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 LA / R satisfies: fdA LA / R < ∞ {long rightwards double arrow} fdA LA / 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 γ2 and the André operation θ{symbol} as it acts on TorR (A, ℓ), ℓ any residue field of A. Second, we prove the conjecture is valid in two cases: when fdR A < ∞ and when R is a Cohen-Macaulay ring. © 2008 Elsevier B.V. All rights reserved.