Projective kernels of extensions of Gorenstein-projective modules

Every short exact sequence with a projective kernel and a Gorenstein-projective quotient splits. This is the vanishing of degree-one extensions of a Gorenstein-projective module by a projective module, and is the input for identifying finite projectives as relative injectives in the Gorenstein-projective exact category. Maps into projectives therefore extend across short exact sequences with Gorenstein-projective quotient.

No noetherian or self-injectivity hypothesis is required, and the ring and module universes are independent. Taking the ring to be Rᵐᵒᵖ gives the result for right R-modules.

References

• Ragnar-Olaf Buchweitz, Maximal Cohen–Macaulay Modules and Tate Cohomology, Section 4.
• Edgar E. Enochs and Overtoun M. G. Jenda, Relative Homological Algebra, Section 10.2.

theorem TauCeti.IsGorensteinProjective.nonempty_splitting_of_projective {A : Type u} [Ring A] {S : CategoryTheory.ShortComplex (ModuleCat A)} (hN : IsGorensteinProjective A S.X₃) (hS : S.ShortExact) [Module.Projective A ↑S.X₁] : Nonempty S.Splitting

A short exact sequence with projective kernel and Gorenstein-projective quotient admits a splitting. The middle term need not be assumed Gorenstein-projective.

theorem TauCeti.IsGorensteinProjective.exists_extend {A : Type u} [Ring A] {S : CategoryTheory.ShortComplex (ModuleCat A)} (hN : IsGorensteinProjective A S.X₃) (hS : S.ShortExact) (Q : ModuleCat A) [Module.Projective A ↑Q] (f : S.X₁ ⟶ Q) : ∃ (g : S.X₂ ⟶ Q), CategoryTheory.CategoryStruct.comp S.f g = f

Maps into a projective module extend across a short exact sequence with Gorenstein-projective quotient. The middle module need not be Gorenstein-projective.