Quillen's finiteness conjecture in positive residual characteristic

Let R → S be a map of Noetherian rings such that m · S = 0 for some integer m ≥ 2. We prove that if the flat dimension of the cotangent complex L_{S/R} over S is finite, then it is at most 2. In particular, this settles a conjecture of Quillen in nonzero characteristics. If R → S is local and essentially of finite type, it suffices to assume that S has positive residue characteristic. The proof is based on a new vanishing criterion for the André–Quillen homology of 1-connected simplicial commutative algebras in arbitrary positive characteristic, combined with nilpotence of homotopy operations under a boundedness hypothesis on module fibers.