Quillen’s Conjecture II in arbitrary characteristic

Determine whether, for every homomorphism of noetherian rings A to B that is essentially of finite type, finite flat dimension of the relative cotangent complex L_{B/A} as a B-module implies fd_B L_{B/A} less than or equal to two, without assuming that B has finite flat dimension as an A-module.

Background

Quillen’s Conjecture II removes the hypothesis that B has finite flat dimension over A from Conjecture I and predicts the universal bound fd_B L_{B/A} less than or equal to two. The paper proves this assertion for homomorphisms of noetherian rings of characteristic two, but explicitly notes that the conjecture remains unresolved in general.

Known cases cited in the paper include complete-intersection base rings and general algebra retracts. The unresolved problem is therefore to establish the conjecture beyond these cases and beyond characteristic two.