bass_conjecture_projective

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Bass conjecture: If R is a Noetherian commutative ring of finite Krull dimension, every finitely generated projective module P over R[x₁,...,xₙ] is extended from R. Quillen-Suslin proved it for R = field; open for general Noetherian R.

Why this node was retired

The posted statement is

import Mathlib

theorem bass_conjecture_projective (R : Type*) [CommRing R] [IsNoetherianRing R]
    (hfin : ∃ n : ℕ, Module.finrank R R ≤ n)
    (P : ModuleCat R) (hP : Module.Projective R P) (hfP : Module.Finite R P) :
    ∃ n : ℕ, P = .of R (Fin n → R) := by
  sorry

The target asserts every finite projective over any Noetherian ring is literally a finite free module; the supposed dimension bound is tautological.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For the extension-of-projectives version suggested by the prose: specify a regular commutative Noetherian base ring R, and conjecture that every finitely generated projective R[x₁,…,xₙ]-module is isomorphic to R[x₁,…,xₙ]⊗_R P₀ for some finitely generated projective R-module P₀. Do not replace Krull dimension by finrank R R, extended by free, or module isomorphism by literal equality. The name Bass is ambiguous and must be checked against the intended source.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.