The finitely presented torsion-free SG-projective modules are not necessarily projective

Abstract

It is proved in [Strongly Gorenstein projective, injective and flat modules, J. Algebra 320 (2008) 2659–2674, Theorem 2.8] that a finitely presented torsion-free module is SG-projective if and only if it is projective. Let L⊆F be an extension of fields with [F:L]=2. Set R=L+XF[X], m=XF[X] and D=R[Y]. Via the study of w-invertibility over infra-Krull domains, we prove that the w-conductor m[Y] of D is SG-projective but not projective, and all maximal w-ideals other than m[Y] are w-invertible. It follows that a finitely presented torsion-free SG-projective module do not need to be projective.