Flat descent for quasi-projective dimension

Establish whether finiteness of quasi-projective dimension descends along flat local homomorphisms, including the identity homomorphism from a commutative Noetherian local ring to itself.

Background

The paper proves that every finitely generated module of finite complete intersection dimension has finite quasi-projective dimension over an arbitrary commutative Noetherian ring. The motivating construction initially produces a quasi-projective resolution only after passing to the flat extension in a quasi-deformation.

The authors note that the general descent implication from a flat local extension remains unresolved, even in the apparently simplest case of the identity map. Resolving this would clarify whether the quasi-projective dimension obtained after a quasi-deformation can be transferred back to the original ring by flat descent.