Hanpeng Gao, Dajun Liu, Yu-zhe Liu
6 min
Abstract
For Artin algebras, we establish a bijective between IE-closed subcategories and canonical intervals in the lattice of torsion classes. Enomoto and Sakai previously achieved a classification of IE-closed subcategories over hereditary algebras using twin rigid modules. However, this result fails for the non-hereditary algebras. In this paper, we generalize this classification to arbitrary $τ$-tilting finite algebras by replacing twin rigid modules with canonical twin support $τ$-tilting modules. We provide a homological characterization of these modules via relative torsion theory, and then obtain a constructive algorithm to canonicalize an arbitrary twin support $τ$-tilting module while preserving its associated heart.
Alex: The conditions make it a unique tight fit.
Sam: Theorem 3.6 confirms the bijection between canonical intervals and subcategory hearts, with inverse via U_C, V_C. For τ-tilting finite algebras, canonical twin support τ-tilting modules label them, generalizing prior work.
Alex: How do they prove recovering C from the interval both ways?
Sam: For IE-closed C, the heart of matches C. C sits inside, as it's in V_C with no Homs from U_C. Conversely, heart objects decompose via relative torsion pair into no-Hom-to-C part (zero, since heart ignores U_C) and C part.
Alex: Decomposition pins it exactly to C. Canonical conditions confirm the right interval?
Sam: Yes. The reverse uses those properties. Without canonical, it's only surjective—multiple intervals give the same heart, like the zero subcategory from mismatched bounds. Canonical prevents duplicates.
Alex: Non-canonical ones overlap. How does functorial finiteness fit?
Sam: If bounds are functorially finite—every module has good approximations from them—the heart is too. For M, a right approximation from U_C plus torsion radical from V_C composites to approximate from heart C. Dually for left. Over τ-tilting finite algebras, all are.
Alex: Practical for finite cases.
Sam: This leads to twin support τ-tilting modules: basic pairs M upper, N lower, Fac N inside Fac M. Canonical if Fac M is torsion closure of heart, Fac N the no-Hom part. Theorem 4.2 bijection via heart.
Alex: How do they confirm the bijection lines up perfectly?
Sam: Commutative diagrams show paths converge: intervals to hearts match twin pairs via facets. Theorem 3.6 ensures one-to-one.
Alex: Without canonical, does it work?
Sam: No—plain twins aren't one-to-one. Example: over two-point quiver, one canonical pair gives heart add{simple at 2}; non-canonical (full module upper, simple at 1 lower) gives same heart but oversized upper, failing canonical.
Alex: Canonical tightens bounds, avoids duplicates. Special cases recover torsion or torsion-free hearts.
Sam: Inside upper Fac M, relative torsion pair splits: heart C extension- and submodule-closed in it, U_V ignoring C. Yields short exact sequence for M: radical (biggest U_V in M) to M to quotient C_M in heart. Torsion closure of heart equals that of C_M.
Alex: One quotient generates the torsion—efficient.
Sam: Theorem 4.6: canonical if upper bound is torsion class of C_M, lower generates radical of M. Practical test via sequence. Remark 4.7: algorithm to canonicalize any twin pair by tight generators M*, N* for same heart.
Alex: Shrinks to tight fit, preserves heart.
Sam: Example over Nakayama algebra (global dimension 2, τ-tilting finite, 12 torsion classes): 52 intervals, but 21 IE-closed subcategories, each with unique canonical interval and twin pair. Tables confirm bijections.
Alex: Clear filter from many to few.
Alex: Solid generalization where classical methods fail. Limits to τ-tilting finite algebras, heavy computation for large ones, but enables full classification.
Sam: This provides a precise toolset for subcategory structure. Thanks for listening to ResearchPod.