Resolving subcategories whose finitely presented module categories are abelian

Resolving subcategories whose finitely presented module categories are abelian

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Let $mathcal{X}$ be an additive full subcategory of an abelian category.It is a classical fact that if $mathcal{X}$ is contravariantly finite, then the 30" Gas Cooktop category $mathsf {mod},mathcal{X}$ of finitely presented right $mathcal{X}$-modules is abelian.In this paper, we consider the Hearing Aids question asking when the converse holds true for a resolving subcategory of the category of finitely generated modules over a commutative noetherian henselian local ring.We give both affirmative answers and negative answers to this question.