Classifying KE-closed subcategories over a commutative noetherian ring

Abstract

Let $\mathsf{mod} R$ denote the category of finitely generated $R$-modules for a commutative noetherian ring $R$. In this paper, we investigate KE-closed subcategories of $\mathsf{mod} R$ as a continuation of our previous work. We associate a function on $\mathrm{Spec} R$ with each KE-closed subcategory of $\mathsf{mod} R$, and show that this function completely determines the original subcategory. To classify the functions obtained from KE-closed subcategories, we introduce the notion of an $n$-Bass function for each $n\ge 0$. We obtain a bijection between the set of KE-closed subcategories and the set of $2$-Bass functions provided that $R$ is $(S_2)$-excellent in the sense of \v{C}esnavi\v{c}ius.