Lattice Theoretic Properties of Aprroximating Ideals

TL;DR

This paper explores the lattice properties of special preenveloping and precovering ideals in exact categories, extending classical approximation results to infinite families and establishing new lemmas related to ideal cotorsion pairs.

Contribution

It proves that finite intersections of special preenveloping or precovering ideals are themselves special, and extends these results to infinite families under certain conditions, introducing an ideal version of the Eklof-Trlifaj Lemma.

Findings

Finite intersections of special preenveloping ideals are special.

Finite intersections of special precovering ideals are special.

The ideal cotorsion pair generated by a small ideal is complete.

Abstract

It is proved that a finite intersection of special preenveloping ideals in an exact category $({\mathcal A}; {\mathcal E})$ is a special preenveloping ideal. Dually, a finite intersection of special precovering ideals is a special precovering ideal. A counterexample of Happel and Unger shows that the analogous statement about special preenveloping subcategories does not hold in classical approximation theory. If the exact category has exact coproducts, resp., exact products, these results extend to intersections of infinite families of special peenveloping, resp., special precovering, ideals. These techniques yield the Bongartz-Eklof-Trlifaj Lemma: if $a \colon A \to B$ is a morphism in ${\mathcal A},$ then the ideal $a^{\perp}$ is special preenveloping. This is an ideal version of the Eklof-Trlifaj Lemma, but the proof is based on that of Bongartz' Lemma. The main consequence is that the ideal cotorsion pair generated by a small ideal is complete.