Isomorphism of one-place functors Ext

Abstract

Let Λ be an associative ring with identity. One considers the category of left (unitary) Λ -modules m and also the contravariant and the covariant functors Ext ¹_Λ ( ,A) and Ext ¹_Λ (A, ):_Λ M→_z M. One proves the following results: (1) If the homomorphism of Λ -modules A→ B induces an isomorphism Ext ¹_Λ ( ,A)→Ext ¹_Λ ( ,B), then there exist injective Λ -modules J₁ and J₂ such that A⊕J₁≈B⊕J₂. (2) Every functorial morphism Ext ¹_Λ ( ,A)→Ext ¹_Λ ( ,B) induces a certain homomorphism of Λ -modules A→B. One also obtains a dual result.