Rigidity properties of the cotangent complex

2020/10/26 by Briggs, Benjamin, Iyengar, Srikanth B. · 2 citations
#13B10 #13D03 (primary) #14A15 #14A30 (secondary) #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2010.13314

Abstract

This work concerns maps φ\colon R→ S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D_n(S/R;-)=0 for some n≥ 2, then D_n(S/R;-)=0 for all n≥ 2 and φ is locally complete intersection. This extends Avramov's theorem that draws the same conclusion assuming D_n(S/R;-) vanishes for all n≫ 0, confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of φ.

Cited by

• A characteristic p analog of formal lifting properties
• Adequate complete intersection homomorphisms

Related

• Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology
• Homological properties of the module of differentials
• A rigidity property of local cohomology modules
• Rigidity of Ext and Tor via flat-cotorsion theory
• On Simplicial Commutative Algebras with Finite Andre-Quillen Homology