrass p), for each Weierstrass pset membership, variant Spec(π0A), which we call the connected component of A at Weierstrass p. These components retain most of the André–Quillen homology of A when the coefficients are k(Weierstrass p)-modules (k(Weierstrass p)=residue field of Weierstrass p in π0A). Thus, these components should carry quite a bit of the homotopy theoretic information for A. Our aim will be to apply this strategy to those simplicial algebras which possess Noetherian homotopy. This allows us to have sophisticated techniques from commutative algebra at our disposal. One consequence of our efforts will be to resolve a more general form of a conjecture of Quillen that was posed in Invent. Math. 142 (3) (2000) 547.