Surjections from a projective module over a semiprimary ring

Over a semiprimary ring a map from a projective module of finite length is determined, up to an automorphism of its source, by its range: if a b : F → E have the same range, then b ∘ θ = a for an automorphism θ of F.

For F free of finite length, for instance free of finite rank over an Artinian ring, this says that two presentations of the same module by the same number of generators are related by a change of generators, the module-theoretic analogue of Gaschütz's lemma. Over the group algebra of a finite group over a field it compares two presentations of the augmentation ideal; lifting from the residue field, it compares them over a local ring as well.

Main results

• TauCeti.exists_linearEquiv_comp_eq_of_range_eq: two maps with the same range from a projective module of finite length over a semiprimary ring differ by an automorphism of the source.

References

• T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer (2001), §24 (projective covers over semiperfect rings).
• K. W. Gruenberg, Relation modules of finite groups, CBMS Regional Conference Series in Mathematics 25, American Mathematical Society (1976).

theorem TauCeti.exists_linearEquiv_comp_eq_of_range_eq {R : Type u} [Ring R] [IsSemiprimaryRing R] {F : Type v} [AddCommGroup F] [Module R F] {E : Type w} [AddCommGroup E] [Module R E] [Module.Projective R F] (hF : IsFiniteLength R F) {a b : F →ₗ[R] E} (h : a.range = b.range) : ∃ (θ : F ≃ₗ[R] F), b ∘ₗ ↑θ = a

Maps from a projective module of finite length with the same range differ by an automorphism. Over a semiprimary ring, if F is projective of finite length and a b : F → E have the same range, then b ∘ θ = a for some automorphism θ of F. In particular two surjections from F onto the same module have isomorphic kernels.