On Some Noteworthy Pairs of Ideals in Mod-R

Abstract

An additive functor $F \colon {\mathcal A}\to{\mathcal B}$ between preadditive categories $\mathcal A$ and $\mathcal B$ is said to be a local functor if, for every morphism $f\colon A\to A'$ in $\mathcal A$, F(f) isomorphism in $\mathcal B$ implies f isomorphism in $\mathcal A$. We show that there exist several pairs $(\mathcal I_1,\mathcal I_2)$ of ideals of $\mathcal A$ for which the canonical functor $\mathcal A\to\mathcal A/\mathcal I_1\times \mathcal A/\mathcal I_2$ is a local functor. In most of our examples, the category $\mathcal A$ is a full subcategory of the category Mod -R of all right modules over a ring R. These pairs of ideals arise in a surprisingly natural way and enjoy several properties. Ideals are kernels of functors, and most of our examples of ideals are kernels of important and well studied functors. E.g., (1) the kernel Δ of the canonical functor P of Mod -R into its spectral category Spec(Mod -R), so that Δ is the ideal of all morphisms with an essential kernel; (2) the kernel Σ of the dual functor F of P, so that Σ is the ideal of all morphisms with a superfluous image; (3) the kernels Δ⁽¹⁾ and Σ₍₁₎ of the first derived functors P ⁽¹⁾ and F ₍₁₎ of P and F, respectively; (4) the kernels of suitable functors Hom and ⊗ and their first derived functors ${\rm Ext}^1_R$ and ${\rm Tor}^R_1$.