Some classes of subcategories of module categories:classiﬁcations and the relation between themHaruhisa Enomoto (Osaka Prefecture University)
partially joint work with Arashi Sakai (Nagoya University)
8 September 2021, Symposium on Ring and Representation Theory

1Overview

Today’s talk• Introduce new classes of subcategories of mod Λ.
• Give classiﬁcation results of these subcategories.
Throughout this talk,• Λ: a ﬁnite-dimensional k-algebra over a ﬁeld k.
• mod Λ: the category of ﬁnitely generated right Λ-modules.

2Motivation

SloganStudy various subcategories of mod Λ!
Question• What kinds of subcategories should we study?
• What “study” means?
(Today’s) Answer
1. Subcategories controlled by torsion pairs.
2. Classify and describe poset structure.

3

Outline

Torsion pairs, Serre and wide subcategories

ICE-closed subcategories and torsion hearts

From tors Λ to other posets

4Torsion pairs, Serre and wide
subcategories

Torsion pairs
Deﬁnition (Dickson 1966)• A subcategory T of mod Λ is a torsion class
if it is closed under extensions and quotients.
• A subcategory F of mod Λ is a torsion-free class
if it is closed under extensions and submodules.
• tors Λ (torf Λ): the poset of torsion(-free) classes in mod Λ.
• tors Λ and torf Λ are anti-isom by (−)⊥ and ⊥ (−).
• If tors Λ is a ﬁnite set (fi -tilting ﬁnite), then there’s a bijection
between tors Λ and support fi -tilting modules[Adachi-Iyama-Reiten 2014]
5

Serre and wide subcategories
Deﬁnition (Serre 1953?, Hovey 2001)• A subcategory S of mod Λ is a Serre subcategory if it isclosed under extensions, quotients, and submodules.
• A subcategory W of mod Λ is a wide subcategory if it isclosed under extensions, cokernels, and kernels.
• Serre Λ (wide Λ): the posets of Serre (wide) subcategoriesof mod Λ.
Clearly Serre Λ = tors Λ ∩ torf Λ and Serre Λ ⊆ wide Λ.
Theorem (Ingalls-Thomas 2009, Marks-Št’ovíček 2017)
Suppose Λ is fi -tilting ﬁnite. Then there is a bijection between
tors Λ and wide Λ (but not poset-isom!).
6Picture???

⊃⊃??

??

⊃⊃

⊃wide Λ⊃tors Λ⊃torf Λ⊃⊃Serre Λ7

Picturetors-heart Λ⊃⊃ICE Λ

IKE Λ⊃⊃⊃wide Λ⊃tors Λ⊃torf Λ⊃⊃Serre Λ8

ICE-closed subcategories andtorsion hearts

ICE-closed subcategories
Deﬁnition (E 2020)
A subcategory C of mod Λ is ICE-closed if it is closed under:• Images (f : C1 → C2 with C1 ; C2 ∈ C ⇒ Im f ∈ C),• Cokernels (... ⇒ Coker f ∈ C), and• Extensions.
ICE Λ: the poset of ICE-closed subcategories of mod Λ.
Dually deﬁne IKE-closed subcategories(Image-Kernel-Extension-closed).
• tors Λ ⊆ ICE Λ,

torf Λ ⊆ IKE Λ.

• ICE Λ ∩ IKE Λ = wide Λ.
9ICE-closed subcategories
Example: kQ for Q : 1 ← 2 → 3

Theorem (E-Sakai 2021)
Let C be a subcategory of mod Λ. Then TFAE.
1. C is an ICE-closed subcategory.
2. There is some wide subcategory W containing C such that
C is a torsion class in W.
10

Classiﬁcation of ICE-closed subcategories
Corollary (E-Sakai 2021)
If Λ is fi -tilting ﬁnite, then there is a bijection between:• ICE-closed subcategories of mod Λ and• wide fi -tilting modules(= fiW -tilting object in some wide subcat W).
This generalizes Adachi-Iyama-Reiten’s bijection!
Corollary (E 2020)
If Q is a Dynkin quiver, then there is a bijection between:• ICE-closed subcategories of mod kQ and• rigid kQ-modules(modules M with Ext1kQ (M; M) = 0).
11(Some questions)• Easy characterization of wide fi -tilting modules(for non-hereditary case)?
• Interpretation of wide fi -tilting modules using the derived
category? (silting complex for usual fi -tilting theory).

12

Torsion hearts
The proof uses the notion of torsion hearts.
Deﬁnition (Demonet-Iyama-Reading-Reiten-Thomas 2017,Tattar 2020, Asai-Pfeifer 2021, E-Sakai 2021, etc)
• To each pair U ⊆ T in tors Λ, its heart is:
H[U;T ] := T ∩ U ⊥ (= ”T − U”);
• A subcategory of this form is called a torsion heart.
• tors-heart Λ: the poset of torsion hearts.
The following subcategories are torsion hearts:• Torsion(-free) classes (by T = H[0;T ] and its dual).
• Wide subcategories [Asai-Pfeifer 2021]• ICE-closed subcategories (and IKE) [E-Sakai 2021].

13

Picture Again

tors-heart Λ⊃⊃ICE ΛIKE Λ⊃⊃⊃wide Λ⊃tors Λ⊃

torf Λ⊃⊃Serre ΛHow to obtain the poset tors-heart Λ?
14

From tors Λ to other posets

From tors Λ to other posets

itv(tors Λ) := {(U; T ) | U; T ∈ tors Λ;U ⊆T}“Taking hearts” gives a surj H(−) : itv(tors Λ) “ tors-heart Λ.
Theorem (E, in preparation)
We can deﬁne a certain equivalence relation ∼, which
depends only on the poset structure of tors Λ, s.t.
itv(tors Λ)' tors-heart Λ:∼This restricts to bijections between {certain intervals}= ∼ andwide Λ or ICE Λ.
The posets tors-heart Λ, ICE Λ, IKE Λ, and wide Λ can becomputed from the poset tors Λ (using computer)!

15

Picture Yet Again

tors-heart Λ⊃⊃
ICE ΛIKE Λ⊃⊃⊃

wide Λ⊃tors Λ⊃torf Λ⊃⊃Serre Λ16

Larger picture?

IE Λ⊃tors-heart Λ

SchurL Λ⊃

SchurR Λ⊃⊃⊃IKE Λ⊃
wide Λ⊃⊃⊃torf Λ⊃⊃tors Λ⊃

ICE ΛSerre Λ• SchurR Λ: right Schur subcategories [E 2020](deﬁned using one-sided Schur’s lemma)• IE Λ: Image-Extension-closed subcategories [E-Sakai, inpreparation] (= T ∩ F for some T ∈ tors Λ and F ∈ torf Λ)17