On the rigidity of the cotangent complex at the prime 2

Abstract

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 → A is a homomorphism of Noetherian commutative rings then the flat dimension, as defined in the derived category of A -modules, of the associated cotangent complex L A / R satisfies: fd A L A / R < ∞ ⟹ fd A L A / R ≤ 2 . The aim of this paper is to initiate an approach for solving this conjecture when R 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 γ 2 and the André operation ϑ as it acts on Tor R ( A , ℓ ) , ℓ any residue field of A . Second, we prove the conjecture is valid in two cases: when fd R A < ∞ and when R is a Cohen–Macaulay ring.