Isomorphism of one-place functors~$\operatorname{Ext}$

M. B. Zvyagina

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 4, Tome 112 (1981), pp. 71-74

Résumé

Let $\Lambda$ be an associative ring with identity. One considers the category of left (unitary) $\Lambda$-modules $\mathfrak M$ and also the contravariant and the covariant functors $\operatorname{Ext}^1_\Lambda(\ ,A)$ and $\operatorname{Ext}^1_\Lambda(A,\ )$: $_\Lambda\mathfrak M\to{}_\mathbb Z\mathfrak M$. One proves the following results: (1) If the homomorphism of $\Lambda$-modules $A\to B$ induces an isomorphism $\operatorname{Ext}^1_\Lambda(\ ,A)\to\operatorname{Ext}^1_\Lambda(\ ,B)$, then there exist injective $\Lambda$-modules $J_1$ and $J_2$ such that $A\oplus J_1\approx B\oplus J_2$. (2) Every functorial morphism $\operatorname{Ext}^1_\Lambda(\ ,A)\to\operatorname{Ext}^1_\Lambda(\ ,B)$ induces a certain homomorphism of $\Lambda$-modules $A\to B$. One also obtains a dual result.