Which canonical algebras are derived equivalent to incidence algebras of posets?

0.15 MB·English

by  Sefi Ladkani

WHICH CANONICAL ALGEBRAS ARE DERIVED
EQUIVALENT TO INCIDENCE ALGEBRAS OF POSETS?
SEFI LADKANI
Abstract. Wegiveafulldescriptionofallthecanonicalalgebras over
an algebraically closed field that are derived equivalent to incidence al-
8 gebras of finiteposets. These are the canonical algebras whose number
0
of weights is either 2 or 3.
0
2

n
This note concerns the characterization of the canonical algebras over an
a
J algebraically closed field that are derived equivalent to incidence algebras of

3 finite partially ordered sets (posets), expressed in the following theorem.

] Theorem. Let Λ be a canonical algebra of type (p,λλλ) over an algebraically
T
closed field. Then Λ is derived equivalent to an incidence algebra of a poset
R
if and only if the number of weights of p is either 2 or 3.
.
h
t
a This theorem can be interpreted both geometrically and algebraically.
m From ageometric viewpoint, byconsideringmodulesover incidencealgebras
[ as sheaves over finite spaces [8] and using the derived equivalence between

the categories of modules over a canonical algebra and coherent sheaves
2
v over a weighted projective line [3], we are able to obtain explicit derived
2 equivalences between the categories of sheaves of finite dimensional vector
1
spacesovercertain finiteT topological spacesandthecategories ofcoherent
4 0
1 sheaves over certain weighted projective lines.
. Fromanalgebraicviewpoint,inanattempttoclassifyallpiecewisehered-
8
0 itaryincidencealgebrasoveranalgebraicallyclosedfield,onefirstaskswhich
7 types of piecewise hereditary categories can actually occur. Happel’s classi-
0
fication [6] tells us that we only need to consider the canonical algebras and
:
v pathalgebras of quivers. For thecanonical algebras thetheorem abovegives
i
X a complete answer, while for path algebras, see the remarks in Section 2.2.
r We finally note that for the constructions of incidence algebras derived
a
equivalent to canonical algebras, the assumption that the base field is alge-
braically closed can be omitted.
Acknowledgement. I would like to thank Helmut Lenzing for useful dis-
cussions related to this paper, and the referee for the helpful comments.
0. Notations
The canonical algebras were introduced in [11]. Let k be a field, p =
(p ,...,p ) be a sequence of t ≥ 2 positive integers (weights), and λλλ =
1 t
(λ ,...,λ ) be a sequence of pairwise distinct elements of k \ {0}. The
3 t
canonical algebra of type (p,λλλ), is the algebra Λ(p,λλλ) = kQ/I where Q is
1
2 SEFILADKANI
the quiver
•1,1 x1 // •1,2 x1 // ... // •1,p1−1
•0 zxz1xzz2zz// ==•2,1 x2 // •2,2 x2 // ... // •2,p2−H1HHxH2xH1HH$$// •ω
4 BB
4 (cid:6)
4 (cid:6)
x4t444444... ... ... ... (cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)x(cid:6)t(cid:6)
4 (cid:6)
•(cid:25)(cid:25)t,1 xt // •t,2 xt // ... // •t,pt−(cid:6)1
and I is the ideal in the path algebra kQ generated by the following linear
combinations of paths from 0 to ω:
I = hxpi −xp2 +λ xp1 : 3 ≤ i ≤ ti
i 2 i 1
As noted in [11], as long as t ≥ 3, one can omit weights equal to 1, and
when t = 2, the ideal I vanishes and the canonical algebra is equal to the
path algebra of Q. In the latter case, one usually writes only the weights
greater than 1, in particular the algebra of type () equals the path algebra
of the Kronecker quiver. Hence when speaking on the number of weights,
we shall always mean the number of p with p ≥ 2.
i i
Let X be a finite partially ordered set (poset). The incidence algebra kX
is the k-algebra spanned by the elements e for the pairs x≤ y in X, with
xy
the multiplication defined by setting exyey′z = exz if y = y′ and exyey′z = 0
otherwise.
A k-diagram F over X consists of finite dimensional vector spaces F(x)
for x ∈ X, together with linear transformations rxx′ : F(x) → F(x′) for all
x ≤ x′, satisfying the conditions rxx = 1F(x) and rxx′′ = rx′x′′rxx′ for all
x ≤ x′ ≤ x′′. The category of finite dimensional right modules over kX is
equivalent to the category of k-diagrams over X, see [8].
Forafinite-dimensionalalgebraΛoverk,wedenotebyCb(Λ)thecategory
ofboundedcomplexesof(right)finite-dimensionalΛ-modules,andbyDb(Λ)
theboundedderivedcategory. ThealgebraΛispiecewise hereditary ifDb(Λ)
is equivalent as triangulated category to Db(H) for a hereditary k-category
H.
1. The necessity of the condition t ≤ 3 in the Theorem
For a k-algebra Λ, denote by HHi(Λ) the i-th Hochschild cohomology of
Λ, which equals Exti (Λ,Λ), where Λ is considered as a Λ-Λ-bimodule
Λ⊗Λop
in the natural way.
Proposition 1.1. Let X be a poset such that kX is piecewise hereditary.
Then HHi(kX) = 0 for any i > 1.
Proof. First, since for any two finite-dimensional k-algebras Λ , Λ we have
1 2
HHi(Λ ⊕Λ ) = HHi(Λ )⊕HHi(Λ ), we may assume that X is connected,
1 2 1 2
as the decomposition of X into connected components X = ⊔r X induces
i=1 i
r
a decomposition of the incidence algebra kX = kX .
i=1 i
LetkX betheconstant diagram on X, definedLbykX(x) = k forall x∈ X
with all maps being the identity on k. By [8, Corollary 3.20], HHi(kX) =
Exti (k ,k ).
X X X
CANONICAL ALGEBRAS DERIVED EQUIVALENT TO INCIDENCE ALGEBRAS 3
(2.1) •1,1 // ... // •1,p1−2 // •1,p1−1 3 ≤ p1 ≤ p2 ≤ p3
•0zDzDDzzDzDzD// ==!!•2,1 // ... // •2,p2−;;2;qq;qq;qq;qq;qq;;qq;qq//;qq•;;88882;(cid:29)(cid:29),p2−Hv1HvHvHvHvHvHv$$//:: •ω
•3,1 // ... // •3,p3−2 // •3,p3−1
(2.2) qqqqqqqq88 •1,1HHHHHHH$$ p1 = 2 < p2 ≤ p3
•0DDDDDD// !!•2,1 // ... // •2,p2−2qqqqqqq//q•882,p2−v1vvvvvv//:: •ω
•3,1 // ... // •3,p3−2 // •3,p3−1
(2.3) •0 eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee22 ••21,,11HHHHHHH//$$ •ω p1 = p2 = 2< p3
DDDDDD!! qqqqqqqq88 vvvvvvv::
•3,1 // ... // •3,p3−2 // •3,p3−1
(2.4) •1,1 p1 = p2 = p3 = 2
zzzzz== DDDDDD
z ""
•0 // •2,1 // •ω
DDD zz<<
D z
D z
D z
!! z
•
3,1
Figure 1. Hasse diagrams of posets derived equivalent to
canonical algebras of type (p ,p ,p ).
1 2 3
Since X is connected, k is indecomposable, and by [4, IV (1.9)], the
X
groups Exti (k ,k ) vanish for i >1, hence HHi(kX) = 0 for i > 1. (cid:3)
X X X
Corollary 1.2. Let k be algebraically closed and let Λ be a canonical algebra
over k of type (p,λλλ) where p = (p ,...,p ). If Db(Λ) ≃ Db(kX) for some
1 t
poset X, then t ≤ 3.
Proof. Assume that t ≥ 4. Then Λ is not of domestic type and by [5,
Theorem 2.4], dim HH2(kX) = t −3, a contradiction to Proposition 1.1.
k
Therefore t ≤ 3. (cid:3)
2. Constructions of posets from canonical algebras
2.1. The case t = 3. Recall that when t = 3, the canonical algebra Λ(p,λλλ)
is independent of the parameter λ , so we may assume that λ = 1, and
3 3
denote the algebra by Λ(p). Let p = (p ,p ,p ) be a triplet of weights with
1 2 3
2 ≤ p ≤ p ≤ p . Attach to p a poset X whose Hasse diagram is shown in
1 2 3 p
Figure 1. Explicitly, use the Hasse diagram of (2.1) if p > 2, (2.2) if p = 2
1 1
and p > 2, (2.3) if p = 2 and p > 2 and (2.4) if p = 2.
2 2 3 3
4 SEFILADKANI
Theorem 2.1. Let k be a field and let p= (p ,p ,p ). Then Λ(p) is derived
1 2 3
equivalent to the incidence algebra of X .
p
Proof. The idea of the proof relies on the notion of a formula introduced
in [7], and we refer to that paper for more details. We shall first construct
a functor F : Cb(kX ) → Cb(Λ(p)) that induces a triangulated functor
p
F : Db(kX ) → Db(Λ(p)), and then prove that F is an equivalence.
p
Note that similarly to the identification in [7] of complexes of diagrams
e e
with diagrams of complexes, we may identify complexes of modules over the
canonical algebra with a (non-commutative) diagram of complexes of vector
spaces satisfying the canonical algebra relations.
For a morphism f : K → L of complexes K = (Ki,di ), L = (Li,di )
K L
f
of vector spaces, denote by C(K −→ L) the cone of f. Recall that C(K →
L)i = Ki+1⊕Li, with the differential acting as the matrix
−di+1 0
K
(cid:18) fi+1 di (cid:19)
L
by viewing the terms as column vectors of length two. Denote by [1] the
shift operator, that is, K[1]i = Ki+1 with di = −di+1.
K[1] K
We will demonstrate the construction of F for the posets of type (2.1),
the other cases being similar. Let K , K and K for 1≤ i ≤ 3, 1 ≤ j < p
0 ω i,j i
e
be complexes in a commutative diagram
K1,1 // ... // K1,p1−2 x11 // K1,p1−1
K0DzDzDzDzDzDz// K""<< 2,1 // ... // K2,p2−::2:rr:xrr:rr2x:xrr2:32rr:21:rr:rr//:rrK:8888::x2(cid:29)(cid:29),1p32−Iv1IvIvyIv2yyIv13IvIv//$$:: Kω
K3,1 // ... // K3,p3−2 x33 // K3,p3−1
Let L = K and L = K for 1≤ i ≤ 3 and 1 ≤ j < p −1. Define
0 0 i,j i,j i
(y1 y3)
L1,p1−1 = C(K1,p1−1⊕K3,p3−1 −−−−−→Kω)[−1]
(y2 y1)
L2,p2−1 = C(K2,p2−1⊕K1,p1−1 −−−−−→Kω)[−1]
(y3 y2)
L3,p3−1 = C(K3,p3−1⊕K2,p2−1 −−−−−→Kω)[−1]
(y1 y2 y3)
Lω = C(K1,p1−1⊕K2,p2−1⊕K3,p3−1 −−−−−−→Kω)[−1]
with the three maps
(x11 −x13 0)T
L1,p1−2 −−−−−−−−−→ L1,p1−1
(−x22 x21 0)T
L2,p2−2 −−−−−−−−−→ L2,p2−1
(x33 −x32 0)T
L3,p3−2 −−−−−−−−−→ L3,p3−1
and the three maps from Li,pi−1 to Lω being the canonical embeddings.
CANONICAL ALGEBRAS DERIVED EQUIVALENT TO INCIDENCE ALGEBRAS 5
Then the following is a (non-commutative) diagram of complexes
L1,1 // ... // L1,p1−2 // L1,p1−1
{{{{{{== HHHHHHH##
L0 CCCCCC//!!L2,1 // ... // L2,p2−2 // L2,p2−v1vvvvvv//;; Lω
L3,1 // ... // L3,p3−2 // L3,p3−1
that satisfies the canonical algebra relation, and we get the required functor
F which induces, by the general considerations in [7], the functor F.
ToprovethatF isanequivalence, weuseBeilinson’sLemma[2,Lemma1]
e
andverifythatforanytwosimpleobjectsS ,S (wherex,y ∈ X)andi ∈ Z,
x y
e
the functor F satisfies
eHom (S ,S [i]) ≃ Hom (FS ,FS [i])
Db(kXp) x y Db(Λ(p)) x y
andmoreover theimages FS generate Db(Λ(p)) aseatriaengulated category.
x
We omit the details of this verification. However, we just mention that
e
FS and FS , for 1 ≤ i ≤ 3 and 1 ≤ j < p −1, are the corresponding
0 i,j i
seimpleΛ(pe)-modules, whileFS1,p1−1,FS2,p2−1,FS3,p3−1 andFSω aregiven
by
e e e e
0 // ... // 0 // k 0 // ... // 0 // 0
(cid:2)(cid:2)(cid:2)(cid:2)@@ ====1(cid:30)(cid:30) (cid:1)(cid:1)(cid:1)(cid:1)@@ ====(cid:30)(cid:30)
0 // 0 // ... // 0 // k // k 0 // 0 // ... // 0 // k // k
====(cid:30)(cid:30) (cid:1)(cid:1)1(cid:1)(cid:1)@@ <<<<(cid:30)(cid:30) (cid:1)(cid:1)1(cid:1)(cid:1)1@@
0 // ... // 0 // 0 0 // ... // 0 // k
FS1,p1−1 FS2,p2−1
e e
0 // ... // 0 // k 0 // ... // 0 // k
(cid:2)(cid:2)(cid:2)(cid:2)@@ ====1(cid:30)(cid:30) (cid:2)(cid:2)(cid:2)(cid:2)@@ ====1(cid:30)(cid:30)
0 // 0 // ... // 0 // 0 // k 0 // 0 // ... // 0 // k // k [−1]
<<<<(cid:30)(cid:30) (cid:1)(cid:1)(cid:1)(cid:1)1@@ (cid:16) <<<<(cid:30)(cid:30) (cid:1)(cid:1)1(cid:1)(cid:1)1@@ (cid:17)
0 // ... // 0 // k 0 // ... // 0 // k
FS3,p3−1 FSω
(cid:3)
e e
Example 2.2 ([9, Example 18.6.2]). Let A2 be the quiver •1 // •2 and
let X = A ×A ×A . Thentheincidence algebra of X is derived equivalent
2 2 2
to the canonical algebra of type (3,3,3).
Remark 2.3. Observe that ω is the unique maximal element in the posets
whoseHassediagramsaregivenin(2.3)and(2.4). HencebytakingY = {ω}
inCorollary4.15of[8], werecover thefactthatthecanonicalalgebraoftype
(2,2,p) is derived equivalent to the path algebra of the extended Dynkin
quiver D .
p+2
Remarek 2.4. Similar applications of [7, Theorem 1.1] and its corollaries
for the posets in (2.2) show that the canonical algebra of type (2,p ,p )
2 3
6 SEFILADKANI
is derived equivalent to the incidence algebras of the posets whose Hasse
diagrams are given by
•1 ... •p2−2 // • oo •p3−2 ... •1 p2,p3 ≥ 3
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
• // • oo •
•
~~~~~~>>•1 // ... // •p2−2FFFFFFF## • p2,p3 ≥ 3
• • •
@@@@@@ xxxxxxx;;
•1 // ... // •p3−2 •
• p ,p ≥ 2
•1 ... •p2−1{{xxxxxxx•(cid:15)(cid:15) FFFFFFF•##p3−1 ... •1 2 3
FFFFFFF## (cid:15)(cid:15) {{xxxxxxx
•
where edges without arrows can be oriented arbitrarily.
2.2. Remarksonpathalgebras. LetQbeafinitequiverwithoutoriented
cycles. Thesetofitsvertices Q hasanaturalpartialorderdefinedbyx≤ y
0
for two vertices x,y ∈ Q if there exists an oriented path from x to y. When
0
Q has the property that any two vertices are connected by at most one
oriented path, the path algebra kQ is isomorphic to the incidence algebra
of the poset (Q ,≤). A partial converse is given by the following lemma.
0
Lemma 2.5 ([10, Theorem 4.2]). Let X be a poset. Then gl.dimkX ≤ 1 if
and only if any two points in the Hasse diagram of X are connected by at
most one path.
For two quivers Q and Q′, we denote Q ∼ Q′ if Q′ can be obtained from
Q by applying a sequence of BGP reflections (at sources or sinks), see [4,
(I.5.7)]. Since BGP reflections preserve the derived equivalence class, we
conclude that if Q is a quiver such that Q ∼ Q′ for a quiver Q′ having the
property that any two vertices are connected by at most one oriented path,
then the path algebra kQ is derived equivalent to an incidence algebra of a
poset.
2.3. The case t = 2. When the number of weights is at most 2, the corre-
sponding canonical algebra is a path algebra of a quiver, however there are
two distinct paths from the source 0 to the sink ω.
When the weight type is (p ,p ) (with p ,p ≥ 2), we can overcome this
1 2 1 2
problem by applying a BGP reflection at the sink ω. The resulting quiver
CANONICAL ALGEBRAS DERIVED EQUIVALENT TO INCIDENCE ALGEBRAS 7
•0 // •p
AAAA yyy<<
AA yyy
•1 // •2 // ... // •p−1
Figure 2. The quiver A – not derived equivalent to any
1,p
incidence algebra.
e
is shown below,
•1,1 // •1,2 // ... // •1,p1−1 2 ≤ p1,p2
zzzzzz== ddHHHHHHH
•0 // •2,1 // •2,2 // ... // •2,p2−1 oo •ω
and its path algebra is an incidence algebra derived equivalent to the canon-
ical algebra of type (p ,p ).
1 2
In the remaining case, where the weight type is either (p) or (), the
corresponding canonical algebra equals the path algebra of the quiver A
1,p
drawn in Figure 2, and we show the following.
e
Proposition 2.6. Let p ≥ 1 and let k be algebraically closed. Then there is
no poset whose incidence algebra is derived equivalent to the path algebra of
the quiver A .
1,p
Proof. AssuemethatthereexistsaposetX suchthatkX isderivedequivalent
to the path algebra of A .
1,p
If gl.dimkX ≤ 1, then by Lemma 2.5, the algebra kX equals the path
e
algebra of its Hasse diagram Q, thus by [4, (I.5.7)], Q is obtained from A
1,p
by a sequence of BGP reflections. But this is impossible since the only
e
possible reflections are at 0 and p, and they give quivers isomorphic to A .
1,p
However, A is not the Hasse diagram of any poset.
1,p e
Hence gl.dimkX ≥ 2. By Lemma 2.5, there exists at least one commu-
e
tativity relation in the quiver Q, so that kX is not a gentle algebra. This
is again impossible since the path algebra of A is gentle and the property
1,p
of an algebra being gentle is invariant under derived equivalence [12]. Al-
e
ternatively, one can use the explicit characterization in [1] of iterated tilted
algebras of type A˜ . (cid:3)
n
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Einstein Institute of Mathematics, The Hebrew University of Jerusalem,
Jerusalem 91904, Israel
E-mail address: [email protected]