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

Abstract Let $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> be a finite-dimensional basic algebra. Sakai recently used certain sequences of image-cokernel-extension-closed (ICE-closed) subcategories of finitely generated $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> -modules to classify certain (generalized) intermediate t -structures in the bounded derived category. We classify these “contravariantly finite ICE-sequences” using concepts from $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -tilting theory. More precisely, we introduce “cogen-preordered $$\tau ^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> -rigid modules” as a generalization of (the dual of) the “TF-ordered $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> -rigid modules” of Mendoza and Treffinger. We then establish a bijection between the set of cogen-preordered $$\tau ^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> -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 $$(m+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -intermediate t -structures whose aisles are homology-determined.