Classifying subcategories of modules over Noetherian algebras

Iyama, Osamu; Kimura, Yuta

doi:10.1016/j.aim.2024.109631

Advances in Mathematics

2024

DOI: 10.1016/j.aim.2024.109631

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Classifying subcategories of modules over Noetherian algebras

Osamu Iyama,

Yuta Kimura

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2024

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Cited by 3 publications

References 39 publications

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When are KE-closed subcategories torsion-free classes?

Kobayashi,

Saito

2024

Math. Z.

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When are KE-closed subcategories torsion-free classes?

Kobayashi,

Saito

2024

Math. Z.

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Tilting and Silting Theory of Noetherian Algebras

Kimura

2023

International Mathematics Research Notices

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We develop silting theory of a Noetherian algebra $\Lambda $ over a commutative Noetherian ring $R$. We study mutation theory of $2$-term silting complexes of $\Lambda $, and as a consequence, we see that mutation exists. As in the case of finite-dimensional algebras, functorially finite torsion classes of $\Lambda $ bijectively correspond to silting $\Lambda $-modules, if $R$ is complete local. We show a reduction theorem of $2$-term silting complexes of $\Lambda $, and by using this theorem, we study torsion classes of the module category of $\Lambda $. When $R$ has Krull dimension one, we describe the set of torsion classes of $\Lambda $ explicitly by using the set of torsion classes of finite-dimensional algebras.

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Semistable torsion classes and canonical decompositions in Grothendieck groups

Asai,

Iyama

2024

Proceedings of London Math Soc

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We study two classes of torsion classes that generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen–Fei. More strongly, for ‐tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. We also answer a question by Derksen–Fei negatively by giving examples of algebras that do not satisfy the ray condition. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type .

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Classifying subcategories of modules over Noetherian algebras

Cited by 3 publications

References 39 publications

When are KE-closed subcategories torsion-free classes?

When are KE-closed subcategories torsion-free classes?

Tilting and Silting Theory of Noetherian Algebras

Semistable torsion classes and canonical decompositions in Grothendieck groups

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