Combinatorics of latticesof subcategories of a module category

Haruhisa Enomoto(Osaka Metropolitan University)� September ����,The 6�th Algebra Symposium�Table of contents

Introduction
Torsion classes
and wide subcategories

Main result�Introduction

Overview
For a ring ⇤,{C | C is a collection of ⇤-modules satisfying (*)}
is a poset by inclusion for a given condition (*).
obtain various posets by changing (*).
Question
How these posets are combinatorially related?
Some important posets in combinatorics can be realized in thisway.
Applications in combinatorics!
�Example

k: a �eld (only consider f.d. modules).
Consider collections C of k-modules which is• closed under direct sums and direct summands(X; Y 2 C , X Y 2 C)
Such C: only 0 and mod k (all k-modules).
poset: 0 < mod k.

�Example
k 2 = k ⇥ k: a product of �eld (only consider f.d. modules).
Consider collections C of k 2 -modules satisfying:• closed under direct sums and direct summands

Every k 2 -module M is decomposed as
M = S1afor unique a; b 2 N, where S1 := kS2b0 and S2 := 0k.

Such C $ choosing S1 and S2

mod k 2{S1a }{S2b }
0�Examplek n : a product of �eld (only consider f.d. modules).
Consider collections C of k n -modules satisfying:• closed under direct sums and direct summands
Every k n -module M is decomposed uniquely as

where Si := 0···M = S1a1

S2a200k

Such C $ subset of {S1 ; : : : ; Sn }!

Snan······

0.

Thus the poset ⇠= powerset 2{1;:::;n} of {1; : : : ; n}.
6Today’s talkToday: Two class of collections of ⇤-modules.
�. Torsion classthe poset tors ⇤�. Wide subcategory

the poset wide ⇤Then construct wide ⇤ from tors ⇤ combinatorially!

�Torsion classes
and wide subcategories

Subcategories of the module category• ⇤: a �nite-dimensional k-algebra over a �eld k.
• mod ⇤: the category of f.g. ⇤-modules.
Collections of ⇤-modules = Subcategories of mod ⇤Theorem (Krull-Schmidt)
Every M 2 mod ⇤ is decomposed uniquely as
M = M1

M2···

Mnsuch that each Mi is indecomposable.
Subcategories (closed under direct sums & summands)= Sets of indecomposable ⇤-modules.
8Torsion classes
De�nition (Dickson ��66)
A subcategory T of mod ⇤ is a torsion class:, closed under extensions and quotients: for any0L

MN0;L; N 2 T ) M 2 T , and M 2 T ) N 2 T .
tors ⇤: the poset of torsion classes (by inclusion)
By Krull-Schmidt, we only have to consider indecomposable
modules in a torsion class.
• tors k = {0; mod k}.
• ⇤: semisimple ) torsion class = subcat closed underdirect sum & summands, tors ⇤ is the powerset of the setof indecomposable (=simple) modules.

�Torsion classes: Working example
torsion class = closed under ext & quotients
Consider ⇤ :=ˆk k ˜0 kˆ

˜= { 0a bc | a; b; c 2 k}.

Then 9 � indec. ⇤-modules A; B; C with exact seq0ABC0:

tors ⇤:{A; B; C}{B; C}{A}{C}{}��Torsion classes: Remarks• A torsion class is a part of a torsion pair:• A torsion pair (T ; F) is a pair of subcats of mod ⇤ which
divides mod ⇤ into two parts:torsion part T and torsion-free part F.
Classicaly: torsion groups and torsion-free groups.
• Naturally appears when considering the derived categoryand t-structures.

��Torsion classes: Combinatorial properties

In the rest, we assume tors ⇤ is �nite.
tors ⇤ has the following properties:• A complete lattice: for any family Ti of torsion classes, 9W
VTTi and Ti (= Ti ) in tors ⇤.
• The Hasse diagram is an n-regular graph(n is the number of simple ⇤-modules).
• #{T | T covers i elements}= #{T | T is covered by i elements}• Semiditributive.

��Torsion classes: Rank � Exampleabtors k(1 ! 2 ! 3)=(ab),calculated by Geuenich’s String Applet
��There is an algebra ⇤ such that tors ⇤ is:• Tamari lattice Tam: the poset of order of binary operation:a(b(cd))a((bc)d)(ab)(cd)(a(bc))d((ab)c)d• Dynkin variants of Tamari lattice (Cambrian lattice).
• Finite Coxeter groups with weak order.
��Wide subcategories (module cat inside module cat!)De�nition (Hovey ����)A subcategory W of mod ⇤ is wide if• closed under kernels and cokernels: for every f : W1 ! W2
with Wi 2 W, we have Ker f ; Coker f 2 W.
• closed under extensions.

wide ⇤: the poset of wide subcategories of mod ⇤.
An wide subcategory is an abelian subcategory.
Example⇤ “ : a nice ring surj ) mod ,! mod ⇤: wide subcat.
Indeed, wide ⇤ ⇠= {⇤ !

| a nice ring epi}.
��Wide subcategories: Example• ⇤: semisimple wide ⇤ = tors ⇤ ⇠= powerset.
ˆk k ˜ˆ˜• Working example: ⇤ := 0 k = { 0a bc | a; b; c 2 k}.
� indec. modules A; B; C with exact seq0A

B0:

Cwide ⇤

{A; B; C}
{A}

{B}
{}

{C}

�6

Wide subcategories: Combinatorial properties

wide ⇤ has the following properties:• A complete lattice.
• Ranked (graded).
• Rank-symmetric:
ai := #{W | W is rank i} = #{W | W is rank ni}:

• Rank-unimodal (a0  a1  · · ·  an=2 · · · an 1
[Aoki-Higashitani-Iyama-Kase-Mizuno ����]an )��Wide subcategories: Rank � Exampleabwide k(1 ! 2 ! 3)=(ab)�8There is an algebra ⇤ such that wide ⇤ is:• Non-crossiong partition lattice NC:from Wikipedia, T. Piesk (CC BY �.�)• Dynkin variants of NC.
• The shard intersection order, a lattice structure on a �nite
Coxeter group de�ned recently [Reading ����],using combinatorics of hyperplane arrangement
��Main result

Main result: Overview
For a given algebra ⇤, we have two posets:• tors ⇤, the poset of torsion classes.
• wide ⇤, the poset of wide subcategories.
We will construct wide ⇤ from tors ⇤ purely combinatorially!
Example⇤: an upper-triangular matrix algebra, then• tors ⇤ ⇠= Tam: Tamari lattice• wide ⇤ ⇠= NC: non-crossing partition latticeNon-trivial relation between them!
��Working Example⇤ :=ˆk k ˜
0 k, � indec. modules A; B; C.
{A; B; C}{A; B; C}
{A}

{B}
{}
wide ⇤{B; C}
{C}

{A}
{C}
{}
tors ⇤Non-isomorphic posets, but the number is same!
��Marks–Št’ovíček’s bijection
Theorem (Marks–Št’ovíček ����)We have a bijection
T : wide ⇤'tors ⇤where T(W) is the smallest torsion class containing W.
Problem
T : wide ⇤ ! tors ⇤ is order-preserving, but not a poset isom.
StrategyGive a new poset str » on tors ⇤ such thatT : wide ⇤ ⇠= (tors ⇤; » ):��Setting
We are given tors ⇤ just as an abstract lattice.
Working Example⇤ :=ˆk k ˜0 k. The Hasse quiver of tors ⇤:1xz
y0

Hasse quiver: a ! b if a > b and 6 9c with a > c > b.
��The join-irreducible labelingDe�nition
For each Hasse arrow a ! b in tors ⇤, we label its arrow bymin{x 2 tors ⇤ | b _ x = a}:This is well-de�ned and called the join-irreducible label.
z1yxzxyzy0
��Kappa map »⇠[Barnard–Todorov–Zhu ����] introduced » : tors ⇤ ! tors ⇤:De�nition

For each x 2 tors ⇤, there’s unique »(x) 2 tors ⇤ satisfying{labels of Hasse arrows starting at x}={labels of Hasse arrows ending at »(x)}(well-de�ned for semidistributive lattices.)•xa•b•c•a••bc»(x)��Kappa map »: Example

Red: the orbit of ».
z

1yx
labels from a = labels into »(a)

zxyz
y0

�6

Kappa map »: Example

Red: the orbit of ».
z

1yx
labels from a = labels into »(a)

zxyz
y0

�6

Kappa map »: Example

Red: the orbit of ».
z

1yx
labels from a = labels into »(a)

zxyz
y0

�6

Kappa map »: Example

Red: the orbit of ».
z

1yx
labels from a = labels into »(a)

zxyz
y0

�6

Kappa map »: Example

Red: the orbit of ».
z1yx
labels from a = labels into »(a)

zxyz
y0

�6

The kappa order and Main TheoremDe�nition (The kappa order)
For x; y 2 tors ⇤, de�nex » y :() x  y and »(x)»(y ).

Theorem (E)T : wide ⇤ ! tors ⇤ induces a poset isomorphism:
wide ⇤ ⇠= (tors ⇤; » )��Working Example⇤ :=ˆk k ˜0 kx » y :() x  y and »(x)tors ⇤»(y ).

(tors ⇤; » ) ⇠= wide ⇤11xzxyzy00
�8

A sketch of proof» has a natural representation-theoretic meaning!
We can dually consider torsion-free class (torf).
tors ⇤T[MS]wide ⇤Fdual of [MS]torf ⇤(⇤)

»tors ⇤(*) is poset anti-isom.
Then for W1 ; W2 2 wide ⇤, we haveT(W1 ) » T(W2 ) : () T(W1 ) ✓ T(W2 ) and »T(W1 ) ◆ »T(W2 )() T(W1 ) ✓ T(W2 ) and F(W1 ) ✓ F(W2 )() W1 ✓ W2��Combinatorial applications

By applying wide ⇤ ⇠= (tors ⇤; » ), we obtain:• Non-crossing partition lattice ⇠= Tamari lattice with » .
• Dynkin variants of the above.

• The shard intersection order on a Coxeter group W
coincide with » w.r.t. the weak order.

��