Sequences of ICE-closed Subcategories via Preordered \(\tau ^{-1}\)-rigid Modules

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 classify 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.