Quillen’s conjecture: vanishing André–Quillen homology implies quasi-complete intersection

Prove that any surjective local ring homomorphism φ: Q → R for which the André–Quillen homology groups D_i(R/Q; −) vanish for all sufficiently large degrees i is necessarily a quasi-complete intersection homomorphism (that is, the homology of the Koszul complex on a minimal generating set of Ker φ is isomorphic, as a graded R-algebra, to the exterior algebra on H_1, with H_1 free over R).

Background

The paper studies Poincaré series under quasi-complete intersection (q.c.i.) homomorphisms φ: Q → R. A homomorphism is q.c.i. when the homology of the Koszul complex E on a minimal generating set of Ker φ is isomorphic to the exterior algebra over R on H_1(E), and H_1(E) is free over R. Equivalently, q.c.i. maps admit a two-step Tate resolution.

The authors note a classical connection to André–Quillen homology: q.c.i. maps can be characterized by vanishing of D_i(R/Q; −) for i ≥ 3. They further recall Quillen’s long-standing conjecture asserting that vanishing of D_i(R/Q; −) in sufficiently high degrees forces φ to be q.c.i. This conjecture is cited as context for the class of maps studied in the paper.