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Semibricks, wide subcategories and recollements Yingying Zhang - - PowerPoint PPT Presentation

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Semibricks, wide subcategories and recollements Yingying Zhang Hohai University August 30, 2019, Nagoya Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories Semibricks.etc 1 Gluing semibricks 2 Reduction of wide

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Semibricks, wide subcategories and recollements

Yingying Zhang

Hohai University

August 30, 2019, Nagoya

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

1

Semibricks.etc

2

Gluing semibricks

3

Reduction of wide subcategories

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

  • 1. Semibricks.etc
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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

  • 1. Semibricks.etc

A : a finite-dimensional algebra over a field

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

  • 1. Semibricks.etc

A : a finite-dimensional algebra over a field Semibricks

1 A module S ∈ mod A is called a brick if EndA(S) is a division

algebra (i.e., the non-trivial endomorphisms are invertible). brickA = {isoclasses of bricks in mod A}.

2 A set of S ∈ mod A of isoclasses of bricks is called a

semibrick if HomA(S1, S2) = 0 for any S1 = S2 ∈ S. sbrickA = {semibricks in mod A}.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Wide subcategories(Hov) A full subcategory C of an abelian category A is called wide if it is abelian and closed under extensions.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Wide subcategories(Hov) A full subcategory C of an abelian category A is called wide if it is abelian and closed under extensions. Put wideA = {wide subcategories of A}. wideA = {wide subcategories of mod A}. wideCA = {wide subcategories of A containing C}.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Support τ-tilting modules (Adachi-Iyama-Reiten) Let (X,P) be a pair with X ∈ mod A and P ∈ proj A. We call (X,P) a support τ-tilting pair if

1 X is τ-rigid, i.e., HomA(X, τX)=0 2 HomA(P, X)=0 3 |X| + |P| = |A|

In this case, X is called a support τ-tilting module. Put sτ-tiltA = {basic support τ-tilting A-modules}.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Related works

1 Representations of K-species and bimodules. (Rin,1976) 2 τ-tilting theory. (AIR,2014) 3 τ-tilting finite algebras, g-vectors and brick-τ-rigid

  • correspondence. (DIJ,2019)

4 Semibricks. (As,2019)

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Recollements(BBD,FP,Ha,K) Let A, B, C be abelian categories. Then a recollement of B relative to A and C, diagrammatically expressed by A

i∗

B

i∗

  • j∗

i!

  • C
  • j!
  • j∗
  • which satisfy the following three conditions:

1 (i∗, i∗), (i∗, i!), (j!, j∗) and (j∗, j∗) are adjoint pairs; 2 i∗, j! and j∗ are fully faithful functors; 3 Imi∗ = Kerj∗.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Remark

1 i∗ and j∗ are both right adjoint functors and left adjoint

functors, therefore they are exact functors of abelian categories.

2 i∗i∗ ∼

= id, i!i∗ ∼ = id, j∗j! ∼ = id and j∗j∗ ∼ = id. Also i∗j! = 0, i!j∗ = 0.

3 Denote by R(A, B, C) a recollement of B relative to A and C

as above and R(A, B, C) a recollement of mod B relative to mod A and mod C.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Remark

1 i∗ and j∗ are both right adjoint functors and left adjoint

functors, therefore they are exact functors of abelian categories.

2 i∗i∗ ∼

= id, i!i∗ ∼ = id, j∗j! ∼ = id and j∗j∗ ∼ = id. Also i∗j! = 0, i!j∗ = 0.

3 Denote by R(A, B, C) a recollement of B relative to A and C

as above and R(A, B, C) a recollement of mod B relative to mod A and mod C. Associated to a recollement there is a seventh funtor j!∗ := Im(j! → j∗) : mod C → mod B called the intermediate extension functor.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Intermediate extension functor

1 i∗j!∗ = 0, i!j!∗ = 0. 2 j∗j!∗ ∼

= id and the functors i∗, j!, j∗ and j!∗ are full embeddings.

3 The functor j!∗ sends simples in mod C to simples in mod B.

There is a bijection between sets of isomorphism classes of simples: (gluing simple modules) {simples ∈ mod A}⊔{simples ∈ mod C} → {simples ∈ mod B} given by mapping a simple ML ∈ mod A to i∗(ML) and a simple MR ∈ mod C to j!∗(MR).

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Related works

1 Analysis and topology on singular spaces. (BBD,1981) 2 Recollements of extension algebras. (CL,2003) 3 One-point extension and recollement. (LL,2008) 4 From recollement of triangulated categories to recollement of

abelian categories. (LW,2010)

5 Weight structures vs. t-structures; weight filtrations, spectral

sequences, and complexes. (B,2010)

6 Gluing silting objects. (LVY,2014) 7 Lifting of recollements and gluing of partial silting sets.

(SZ,arXiv2018)

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Connection(Rin,Asai) Bijections: Ringel’s bijection: sbrickA − →wideA Asai’s bijection: sτ-tiltA − → fL−sbrickA via M − →ind(M/radBM) If A is τ-tilting finite, fL−sbrickA =sbrickA and there is a bijection sτ-tiltA − →sbrickA

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

  • 2. Gluing semibricks
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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

  • 2. Gluing semibricks

Lemma If F : mod A → mod B is a fully faithful functor, then we have F(brickA) ⊆brickB and F(sbrickA) ⊆sbrickB. Proposition Let R(A, B, C) be a recollement.

1 i∗(brickA) ⊆brickB and i∗(sbrickA) ⊆sbrickB; 2 j!(brickC) ⊆brickB and j!(sbrickC) ⊆sbrickB; 3 j∗(brickC) ⊆brickB and j∗(sbrickC) ⊆sbrickB; 4 j!∗(brickC) ⊆brickB and j!∗(sbrickC) ⊆sbrickB.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Theorem{Gluing semibricks} Let R(A, B, C) be a recollement. i∗(sbrickA) ⊔ j!∗(sbrickC) ⊆sbrickB. There is an injection between sets of isomorphism classes of semibricks: sbrickA ⊔ sbrickC → sbrickB through a semibrick SL ∈ mod A and a semibrick SR ∈ mod C into i∗(SL) ⊔ j!∗(SR).

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Theorem Let R(A, B, C) be a recollement. If B is τ-tilting finite, A and C are τ-tilting finite. Corollary Let A be a finite dimensional algebra and e an idempotent element

  • f A. If A is τ-tilting finite, it follows that eAe and A/e are

τ-tilting finite.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Let R(A, B, C) be a recollement of module categories and B a τ-tilting finite algebra. Since semibricks can be glued via a recollement, the natural question is the following:

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Let R(A, B, C) be a recollement of module categories and B a τ-tilting finite algebra. Since semibricks can be glued via a recollement, the natural question is the following: Question Given a recollement of module categories, support τ-tilting modules MA and MC in mod A and mod C, is it possible to construct a support τ-tilting module in mod B corresponding to the glued semibrick?

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Let R(A, B, C) be a recollement of module categories and B a τ-tilting finite algebra. Since semibricks can be glued via a recollement, the natural question is the following: Question Given a recollement of module categories, support τ-tilting modules MA and MC in mod A and mod C, is it possible to construct a support τ-tilting module in mod B corresponding to the glued semibrick? Answer Yes, there exists a unique support τ-tilting B-module MB which is associated with the induced semibrick i∗(SA) ⊔ j!∗(SC).

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Example {Gluing support τ-tilting modules over τ-tilting finite algebras} Let A be the path algebra over a field of the quiver 1 → 2 → 3. If e is the idempotent e1 + e2, then as a right A-module A/e is isomorphic to S3 and eAe is the path algebra of the quiver 1 → 2. In this case, there is a recollement as follows: mod (A/e)

i∗

mod A

i∗

  • j∗

i!

  • mod (eAe)
  • j!
  • j∗
  • where i∗ = − ⊗A A/e, j! = − ⊗eAe eA, i! = HomA(A/e, −),

i∗ = − ⊗A/e A/e, j∗ = − ⊗A Ae, j∗ = HomeAe(Ae, −).

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Table:

sτ-tilt (A/e) sτ-tilt A sτ-tilt (eAe)

3 3 2 3 1 2 3 1 2 2 3 3 1 2 3 1 1 2 1 3 3 2 3 2 3 3 1 1 3 3 1 2 2 1 2 2 1 2 1 1 2 1 2 2 1 1

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Example Let A be the preprojective algebra of type A3 which is given by the following quiver and relation aa′ = 0, b′b = 0, bb′ = a′a. 1

a

2

a′

  • b

3

b′

  • Let e = e1 + e3. Then as a right A-module A/e is isomorphic to

S2 and eAe is the preprojective algebra of type A2. Then there is a recollement R(A/e, A, eAe) induced by the idempotent e.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

  • 3. Reduction of wide subcategories
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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

  • 3. Reduction of wide subcategories

Theorem There is a bijection widei∗(A)B ↔ wideC given by widei∗(A)B ∋ C → j∗(C) ∈ wideC and wideC ∋ W → C = {M ∈ B|j∗(M) ∈ W} ∈ widei∗(A)B.

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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Theorem If C ⊂ B is wide and satisfies i∗(A) ⊂ C, then we can get a recollement of wide subcategories as follows: A

i∗

C

i∗

  • j∗

i!

  • j∗(C)
  • j!
  • j∗
  • .
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Outline Semibricks.etc Gluing semibricks Reduction of wide subcategories

Example Let A be the path algebra over a field of the quiver 1 ← 2 → 3, of type A3. If e is the idempotent e2 + e3, then as a right A-module A/AeA is isomorphic to S1. In this case, there is a recollement as follows: mod (A/AeA)

i∗

mod A

i∗

  • j∗

i!

  • mod (eAe)
  • j!
  • j∗
  • where i∗ = − ⊗A A/e, j! = − ⊗eAe eA, i! = HomA(A/e, −),

i∗ = − ⊗A/e A/e, j∗ = − ⊗A Ae, j∗ = HomeAe(Ae, −).

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Table:

mod (A/AeA) C j∗(C)

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Thanks for your attention!