leavitt path algebra via the singularity category of a radical … · 2020. 7. 15. · leavitt path...

Leavitt path algebra via the singularity category of

a radical-square-zero algebra

Xiao-Wu Chen, USTC

Western Sydney University Abend Seminars, Sydney

July 9, 2020

Xiao-Wu Chen, USTC LPA via Singularity Category

Overview

Leavitt path algebras might be traced back to the 1962 paper

of William G. Leavitt (1916-2013)

interesting connections with C ∗-algebras, symbolic dynamic

systems and noncommutative (differential/algebraic)

geometry.

Our focus: the link of Leavitt path algebras to (various)

singularity categories of some finite dimensional algebras

We work over a fixed field k.

Overview

geometry.

Overview

geometry.

Overview

geometry.

The content

Quivers and Leavitt path algebras

An introduction to singularity categories

Other links via categorical equivalences

Keller’s conjecture on singular Hochschild cohomology

Quivers

Q = (Q0,Q1; s, t : Q1 → Q0) a finite quiver (= orientedgraph)

Q0 = the set of vertices, Q1 = the set of arrows

visualize an arrow α as s(α)α−→ t(α)

a vertex i is called a sink, if s−1(i) = ∅; for simplicity, weassume that Q has no sinks.

Quivers

a vertex i is called a sink, if s−1(i) = ∅;

for simplicity, we

assume that Q has no sinks.

Quivers

Examples

Example

Let Q be the following rose quiver with two petals

·1 βffα 88

Then Q0 = {1}, Q1 = {α, β}.

Example

Let Q ′ be the following quiver

1·α 88γ //

2·δoo βff

Then Q ′0 = {1, 2}, Q ′1 = {α, β, γ, δ}, s(γ) = 1 for example.

Examples

Example

·1 βffα 88

Then Q0 = {1}, Q1 = {α, β}.

Example

1·α 88γ //

2·δoo βff

Examples

Example

·1 βffα 88

Then Q0 = {1}, Q1 = {α, β}.

Example

1·α 88γ //

2·δoo βff

Examples

Example

·1 βffα 88

Then Q0 = {1}, Q1 = {α, β}.

Example

1·α 88γ //

2·δoo βff

Path algebras

a finite path in Q is p = αn · · ·α2α1 of length n

· α1−→ · α2−→ · · · · · αn−→ ·

In this case, we set s(p) = s(α1) and t(p) = t(αn).

paths of length one = arrows; paths of length zero = vertices

(for i ∈ Q0, we associate a trivial path ei .)

The path algebra kQ: k-basis = paths in Q, the multiplication

= concatenation of paths. More precisely, for two paths p and

q in Q, p · q = pq if s(p) = t(q), otherwise, p · q = 0.For example, eiej = δi ,jei , eip = δi ,t(p)p, pei = δs(p),ip.

Path algebras

· α1−→ · α2−→ · · · · · αn−→ ·

Path algebras

· α1−→ · α2−→ · · · · · αn−→ ·

Path algebras

· α1−→ · α2−→ · · · · · αn−→ ·

= concatenation of paths.

More precisely, for two paths p and

Path algebras

· α1−→ · α2−→ · · · · · αn−→ ·

q in Q, p · q = pq if s(p) = t(q), otherwise, p · q = 0.

For example, eiej = δi ,jei , eip = δi ,t(p)p, pei = δs(p),ip.

Path algebras

· α1−→ · α2−→ · · · · · αn−→ ·

Path algebras, continued

Qn = the set of paths in Q of length n.

Then kQ =⊕

n≥0 kQn is naturally N-graded.

The unit 1kQ =∑

i∈Q0 ei has a decomposition into pairwise

orthogonal idempotents.

Set J =⊕

n≥1 kQn, the two-sided ideal of kQ generated by

arrows.

Then kQ =⊕

The unit 1kQ =∑

Set J =⊕

arrows.

Then kQ =⊕

The unit 1kQ =∑

Set J =⊕

arrows.

Then kQ =⊕

The unit 1kQ =∑

Set J =⊕

arrows.

On the terminology “quiver”

Representations of quivers, the same as modules over path

algebras, were initiated by a classical 1972 paper of Peter

Gabriel.

Over an algebraically closed field, any finite dimensional

algebra is Morita equivalent to kQ/I for some ideal

Jd ⊆ I ⊆ J2 (whose Jacobson radical is J/I !)

Quivers appear naturally in Lie theory, quantum groups,

cluster theory, noncommutative geometry, mathematical

physics ...

Gabriel.

physics ...

Gabriel.

physics ...

The Leavitt path algebra, the very definition

Q̄ = the double quiver of Q, that is, for each arrow α : i → jin Q, we add a new arrow α∗ : j → i .

Definition (Abrams-Aranda Pino 2005/Ara-Moreno-Pardo 2007)

The Leavitt path algebra L(Q) of Q is the quotient algebra of kQ̄

by the two-sided ideal generated by the following elements

(CK1) αβ∗ − δα,βet(α), for all α, β ∈ Q1;

(CK2)∑{α∈Q1 | s(α)=i} α

∗α− ei , for all i ∈ Q0.

Here, CK stands for Cuntz-Krieger.

(CK2)∑{α∈Q1 | s(α)=i} α

Example: The Leavitt algebra

Example

Let Q be the rose quiver with two petals. Then we have an

isomorphism

L(Q) ' k〈x1, x2, y1, y2〉〈xiyj − δi ,j , y1x1 + y2x2 − 1〉

.

The latter algebra is called the Leavitt algebra L2 of order two.

Example

isomorphism

.

Example

isomorphism

.

Nice properties of the Leavitt path algebra

The Leavitt path algebra L(Q) is naturally Z-graded asL(Q) =

⊕n∈Z L(Q)n with ei ∈ L(Q)0, α ∈ L(Q)1 and

α∗ ∈ L(Q)−1.

L(Q)n · L(Q)m = L(Q)n+m, that is, L(Q) is strongly graded.

The zeroth component subalgebra L(Q)0 is a direct limit of

finite products of full matrix algebras; in particular, it is von

Neumann regular.

α∗ ∈ L(Q)−1.

Neumann regular.

α∗ ∈ L(Q)−1.

Neumann regular.

Some consequences

Consider the category L(Q)-grproj of finitely generated

Z-graded projective L(Q)-modules.

Proposition

The category L(Q)-grproj is a semisimple abelian category.

The proof: strongly gradation implies that

L(Q)-grproj ' L(Q)0-proj.

Now, use the von Neumann regularity of L(Q)0.

Some consequences

Consider the category L(Q)-grproj of finitely generated

Z-graded projective L(Q)-modules.

Proposition

The category L(Q)-grproj is a semisimple abelian category.

The proof: strongly gradation implies that

L(Q)-grproj ' L(Q)0-proj.

Now, use the von Neumann regularity of L(Q)0.

Some consequences, continued

The degree-shift functor on L(Q)-grproj: P 7→ P(1), whereP(1)m = Pm+1.

Proposition

In L(Q)-grproj, there is an isomorphism

(L(Q)ei )(1) '⊕

{α∈Q1 | s(α)=i}

L(Q)et(α)

The map sends ei to∑α∗, and the inverse sends et(α) to α. Use

the CK relations!

Proposition

(L(Q)ei )(1) '⊕

{α∈Q1 | s(α)=i}

L(Q)et(α)

the CK relations!

Proposition

(L(Q)ei )(1) '⊕

{α∈Q1 | s(α)=i}

L(Q)et(α)

the CK relations!

The content

The stable module category

Let A be a finite dimensional algebra

A-proj ⊆ A-mod

The stable module category A-mod = A-mod/[A-proj]: kills

morphisms factoring through projective

A-proj ⊆ A-mod

The stable module category, continued

The stable module category A-mod is left triangulated

The syzygy functor Ω: A-mod −→ A-mod (usually not anequivalence!)

Short exact sequences induce exact triangles:

Ω(N)

��

// P(N)

��

// N

L // M // N

Ω(N)

��

// P(N)

��

// N

L // M // N

Ω(N)

��

// P(N)

��

// N

L // M // N

Our concerns: radical-square-zero algebras

For a quiver Q, we set AQ = kQ/J2 to be the quotient

algebra, which is finite dimensional and radical-square-zero.

Indeed, AQ has a basis {ei | i ∈ Q0} ∪ {α | α ∈ Q1}, themultiplication rule is given by

eiej = δi ,jei , eiα = δi ,t(α)α, βej = δs(β),jβ, αβ = 0.

Examples

Example

Let Q be the rose quiver with two petals. Then kQ ' k〈α, β〉 thefree algebra with two variables, and AQ is a three dimensional

algebra with basis {1 = e1, α, β}.

Example

Let Q ′ be the quiver as above. Then AQ′ is a six dimensional

algebra with basis {e1, e2, α, β, γ, δ}, such that 1 = e1 + e2 is theunit.

Examples

Example

Let Q be the rose quiver with two petals. Then kQ ' k〈α, β〉 thefree algebra with two variables, and AQ is a three dimensional

algebra with basis {1 = e1, α, β}.

Example

Let Q ′ be the quiver as above. Then AQ′ is a six dimensional

algebra with basis {e1, e2, α, β, γ, δ}, such that 1 = e1 + e2 is theunit.

Examples, continued

Example

A module over AQ for the rose quiver Q takes the form

V1 VβiiVα 55

V1 a vector space, linear maps Vα and Vβ with zero relations.

Example

A module over AQ′ takes the form

V1Vα 55

Vγ //V2

Vδoo Vβii

Remarks: classical representation theory concerns indecomposable

modules over AQ and AQ′ .

Examples, continued

Example

V1 VβiiVα 55

Example

V1Vα 55

Vγ //V2

Vδoo Vβii

modules over AQ and AQ′ .

Examples, continued

Example

V1 VβiiVα 55

Example

V1Vα 55

Vγ //V2

Vδoo Vβii

modules over AQ and AQ′ .Xiao-Wu Chen, USTC LPA via Singularity Category

The syzygy modules

for each i ∈ Q0, we have the one-dimensional simpleAQ-module Si and the projective module Pi = AQei

We have a short exact sequence

0 −→⊕

{α∈Q1 | s(α)=i}

St(α) −→ Pi −→ Si −→ 0.

Therefore, we have an isomorphism

Ω(Si ) '⊕

{α∈Q1 | s(α)=i}

St(α)

The SAME formula as in L(Q)-grproj!

The syzygy modules

0 −→⊕

{α∈Q1 | s(α)=i}

St(α) −→ Pi −→ Si −→ 0.

Ω(Si ) '⊕

{α∈Q1 | s(α)=i}

St(α)

The syzygy modules

0 −→⊕

{α∈Q1 | s(α)=i}

St(α) −→ Pi −→ Si −→ 0.

Ω(Si ) '⊕

{α∈Q1 | s(α)=i}

St(α)

Comparison between Ω and (1)

For the comparison, we recall

Proposition

(L(Q)ei )(1) '⊕

{α∈Q1 | s(α)=i}

L(Q)et(α)

The syzygy modules, continued

BUT, the categories AQ-mod and L(Q)-grproj are very

different!

The category AQ-mod is NOT semisimple abelian!

The syzygy functor Ω is NOT an equivalence, but the

degree-shift functor (1) is!

Use the stabilization S(AQ-mod) in the sense of [A. Heller1968]; also see [E. Spanier-G. Whitehead 1953]

different!

The first definition via the stabilization

Definition (Buchweitz 1986/Keller-Vossieck 1987/Beligiannis 2000)

For a finite dimensional algebra A, its singularity category is

defined to be

Dsg(A) = S(A-mod).

S(A-mod) is obtained from A-mod by formally inverting Ω!

More precisely, the objects are (M, n), with an A-module M

and n ∈ Z; the morphisms are givenHom((M, n), (L,m)) = colim HomA(Ω

i−n(M),Ωi−m(L))

Ω now becomes M = (M, 0) 7→ (M,−1), an automorphism onS(A-mod)! Then S(A-mod) is triangulated in the sense of[J.L. Verdier 1963]; compare [D. Puppe 1962].

defined to be

Dsg(A) = S(A-mod).

defined to be

Dsg(A) = S(A-mod).

defined to be

Dsg(A) = S(A-mod).

Ω now becomes M = (M, 0) 7→ (M,−1), an automorphism onS(A-mod)!

Then S(A-mod) is triangulated in the sense of[J.L. Verdier 1963]; compare [D. Puppe 1962].

defined to be

Dsg(A) = S(A-mod).

A first link

Theorem (Smith 2012)

There is an equivalence (of triangulated categories)

Dsg(AQ) ' L(Q)-grproj

sending Si to L(Q)ei , with Ω corresponding to (1)!

The proof: use some argument of the stabilization in [C. 2011].

Remark: the suspension functor Σ corresponds to (−1).

Remark: constructing singular equivalences, this equivalence is

used to describe the singularity category of other algebras (trivial

extensions [C. 2016] and quadratic monomial algebras [C. 2018])

A first link

Examples revisited

Example

Q = ·1 βffα 88

Q ′ = 1·α 88γ //

2·δoo βff

Then L(Q) and L(Q ′) are graded Morita equivalent; (using the

strong shift equivalence in [P. Smith 2011], or [G. Abrmas-A.

Louly-E. Pardo-C. Smith 2011])

Alternatively, there is a singular equivalence Dsg(AQ) ' Dsg(AQ′)by [A.R. Nasr-Isfahani 2016] or [C. 2016/C.2018].

Examples revisited

Example

Q = ·1 βffα 88

Q ′ = 1·α 88γ //

2·δoo βff

Examples revisited

Example

Q = ·1 βffα 88

Q ′ = 1·α 88γ //

2·δoo βff

The second definition via Verdier quotient

the bounded derived category Db(A-mod): bounded

complexes of modules, with quasi-isomorphisms inverted

Key formula: ExtnA(M,N) = Hom(M,Σn(N))

Kb(A-proj) is a triangulated subcategory of Db(A-mod)

The second definition via Verdier quotient, continued

Fact: Kb(A-proj) = Db(A-mod) if and only if gl.dim(A)

The third definition and variations

The third definition of Dsg(A) via the Tate-Vogel cohomology

[Mislin 1994];

its relation to Gorenstein projective modules

The compact completion: the stable derived category of A [H.

Krause 2005] = the homotopy category Kac(A-Inj) of

unbounded acyclic complexes of (not necessarily finite

dimensional) injective A-modules; having arbitrary

coproducts, containing (in a NONTRIVIAL manner) Dsg(A)

as compacts.

The dg singularity category Sdg(A) [Keller 2018]: using dg

quotient of [B. Keller 1999] and [V. Drinfeld 2004]; a dg

category with its zeroth cohomology H0(Sdg(A)) = Dsg(A).

[Mislin 1994]; its relation to Gorenstein projective modules

as compacts.

dimensional) injective A-modules;

having arbitrary

as compacts.

The content

Completing Smith’s equivalence

Recall that L(Q) is Z-graded, viewed as a dg algebra with 0 diff.;D(L(Q)op) = der. cat. of (right) dg L(Q)-modules.

Completions:

Dsg(AQ) ⊆ Kac(AQ-Inj), and L(Q)-grproj ⊆ D(L(Q)op)

Theorem (C.-Yang 2015)

There is a triangle equivalence

Kac(AQ-Inj) ' D(L(Q)op),

whose restriction to compacts yields Smith’s equivalence.

The ingredients: Koszul duality between AQ and kQ, and the map

ι : kQ −→ L(Q)

is a universal localization in the sense of Cohen-Schofield.

Recall that L(Q) is Z-graded, viewed as a dg algebra with 0 diff.;D(L(Q)op) = der. cat. of (right) dg L(Q)-modules. Completions:

ι : kQ −→ L(Q)

is a universal localization in the sense of Cohen-Schofield.Xiao-Wu Chen, USTC LPA via Singularity Category

The compact generator

Theorem (Li, 2018)

There is an explicit complex I in Kac(AQ-Inj), which is a compact

generator, and whose dg endomorphism algebra is quasi-isomorphic

to L(Q).

The construction of I as a dg AQ-L(Q)-bimodule is inspired by the

basis in [A. Alahmadi-H. Alsulami-S.K. Jain-E. Zelmanov, 2012];

the existence of I implies (and actually enhances) C.-Yang’s

equivalence.

The compact generator

Theorem (Li, 2018)

There is an explicit complex I in Kac(AQ-Inj), which is a compact

generator, and whose dg endomorphism algebra is quasi-isomorphic

to L(Q).

The construction of I as a dg AQ-L(Q)-bimodule is inspired by the

basis in [A. Alahmadi-H. Alsulami-S.K. Jain-E. Zelmanov, 2012];

the existence of I implies (and actually enhances) C.-Yang’s

equivalence.

Enhancing Smith’s equivalence

The dg level contains more rigid information, for example, the

Hochschild cohomology.

Enhancements:

Dsg(AQ) Sdg(AQ) and L(Q)-grproj perdg(L(Q)op)

Proposition (C.-Li-Wang)

There is a zigzag of quasi-equivalences between

Sdg(AQ) ' perdg(L(Q)op).

Taking H0, we recover Smith’s equivalence.

The proof: enhance a result of [H. Krause 2005] and use H. Li’s

injective Leavitt complex.

Remark: Sdg(AQ) has the same Hochschild cohomology with L(Q).

Hochschild cohomology. Enhancements:

Remark: Sdg(AQ) has the same Hochschild cohomology with L(Q).Xiao-Wu Chen, USTC LPA via Singularity Category

The content

Keller’s theorem

The enveloping algebra Ae = A⊗ Aop

The singular Hochschild cohomology

HH∗sg(A,A) = HomDsg(Ae)(A,Σ∗(A))

Theorem (Keller 2018)

Assume that A/radA is separable. Then there is an isomorphism of

Z-graded algebras

HH∗(Sdg(A),Sdg(A)) ' HH∗sg(A,A).

Remark: important in Hua-Keller’s work on Donovan-Wemyss’s

conjecture.

Keller’s theorem

Z-graded algebras

conjecture.

Keller’s theorem

Z-graded algebras

conjecture.Xiao-Wu Chen, USTC LPA via Singularity Category

Singular Hochschild cochain complex

in the normalized bar resolution of A, we have Ωp, the

(graded) bimodule of noncommutative differential p-forms

There are natural maps

θp : C∗(A,Ωp) −→ C ∗(A,Ωp+1)

between the Hochschild cochain complexes.

The colimit, denoted by C ∗sg(A,A), is called the singular

Hochschild cochain complex of A; it computes HH∗sg(A,A).

Theorem (Wang 2018)

There is a natural B∞-algebra structure on C∗sg(A,A).

θp : C∗(A,Ωp) −→ C ∗(A,Ωp+1)

Theorem (Wang 2018)

θp : C∗(A,Ωp) −→ C ∗(A,Ωp+1)

Theorem (Wang 2018)

θp : C∗(A,Ωp) −→ C ∗(A,Ωp+1)

Theorem (Wang 2018)

A few words on B∞-algebras

1 The notion of a B∞-algebra is due to [E. Getzler-J.D.S. Jones,

1994].

2 Roughly speaking, a B∞-algebra B is a graded Poisson algebra

up to homotopy; its cohomology H0(B) is a Gerstenhaber

algebra.

3 a B∞-algebra is an A∞-algebra with µp,q with p, q ≥ 0.

4 Our concern: brace B∞-algebra, with dg algebra and µp,q = 0

for p > 1.

5 The Hochschild cochain complex C ∗(A,A) is a braceB∞-algebra, important in deformation theory of categories.

1994].

algebra.

for p > 1.

1994].

algebra.

for p > 1.

1994].

algebra.

for p > 1.

1994].

algebra.

for p > 1.

Keller’s conjecture

Two (brace) B∞-algebras for A: the classical one

C ∗(Sdg(A),Sdg(A)), and the singular one C∗sg(A,A)

Keller’s theorem says that they have the same cohomology

Conjecture (Keller 2018)

There is an isomorphism in the homotopy category of B∞-algebras

C ∗(Sdg(A),Sdg(A)) ' C ∗sg(A,A).

In particular, the isomorphism on the cohomology respects the

Gerstenhaber structures.

The radical-square-zero case

Theorem (C.-Li-Wang)

Assume that Q has no sinks. Then there are isomorphisms of

B∞-algebras

C ∗sg(AQ ,AQ)Υ−→ C ∗(L(Q), L(Q)) ∆−→ C ∗(Sdg(AQ),Sdg(AQ)).

The ingredients of ∆: the dg enhancement of Smith’s equivalence,

and the fact that C ∗(−,−) behaves well with respect toquasi-equivalences.

The radical-square-zero case

Assume that Q has no sinks. Then there are isomorphisms of

B∞-algebras

C ∗sg(AQ ,AQ)Υ−→ C ∗(L(Q), L(Q)) ∆−→ C ∗(Sdg(AQ),Sdg(AQ)).

The ingredients of ∆: the dg enhancement of Smith’s equivalence,

and the fact that C ∗(−,−) behaves well with respect toquasi-equivalences.

The ingredients of Υ

We introduce two new and explicit B∞-algebras:

(1) the combinatorial B∞-algebra C∗sg(Q,Q) via parallel paths

in Q

(2) the Leavitt B∞-algebra Ĉ∗(L, L), whose algebra structure

is a trivial extension of a subalgebra of L = L(Q)⊕i∈Q0 eiLei ⊕ s

−1 ⊕i∈Q0 eiLei

So, we have

C ∗sg(AQ ,AQ)κ−→ C ∗sg(Q,Q)

ρ−→ Ĉ ∗(L, L)

strict B∞-isomorphisms, where ρ sends a parallel path (p, q)

to q∗p ∈ L!

in Q

So, we have

ρ−→ Ĉ ∗(L, L)

to q∗p ∈ L!

in Q

So, we have

ρ−→ Ĉ ∗(L, L)

to q∗p ∈ L!

in Q

So, we have

ρ−→ Ĉ ∗(L, L)

to q∗p ∈ L!Xiao-Wu Chen, USTC LPA via Singularity Category

The ingredients of Υ, continued

an explicit bimodule projective resolution of L, together with a

homotopy deformation retract (in particular, L is quasi-free in

the sense of [J. Cuntz-D. Quillen 1995])

the homotopy transfer theorem for dg algebras yields an

A∞-quasi-morphism

(Φ1,Φ2, · · · ) : Ĉ ∗(L, L) −→ C ∗(L, L)

each Φi is explicit; by manipulation on brace B∞-algebras, we

verify that it is a B∞-morphism, as required.

A∞-quasi-morphism

(Φ1,Φ2, · · · ) : Ĉ ∗(L, L) −→ C ∗(L, L)

A∞-quasi-morphism

(Φ1,Φ2, · · · ) : Ĉ ∗(L, L) −→ C ∗(L, L)

The isomorphisms, in summary

The isomorphisms in the proof:

C ∗sg(AQ ,AQ)

��

κ //

Υ

**

C ∗sg(Q,Q)ρ // Ĉ ∗(L, L)

(Φ1,Φ2,··· )

��C ∗(Sdg(AQ),Sdg(AQ)) C

∗(L, L)∆

oo

Remark: ∆ is categorical and somehow implicit, while Υ is

combinatorial and explicit.

The isomorphisms, in summary

The isomorphisms in the proof:

C ∗sg(AQ ,AQ)

��

κ //

Υ

**

C ∗sg(Q,Q)ρ // Ĉ ∗(L, L)

(Φ1,Φ2,··· )

��C ∗(Sdg(AQ),Sdg(AQ)) C

∗(L, L)∆

oo

Remark: ∆ is categorical and somehow implicit, while Υ is

combinatorial and explicit.

Removing the sinks: an invariance theorem

Keller’s conjecture is invariant under one-point (co)extensions and

singular equivalences with levels. In particular, Keller’s conjecture

is invariant under derived equivalences.

Consequently, Keller’s conjecture holds for AQ = kQ/J2 with any

finite quiver Q (possibly with sinks)!

Removing the sinks: an invariance theorem

Keller’s conjecture is invariant under one-point (co)extensions and

singular equivalences with levels. In particular, Keller’s conjecture

is invariant under derived equivalences.

Consequently, Keller’s conjecture holds for AQ = kQ/J2 with any

finite quiver Q (possibly with sinks)!

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Thank You!

http://home.ustc.edu.cn/∼xwchen

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