global dimensions of endomorphism algebras of …math.sjtu.edu.cn/conference/scrta/report/zhang,...

Global dimensions of endomorphism algebras of

generator-cogenerators over m-replicated algebras

Shunhua ZhangSchool of Mathematics, Shandong University

This is a joint with Hongbo Lv

2011.10.6 Shanghai Jiao Tong University

Contents

Motivation 1

Notations 2

Main results 3

Motivation

The endomorphism algebras of generator-cogenerators haveattracted a lot of interest, and these are just the artin algebrasof dominant dimension at least two.

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Motivation


The smallest value of the global dimensions of theendomorphism algebras of generator-cogenerators was definedto be the representation dimension by M.Auslander.

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Motivation


The smallest value of the global dimensions of theendomorphism algebras of generator-cogenerators was definedto be the representation dimension by M.Auslander.

In particular, M.Auslander proved that an artin algebra isrepresentation-finite if and only if its representation dimensionis at most two.

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Motivation

2003, O.Iyama has shown that the representation dimensionof an artin algebra is always finite.

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Motivation


2006, R.Rouquier has shown that there is no upper bound forthe representation dimensions of artin algebras.

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Motivation



These motivate the investigation on the possibilities for theglobal dimensions of the endomorphism algebras ofgenerator-cogenerators.

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Motivation



These motivate the investigation on the possibilities for theglobal dimensions of the endomorphism algebras ofgenerator-cogenerators.

In general, it is not easy to compute the global dimension ofEnd(M) whenever M is a generator-cogenerator. Henceconstructions of generator-cogenerators with a fixed globaldimension have an independent interest.

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Motivation

2008, V.Dlab and C.M.Ringel described the possibilities forthe global dimensions of the endomorphism algebras ofgenerator-cogenerators for a hereditary algebra.

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Motivation

2008, V.Dlab and C.M.Ringel described the possibilities forthe global dimensions of the endomorphism algebras ofgenerator-cogenerators for a hereditary algebra.

We follow the idea of V.Dlab and C.M.Ringel and investigatethe possible values for the global dimensions of theendomorphism algebras of generator-cogenerators form-replicated algebra.

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Notations

Let Λ be an artin algebra.gl.dim Λ is the global dimension of Λ.τΛ is the Auslander-Reiten translation of Λ.Let M be a Λ-module.pd M is the projective dimension of M.Ω−k

Λ M is the kth cosyzygy of M.

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Notations

Let Λ be an artin algebra.gl.dim Λ is the global dimension of Λ.τΛ is the Auslander-Reiten translation of Λ.Let M be a Λ-module.pd M is the projective dimension of M.Ω−k

Λ M is the kth cosyzygy of M.

M is called a generator-cogenerator if all indecomposableprojective modules and indecomposable injective modules arein add M.Let M be a generator-cogenerator.If gl.dimEndΛ(M) = d , then M is also called agenerator-cogenerator with global dimension d .

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Notations

From now on, let A be a hereditary algebra.

A(m) =

A0 0Q1 A1

Q2 A2

. . .. . .

0 Qm Am

.

is the m-replicated algebra of A, where Ai = A and Qi = DA,D is the standard duality between mod A and mod Aop,multiplication is induced from the canonical isomorphismsA ⊗A DA ∼=DA ∼= DA ⊗A A and the zero morphismDA ⊗A DA −→ 0.

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Notations

Let Σk = Ω−kA′ Σ0 = Ω−k

A′ X | X ∈ Σ0 for k ≥ 0.Σ0 is the set of all non-isomorphic indecomposable projectiveA-modules.

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Notations



Let Uk be the direct sum of all the indecomposable modulesin Σk

⋂

ind A(m) for k ≥ 0.

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Notations



Let Uk be the direct sum of all the indecomposable modulesin Σk

⋂

ind A(m) for k ≥ 0.

Let P be the direct sum of all indecomposableprojective-injective A(m)-modules.

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Main results

Theorem 1. Let d be an integer with d ≥ 2 and A be a

representation finite hereditary Artin algebra. There exists an

A(m)- module M which is a generator-cogenerator with global

dimension d if and only if there exists a τA(m)-orbit of

cardinality at least d.

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Main results

Theorem 1. Let d be an integer with d ≥ 2 and A be a

representation finite hereditary Artin algebra. There exists an

A(m)- module M which is a generator-cogenerator with global

dimension d if and only if there exists a τA(m)-orbit of

cardinality at least d.

Remark. In this finite type case, let s be the maximal lengthof all τA(m)-orbits. Then for any d with 2 ≤ d ≤ s, one canfind a generator-cogenerator A(m)-module M such thatgl.dim EndA(m) (M) = d .

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Generator-cogenerators with dimension i + 2

Assume that gl.dim A(m) = t.

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Now I can tell you how to construct a generator-cogeneratorwith special global dimension.

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Proposition 1. Let Ei = A ⊕ DAm ⊕ P ⊕t−1⊕

k=iUk for

1 ≤ i ≤ t − 1. Then Ei is a generator-cogenerator

A(m)-module and gl.dim EndA(m) (Ei ) = i + 2.

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Proposition 1. Let Ei = A ⊕ DAm ⊕ P ⊕t−1⊕

k=iUk for

1 ≤ i ≤ t − 1. Then Ei is a generator-cogenerator

A(m)-module and gl.dim EndA(m) (Ei ) = i + 2.

Corollary 2. Let d be an integer with 3 ≤ d ≤ t + 1. Then

there exists a generator-cogenerator M in mod A(m) with

global dimension d. In particular, the representation

dimension of A(m) is at most 3.

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Generator-cogenerators with dimension d

Proposition 3. Let d be an integer with d ≥ 2m + 3. Let Z

be an indecomposable non-injective A(m)-module such that

τd−(2m+2)Z is a simple and projective A-module. Let

0 −→ τZ −→ ⊕Yj −→ Z −→ 0

be the Auslander-Reiten sequence ending in Z , with

indecomposable modules Yj . Let

M = A ⊕ DAm ⊕d−(2m+3)

⊕i=0

(⊕τiYj) ⊕ P.

Then M is a generator-cogenerator A(m)-module and

gl.dim EndA(m) (M) = d.

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Generator-cogenerators with dimension ∞

Proposition 4. Let N be an indecomposable A-module whose

endomorphism algebra is a division ring and such that there is

a non-split sequence 0 → Nu

−→ N ′ v−→ N → 0. Let

M = A ⊕ DAm ⊕ P ⊕ N ′. Then M is a generator-cogenerator

in mod A(m) and gl.dim EndA(m) (M) = ∞.

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Main results

According to Proposition 1, Proposition 3 and Proposition 4,we get the following theorem.

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Main results


Theorem 2. Let A be a representation infinite hereditary

Artin algebra and d be either an integer with d ≥ 3 or else

the symbol ∞. Then there exists a generator-cogenerator

A(m)-module M with gl.dim EndA(m) (M) = d.

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Main results


Theorem 2. Let A be a representation infinite hereditary

Artin algebra and d be either an integer with d ≥ 3 or else

the symbol ∞. Then there exists a generator-cogenerator

A(m)-module M with gl.dim EndA(m) (M) = d.

Remark. In this infinite type case, for any d with 3 ≤ d ≤ ∞,we explicitly construct a generator-cogenerator A(m)-moduleM with gl.dim EndA(m) (M) = d .

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Main References

[ABST ] I.Assem, T.Brustle, R.Schiffer, G.Todorov, m-clustercategories and m-replicated algebras. Journal of pure and applied

Algebra 212(2008), 884-901.

[DR ] V.Dlab, C.M.Ringel, The global dimension of theendomorphism ring of a generator-cogenerator for a hereditaryartin algebra. Mathematical Reports of the Academy of Science of

the Royal Society of Canada, 30(3)(2008), 89-96.

[LZ ] H.Lv, S.Zhang, Global dimensions of endomorphism algebrasfor generator-cogenerators over m-replicated algebras.Comm.Algebra, 39(2011), 560-571.

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THANK YOU!

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global dimensions of endomorphism algebras of …math.sjtu.edu.cn/conference/scrta/report/zhang,...

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