On the rigidity of the cotangent complex at the prime 2

In [D. Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65–87; L. Avramov, Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Annals of Math. 2 (150) (1999) 455–487] a conjecture was posed to the effect that if R→AR→A is a homomorphism of Noetherian commutative rings then the flat dimension, as defined in the derived category of AA-modules, of the associated cotangent complex LA/RLA/R satisfies: fdALA/R<∞⟹fdALA/R≤2. The aim of this paper is to initiate an approach for solving this conjecture when RR has characteristic 2 using simplicial algebra techniques. To that end, we obtain two results. First, we prove that the conjecture can be reframed in terms of certain nilpotence properties for the divided square γ2γ2 and the André operation ϑϑ as it acts on TorR(A,ℓ)TorR(A,ℓ), ℓℓ any residue field of AA. Second, we prove the conjecture is valid in two cases: when fdRA<∞ and when RR is a Cohen–Macaulay ring.