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.
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.
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.
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
The content
Quivers and Leavitt path algebras
An introduction to singularity categories
Other links via categorical equivalences
Keller’s conjecture on singular Hochschild cohomology
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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, we
assume that Q has no sinks.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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 ...
Xiao-Wu Chen, USTC LPA via Singularity Category
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 ...
Xiao-Wu Chen, USTC LPA via Singularity Category
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 ...
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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!
Xiao-Wu Chen, USTC LPA via Singularity Category
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!
Xiao-Wu Chen, USTC LPA via Singularity Category
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!
Xiao-Wu Chen, USTC LPA via Singularity Category
The content
Quivers and Leavitt path algebras
An introduction to singularity categories
Other links via categorical equivalences
Keller’s conjecture on singular Hochschild cohomology
Xiao-Wu Chen, USTC LPA via Singularity Category
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
Xiao-Wu Chen, USTC LPA via Singularity Category
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
Xiao-Wu Chen, USTC LPA via Singularity Category
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
Xiao-Wu Chen, USTC LPA via Singularity Category
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
Xiao-Wu Chen, USTC LPA via Singularity Category
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
Xiao-Wu Chen, USTC LPA via Singularity Category
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
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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′ .
Xiao-Wu Chen, USTC LPA via Singularity Category
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′ .
Xiao-Wu Chen, USTC LPA via Singularity Category
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′ .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!
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!
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!
Xiao-Wu Chen, USTC LPA via Singularity Category
Comparison between Ω and (1)
For the comparison, we recall
Proposition
In L(Q)-grproj, there is an isomorphism
(L(Q)ei )(1) '⊕
{α∈Q1 | s(α)=i}
L(Q)et(α)
Xiao-Wu Chen, USTC LPA via Singularity Category
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]
Xiao-Wu Chen, USTC LPA via Singularity Category
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]
Xiao-Wu Chen, USTC LPA via Singularity Category
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]
Xiao-Wu Chen, USTC LPA via Singularity Category
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]
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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])
Xiao-Wu Chen, USTC LPA via Singularity Category
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])
Xiao-Wu Chen, USTC LPA via Singularity Category
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])
Xiao-Wu Chen, USTC LPA via Singularity Category
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])
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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].
Xiao-Wu Chen, USTC LPA via Singularity Category
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)
Xiao-Wu Chen, USTC LPA via Singularity Category
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)
Xiao-Wu Chen, USTC LPA via Singularity Category
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)
Xiao-Wu Chen, USTC LPA via Singularity Category
The second definition via Verdier quotient, continued
Fact: Kb(A-proj) = Db(A-mod) if and only if gl.dim(A)
The second definition via Verdier quotient, continued
Fact: Kb(A-proj) = Db(A-mod) if and only if gl.dim(A)
The second definition via Verdier quotient, continued
Fact: Kb(A-proj) = Db(A-mod) if and only if gl.dim(A)
The second definition via Verdier quotient, continued
Fact: Kb(A-proj) = Db(A-mod) if and only if gl.dim(A)
The second definition via Verdier quotient, continued
Fact: Kb(A-proj) = Db(A-mod) if and only if gl.dim(A)
The second definition via Verdier quotient, continued
Fact: Kb(A-proj) = Db(A-mod) if and only if gl.dim(A)
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
The content
Quivers and Leavitt path algebras
An introduction to singularity categories
Other links via categorical equivalences
Keller’s conjecture on singular Hochschild cohomology
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.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.
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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).Xiao-Wu Chen, USTC LPA via Singularity Category
The content
Quivers and Leavitt path algebras
An introduction to singularity categories
Other links via categorical equivalences
Keller’s conjecture on singular Hochschild cohomology
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.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).
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).
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).
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).
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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!
Xiao-Wu Chen, USTC LPA via Singularity Category
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!
Xiao-Wu Chen, USTC LPA via Singularity Category
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!
Xiao-Wu Chen, USTC LPA via Singularity Category
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!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.
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.
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
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.
Xiao-Wu Chen, USTC LPA via Singularity Category
Removing the sinks: an invariance theorem
Theorem (C.-Li-Wang)
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)!
Xiao-Wu Chen, USTC LPA via Singularity Category
Removing the sinks: an invariance theorem
Theorem (C.-Li-Wang)
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)!
Xiao-Wu Chen, USTC LPA via Singularity Category
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Xiao-Wu Chen, USTC LPA via Singularity Category
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Theor. 21 (2018), 833-858.Xiao-Wu Chen, USTC LPA via Singularity Category
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Xiao-Wu Chen, USTC LPA via Singularity Category
Thank You!
http://home.ustc.edu.cn/∼xwchen
Xiao-Wu Chen, USTC LPA via Singularity Category