Mittag-Leffler modules and definable subcategories. II

arXiv2206.14308MaRDI QIDQ6403463

Philipp Rothmaler

Publication date: 28 June 2022

Abstract: In this note I take the opportunity to correct the last statement of Part I of same title and continue the study of uniform purity of epimorphisms in order to derive the main result, which states that--provided $R_{R}inlanglecalKangle$, equivalently, $langlecalLangle$ (the definable subcategory generated by $calL$) contains all absolutely pure left modules--every countably generated $calK$-Mittag-Leffler module in $langlecalLangle$ is a direct summand of a $langlecalLangle$-preenvelope of a union of an $calL$-pure $omega$-chain of finitely presented modules. In conclusion I present a number of examples that starts with and grew out of the study of $calL$-purity (of monomorphisms in $BbbZ$-Mod) for $calL$, the definable subcategory of divisible abelian groups. Rings that get particular attention in this are RD-rings, Warfield rings and (the newly introduced) high rings.

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