Quillen's vanishing André–Quillen homology conjecture for local rings

Let RR be a Noetherian commutative local ring with residue field F\Bbb{F}, and write

Ds(FR)=Ds(FR;F),s0.D_s(\Bbb{F}|R)=D_s(\Bbb{F}|R;\Bbb{F}),\qquad s\geq 0.

Quillen's conjecture. The following are equivalent: Ds(FR)=0D_s(\Bbb{F}|R)=0 for s0s\gg 0; Ds(FR)=0D_s(\Bbb{F}|R)=0 for s3s\geq 3; and RR is a complete intersection. This conjecture characterizes local complete intersections by the vanishing of André–Quillen homology and was proven by Avramov, including in greater generality.

Sources & referencesView supporting material

Primary source

James M. Turner, “On Simplicial Commutative Rings with Vanishing André-Quillen Homology”, arXiv:alg-geom/9707018 (1997).

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