Perverse Morita equivalences are everywhere
Perverse Morita equivalences
are everywhere
Sefi Ladkani
Institut des Hautes Études Scientifiques, Bures-sur-Yvette
http://www.ihes.fr/~sefil/
1
Perverse Morita equivalences are everywhere
Perspective
Triangulated and derived categories can relate objects of different
nature:
• Coherent sheaves over algebraic varieties and modules over noncommutative algebras [Beilinson 1978, Kapranov 1988]
• Homological mirror symmetry conjecture [Kontsevich 1994]
. . . but also relate non-isomorphic objects of the same nature:
• Morita theory for derived categories of modules [Rickard 1989]
• Derived categories of coherent sheaves [Bondal-Orlov 2002]
• Broué’s conjecture on blocks of group algebras [Broué 1990]
2
Perverse Morita equivalences are everywhere
Derived equivalence
Theorem [Rickard 1989]. Let R, S be rings. Then
D(Mod R) ' D(Mod S)
(R, S are derived equivalent, R ∼ S)
if and only if there exists a tilting complex T ∈ D(Mod R):
• exceptional: HomD(Mod R)(T, T [i]) = 0 for i 6= 0,
• compact generator : hadd T i = per R,
such that S ' EndD(Mod R)(T ).
Problems.
• No decision process.
• No constructive method.
3
Perverse Morita equivalences are everywhere
Example – BGP Reflections at sinks/sources
Q – quiver without oriented cycles,
s – sink in Q, i.e. no outgoing arrows from s.
σsQ – the BGP reflection with respect to s, obtained from Q by
inverting all arrows incident to s, so that s becomes a source.
Theorem [Bernstein-Gelfand-Ponomarev]. KQ ∼ KσsQ.
Example.
•
>
~~
~
~~
~~
• @@@
@@
@@
∼
•
•
>
~~
~
~~
~~
• `@@@
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@@
•
Remark. Generalized by [Auslander-Platzeck-Reiten] to sinks in quivers
of arbitrary finite-dimensional algebras.
4
Perverse Morita equivalences are everywhere
Outline
BGP reflection is a combinatorial, local operation at sinks/sources
producing derived equivalences for path algebras of quivers.
We will present generalizations for
• Arbitrary finite-dimensional algebras,
• at arbitrary vertices;
and explore their connections with
• Perverse Morita equivalences [Chuang-Rouquier],
• Quiver mutation [Fomin-Zelevinsky 2002].
5
Perverse Morita equivalences are everywhere
From vertices to complexes
K – algebraically closed field,
A = KQ/I – quiver with relations,
vertex i
projective Pi,
arrow i → j
map Pj → Pi
k – vertex in Q without loops,
M
M
−
Tk = Pk →
Pj ⊕
Pi,
j→k
i6=k
Tk+ =
M
k→j
Pj → Pk ⊕
M
Pi
i6=k
Are these tilting complexes?
• Always compact generators,
• Exceptionality is expressed in terms of the combinatorial data.
6
Perverse Morita equivalences are everywhere
Mutations of algebras
If Tk− is a tilting complex, the negative mutation at k is defined as
−
µ−
(A)
=
End
(T
D(A) k )
k
If Tk+ is a tilting complex, the positive mutation at k is defined as
+
µ+
(A)
=
End
(T
D(A)
k
k )
• There are up to two mutations at a vertex,
• Mutations yield derived equivalent algebras,
• Closely related to the Brenner-Butler tilting modules,
• Mutations are perverse Morita equivalences [Chuang-Rouquier].
7
Perverse Morita equivalences are everywhere
Mutations of algebras – Example
•2 D
A=
•1
µ−
1 (A) is not defined
µ−
2 (A) =
µ−
3 (A) =
•2
•1
zz
zz
z
z
z| z
/
•3
•2 bD
•1
<
zz
zz
z
z
zz
DD
DD
DD
D
•3
z<
zz
z
zz
zz
DD
DD
DD
D"
•3
µ+
1 (A) =
µ+
2 (A) =
•2 D
•1
z
zz
zz
z
z| z
DD
DD
DD
D"
•3
•2 bD
•1
DD
DD
DD
D
/
•3
µ+
3 (A) is not defined
−
0 ) nor µ+ (A0 ) are defined.
Remark. For A0 = µ−
(A),
neither
µ
(A
2
1
1
8
Perverse Morita equivalences are everywhere
Brenner-Butler tilting and algebra mutations
−
TkBB = τA
Sk ⊕
M
Pi
i6=k
If TkBB is a tilting module, the BB-mutation at k is defined as
BB )
µBB
(A)
=
End
(T
A k
k
• BB-mutation is the negative mutation,
• Under mild conditions, the converse holds.
n-BB-tilting [Hu-Xi 2008] and n-APR-tilting [Iyama-Oppermann 2009] can
be written as composition of n negative mutations at the same vertex
A0 ∼ A
n−BB
⇒
−
−
A0 ' µ−
µ
.
.
.
µ
k k
k (A)
9
Perverse Morita equivalences are everywhere
Perverse Morita equivalences
[Chuang-Rouquier]
∼
F : Db(A) −
→ Db(A0) is a perverse Morita equivalence if there are:
• Filtrations of the simples φ ⊂ S1 ⊂ · · · ⊂ Sr , φ ⊂ S10 ⊂ · · · ⊂ Sr0 ,
• Perversity function p : {1, . . . , r} → Z,
such that F induces equivalences
b (A) ' D b (A0 )
F : DA
A0
i
i
F [−p(i)] : Ai/Ai−1 ' A0i/A0i−1
where Ai, A0i are the Serre subcategories generated by Si, Si0.
∼
−
b(A) −
b(µ− (A)) is
When µ−
(A)
is
defined,
then
R
Hom
(T
,
−)
:
D
→
D
A k
k
k
a perverse Morita equivalence, with
• Filtration φ ⊂ {k} ⊂ {1, 2, . . . , n},
• Perversity {−1, 0}.
10
Perverse Morita equivalences are everywhere
Quiver mutation
[Fomin-Zelevinsky]
(
Q – quiver without loops ( • ) and 2-cycles ( • h
k – any vertex in Q.
• ),
The mutation of Q at k, denoted µk (Q), is obtained as follows:
α
[αβ]
β
1. For any pair i −
→k−
→ j, add new arrow i −−−→ j,
2. Invert the incoming and outgoing arrows at k,
3. Remove a maximal set of 2-cycles.
Example. > • @
@@
~~
•
•
~
~~
~~
~
~~
~
~
~~
~
@@
@@
1
−
→
•
• `@@@
@@
@@
/
•
1
−
→
•
•
>
~~
~
~~
~~
~
~~
~
~
~~ ~
g
• @@@
@@
@@
/
2
−
→
•
• `@@@
@@
@@
/
•
2
−
→
•
•
~
~~
~
~
~~ ~
>
~~
~
~~
~~
g
• `@@@
@@
@@
/
•
• @@@
@@
@@
/
3
−
→
•
•
>
~~
~
~~
~~
• @@@
@@
@@
•
11
Perverse Morita equivalences are everywhere
Quiver mutation – matrix version
Q
BQ, via (BQ)ij = |{arrows j → i}| − |{arrows i → j}|, induces
{quivers, no loops and 2-cycles} ↔ {anti-symmetric integral matrices}
Lemma [FZ, Geiss-Leclerc-Schröer]. If Q has no loops and 2-cycles,
(rk−)T · BQ · rk− = Bµk (Q) = (rk+)T · BQ · rk+
where

1

rk− =  ∗

...

∗ −1 ∗ ∗ 
...
1
(rk−)kj = |{arrows j → k}|,

1

rk+ =  ∗

...
∗ −1 ∗
...

∗
1
(rk+)kj = |{arrows k → j}|,
(j 6= k).
12
Perverse Morita equivalences are everywhere
Mutations, Cartan matrices and Euler forms
CA – the Cartan matrix of A, defined by (CA)ij = dimK HomA(Pi, Pj ).
The bilinear form defined by CA is invariant under derived equivalence.
Lemma.
Cµ−(A) = rk− · CA · (rk−)T
k
C
µ+
k (A)
= rk+ · CA · (rk+)T
whenever the mutations are defined.
−T
When A has finite global dimension, its Euler form is cA = CA
, and
cµ−(A) = (rk−)T · cA · rk−
k
c
µ+
k (A)
= (rk+)T · cA · rk+
whenever the mutations are defined.
13
Perverse Morita equivalences are everywhere
Mutations of algebras as mutations of quivers
When the discrete data associated to an algebra – quiver and Euler
form, are “compatible”, mutation of algebras is mutation of quivers.
This happens for:
• Algebras of global dimension 2 ,
• Endomorphism algebras of cluster-tilting objects in stably 2-CY
Frobenius categories [Buan-Iyama-Reiten-Scott 2009, GLS, Palu 2009].
14
Perverse Morita equivalences are everywhere
Mutations of algebras of global dimension 2
A – finite-dimensional K-algebra of global dimension ≤ 2.
e [Assem-Brüstle-Schiffler 2008, Keller] has
The extended quiver Q
A
2
|{arrows i → j}| = dimK Ext1
A(Si, Sj ) + dimK ExtA(Sj , Si)
T
so that BQ
e A = cA − cA is the anti-symmetrization of cA.
e is without loops and 2-cycles.
Theorem [L]. Assume that Q
A
−
e
e
If µk (A) is defined and gl. dim µ−
k (A) ≤ 2, then Qµ− (A) = µk (QA).
k
+
e
e
If µ+
(A)
is
defined
and
gl.
dim
µ
k
k (A) ≤ 2, then Qµ+ (A) = µk (QA).
k
Remark. Not all quiver mutations correspond to algebra mutations.
15
Perverse Morita equivalences are everywhere
Example – Sequence of mutations
e
Theorem [L]. Q
e
−−→ and Q
K(An×A2n+1 ) are mutation equivalent.
Aus(A2n)
•O4 Eo E •O5 Eo E •O6
•
; 6 GG
GG
ww
ww
#
o
5
=
GG
; 11GG
GG
GG
ww
zz
w
z
z
#
#
w
o
o
4 DD
10
GG
;
; 15GG
DD
GG
GG
ww
ww
w
w
!
#
#
w
w
o
o
9
14
=
;
; 18GG
GG
GG
GG
GG
GG
ww
ww
zz
w
w
z
z
w
w
#
#
#
o
o
8 Do D
13
17
;
GG
;
GG
; 20GG
DD
GG
GG
GG
ww
ww
ww
w
w
w
!
w
#
w
#
w
#
12 o
16 o
19 o
21
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2@@o
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1o
7o
•
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•
•
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o •11
•O3 Eo E •10
O EE
O
•O2 Eo E •O9 Eo E •15
O
•O1 Eo E •O8 Eo E •14
O
o •18
•O7 Eo E •13
O EE
O
o •17 o •20
•12
O EE
O EE
O
•16 o •19 o •21
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 12, 13, 14, 15, 7, 8, 9, 10, 1, 2, 3, 4,
21, 19, 16, 20, 17, 12, 18, 13, 7, 21, 19, 16, 20, 17, 12, 21, 19, 16
16
Perverse Morita equivalences are everywhere
Mutations in (stably) 2-Calabi-Yau categories
C – Hom-finite, triangulated 2-CY or Frobenius stably 2-CY ,
U – cluster-tilting object in C [Buan-Marsh-Reineke-Reiten-Todorov, GLS]
Three notions of mutation:
U
Λ = EndC (U )
QΛ
µk (U )
mutation of CT objects,
Λ0 = EndC (µk (U ))
mutation of algebras?
QΛ0
mutation of quivers.
studied in [BIRSc, BIRSm, BMRRT 2006, GLS, Iyama-Yoshino 2008].
17
Perverse Morita equivalences are everywhere
Mutations in Frobenius stably 2-CY categories
In the Frobenius case, mutation of cluster-tilting objects leads to
mutation of their endomorphism algebras:
Theorem [L]. If QΛ and QΛ0 have no loops at k, then
+
−
Λ0 ' µBB
(Λ)
'
µ
(Λ)
'
µ
k
k
k (Λ)
• Generalizes [GLS],
• Builds on [Hu-Xi 2008],
• Reminiscent of [Iyama-Reiten 2008] for 3-CY algebras,
• The matrix cΛ is “partially” skew-symmetric [Keller-Reiten 2007].
18
Perverse Morita equivalences are everywhere
Mutations for 2-CY-tilted algebras
In the triangulated case, mutation of cluster-tilting objects does not
always lead to mutation of their endomorphism algebras:
•1
<
xx
xx
•2 F
FF
F"
µ2
•3
↔
•1
xx
x| x
•2 bF
FF
F
/
•3
•1
/
•2
<
xx
xx
o
•3 F
FF
F"
µ2
•4
↔
•1
<
xx
xx
o
•3 F
FF
F"
•2
/
•4
BB
BB 0op
Theorem [L]. Λ0 ' µBB
k (Λ) ⇐⇒ µk (Λ) and µk (Λ ) are defined.
• There is an effective algorithm that decides whether Λ0 ' µBB
k (Λ).
• Has been applied in [Bastian-Holm-L.] for the derived equivalence
classification of cluster-tilted algebras [BMR 2007] of various classes.
• Applicable also for finite-dimensional Jacobian algebras [Amiot 2009,
Derksen-Weyman-Zelevinsky 2008].
19
Perverse Morita equivalences are everywhere
More perverse Morita equivalences
• n-BB-tilting: φ ⊂ {k} ⊂ S, perversity {−n, 0}.
• Hughes-Waschbüsch reflection: φ ⊂ S \ {k} ⊂ S, perversity {−1, 0}.
• Example. φ ⊂ {2, 4, 6} ⊂ {1, 2, 3, 4, 5, 6}, perversity {−1, 0},
•1]:
•2 o
::
::
::
•6
•3:
::
::
::
•4
/
B
•5
•B 2
/
o
o
ooo
ooooo
woooo
]::OOOO
:: OOO
OOO
::
OOO
:
O'
o
•3
•4
•1
O :
::
::
::
•6
•5
Generally, for any φ ⊂ I ⊂ S and any perversity, one can construct a
complex such that if it is tilting, it induces perverse equivalence.
Question. Can any two derived equivalent algebras be connected by
a sequence of perverse Morita equivalences?
20