Sequences of ICE-closed subcategories via preordered $\tau^{-1}$-rigid modules

TL;DR

This paper introduces cogen-preordered τ^{-1}-rigid modules to classify ICE-closed subcategory sequences, linking them to t-structures in derived categories, advancing the understanding of module categories and t-structure classifications.

Contribution

It generalizes τ-tilting theory concepts and establishes a bijection between cogen-preordered τ^{-1}-rigid modules and ICE-sequences, connecting module theory with t-structure classification.

Findings

Established a bijection between cogen-preordered τ^{-1}-rigid modules and torsion-free class sequences.

Connected ICE-sequences with intermediate t-structures via module-theoretic concepts.

Extended τ-tilting theory to classify subcategory sequences in derived categories.

Abstract

Let $\Lambda$ be a finite-dimensional basic algebra. Sakai recently used certain sequences of image-cokernel-extension-closed (ICE-closed) subcategories of finitely generated $\Lambda$-modules to classify certain (generalized) intermediate $t$-structures in the bounded derived category. We classifying these "contravariantly finite ICE-sequences" using concepts from $\tau$-tilting theory. More precisely, we introduce "cogen-preordered $\tau^{-1}$-rigid modules" as a generalization of (the dual of) the "TF-ordered $\tau$-rigid modules" of Mendoza and Treffinger. We then establish a bijection between the set of cogen-preordered $\tau^{-1}$-rigid modules and certain sequences of intervals of torsion-free classes. Combined with the results of Sakai, this yields a bijection with the set of contravariantly finite ICE-sequences (of finite length), and thus also with the set of $(m+1)$-intermediate $t$-structures whose aisles are homology-determined.