Annals of Representation Theory

Reineke, Markus

Markus Reineke
Wild quantum dilogarithm identities
Volume 1, issue 3 (2024), p. 385-391
https://doi.org/10.5802/art.12
Communicated by Henning Krause.
© The author, 2024
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C EN T R E
MER S ENN E

Annals of Representation Theory
Volume 1, issue 3 (2024), p. 385–391
https://doi.org/10.5802/art.12
Wild quantum dilogarithm identities
Markus Reineke
Abstract. We exhibit and discuss “wild” analogues of the five-term quantum dilogarithm identity.
We derive these from the representation theory of quivers, using motivic wall-crossing, the geometric-
ity of motivic Donaldson–Thomas invariants, and special properties of Kronecker moduli.
1. Introduction
The quantum dilogarithm is a q-series with many remarkable properties [16], including
the famous five-term identity [2]. Cluster algebra theory and wall-crossing of motivic
invariants of quivers have led to vast generalizations of such dilogarithm identities [4].
In this note, we explore the outer limits of this circle of ideas, by investigating “wild”
dilogarithm identities, those arising from wild quivers. For this we use again motivic wall-
crossing, interpret the resulting series in terms of motivic Donaldson–Thomas invariants,
and use the geometric interpretation of the latter in terms of intersection homology of
quiver moduli spaces [6] to establish very strong positivity properties. In the rank two
case, originating from generalized Kronecker quivers, the well-explored and very special
symmetries of Kronecker moduli yield many additional explicit properties of the individual
terms of our highly infinite dilogarithm identities; see Theorem 2.7.
To derive this identity, we collect in Section 3 the available material on wall-crossing
of motivic invariants (see [7] for an introduction), and adapt it to the present notation
and special setting in Section 4. Although similar approaches are used, for example, in
the context of the tropical vertex [3, 12], it it desirable to state the nature of such wild
identities as explicitly as possible.
Manuscript received 2023-03-17, revised 2024-05-17 and accepted 2024-06-24.
Keywords. quantum dilogarithm, quiver representation, quiver moduli, motivic Donaldson-Thomas
invariants.
2020 Mathematics Subject Classification. 16G20, 14D20, 14N35, 17B37.
This work is supported by the DFG CRC-TRR 191 “Symplectic structures in geometry, algebra and
dynamics44 (281071066).
385
386 Markus Reineke
2. Quantum dilogarithm identities
We first define the quantum dilogarithm, and state the classical five-term identity. The
coefficient ring Q(q1/2)[[x]], and in particular the twist by half-powers of q, will become
natural in the context of motivic Donaldson–Thomas invariants.
Definition 2.1. We define the quantum dilogarithm Φ(x) ∈ Q(q1/2)[[x]] as
Φ(x) =
∑
n ≥ 0
qn/2xn
(1− q) · . . . · (1− qn) = exp
∑
n ≥ 1
xn
n ·
(
q−n/2 − qn/2
)
 = ∏
n ≥ 0
1
1− qn+1/2 · x
.
Remark 2.2. The translation from the present definition to the ones in the literature is
straightforward. For example, [2, 14] use the definition (x; q)∞ = Φ(q−1/2x)−1, and [4]
uses E(x) = Φ(−qx). The classical Euler dilogarithm [16] arises as the following limit:
(
q−1/2 − q1/2
)
log Φ(x) =
∑
n ≥ 1
xn ·
(
q−1/2 − q1/2
)
n ·
(
q−n/2 − qn/2
) q→1−→ ∑
n ≥ 1
xn
n2
= Li2(x).
Definition 2.3. For a positive integer m, we define Q(q1/2)qm [[x, y]] as the skew formal
power series ring with skew commutativity relation xy = qmyx.
We can now formulate the five-term quantum dilogarithm identity:
Theorem 2.4 (Schützenberger, Faddeev, Kashaev, Volkov). In Q(q1/2)q[[x, y]], we have
Φ(x)Φ(y) = Φ(y)Φ
(
−q−1/2xy
)
Φ(x).
For m = 2, we will obtain the following identity as a special case of Theorem 2.7.
Theorem 2.5. In Q(q1/2)q2 [[x, y]], we have
Φ(x)Φ(y) =Φ(y)Φ
(
q−2xy2
)
Φ
(
q−6x2y3
)
Φ
(
q−12x3y4
)
. . . ·
· Φ
(
q−1/2xy
)−1
Φ
(
q−3/2xy
)−1
·
· . . . Φ
(
q−12x4y3
)
Φ
(
q−6x3y2
)
Φ(q−2x2y)Φ(x).
To make such identities more readable, we will now introduce a shorthand notation,
which again will be motivated later by the framework of Donaldson–Thomas invariants:
Definition 2.6. For a, b ≥ 0, define
Φ(a,b) = Φ
(
(−1)mabq(a2+b2−2mab−1)/2xayb
)(−1)a2+b2−mab−1
.
More generally, for a Laurent polynomial P (q) =
∑
k ck(−q1/2)k ∈ Q[q±1/2], define
Φ◦P(a,b) =
∏
k
Φ
(
(−1)mabq(a2+b2−2mab−1+k)/2xayb
)(−1)a2+b2−mab−1+k·ck
.
Then the previous identities simplify to
Φ(1,0)Φ(0,1) = Φ(0,1)Φ(1,1)Φ(1,0) for m = 1,
Φ(1,0)Φ(0,1) = Φ(0,1)Φ(1,2)Φ(2,3) . . . · Φ
◦(q+1)
(1,1) · . . . Φ(3,2)Φ(2,1)Φ(1,0)
for m = 2.
Ann. Repr. Th. 1 (2024), 3, p. 385–391 https://doi.org/10.5802/art.12
Wild quantum dilogarithm identities 387
In the case m ≥ 3, we can no longer give explicit identities, but we will obtain a rather
complete qualitative description.
To formulate this main result, we need three more definitions. We denote by σ the
operator on Z2 given by σ(a, b) = (b, mb− a), which generates an infinite dihedral group
together with the involution (a, b) 7→ (b, a). We also define µ± = (m±
√
m2 − 4)/2, the
two roots of the quadratic equation x2 − mx + 1 = 0. Finally, we note that ordered
products in Q(q1/2)qm [[x, y]] of the form
→∏
s ∈ Q
increasing
Fs
are well-defined for series Fs ∈ Q(q1/2)[[xayb]] with constant term one, where s = a/b.
Theorem 2.7. In Q(q1/2)qm [[x, y]] for m ≥ 3, we have
Φ(1,0)Φ(0,1) = Φ(0,1)Φσ(0,1)Φσ2(0,1)Φσ3(0,1) · . . . ·
·
→∏
µ− ≤ a/b ≤ µ+
increasing
Φ◦P(a,b)(a,b) ·
· . . . Φσ−3(1,0)Φσ−2(1,0)Φσ−1(1,0)Φ(1,0),
where the P(a,b) satisfy the following properties:
(1) “Wildness”/Completeness: We have the non-vanishing property
P(a,b) ̸= 0 for all µ− ≤ a/b ≤ µ+.
(2) Dihedral symmetry: We have the symmetries
P(b,a) = P(a,b), Pσ(a,b) = P(a,b).
(3) Positivity: We have
P(a,b) ∈ N[q], of degree d = mab− a2 − b2 + 1 > 0.
(4) Unimodality: The polynomial P(a,b) =
∑
k ckq
k is palindromic and unimodal:
cd−k = ck and 1 = c0 ≤ c1 ≤ . . . ≥ cd−1 ≥ cd = 1.
(5) Lowest order terms:We have
P(1,k) =
[
m
k
]
q
.
(6) Special value: Using the number-theoretic Möbius function µ, we have
P(k,k)(1) =
1
(m− 2)k2
∑
d|k
µ
(
k
d
)
(−1)md+1
(
(m− 1)2d− 1
d
)
.
As a consequence of the dihedral symmetry property, we see that all P(a,b) are deter-
mined by those for a ≤ b ≤ m2 a. All properties will follow from interpreting the P(a,b)
as the Poincaré polynomials in intersection homology of Kronecker moduli [1], a class of
projective varieties parametrizing certain tuples of matrices up to base change.
Ann. Repr. Th. 1 (2024), 3, p. 385–391 https://doi.org/10.5802/art.12
388 Markus Reineke
3. Quiver setup
In this section, we recall the necessary terminology on quiver representations and their
moduli spaces, and we formulate the main ingredients for general quiver dilogarithm iden-
tities, namely the motivic wall-crossing formula, and the definition and geometricity of
motivic Donaldson–Thomas invariants. The reader is referred to [7] for a detailed intro-
duction into motivic wall-crossing for quivers, and to [10] for a short summary.
Let Q be a finite acyclic quiver. We order the set of vertices Q0 = {i1, . . . , in} such
that ik → il implies k > l. The Euler form of Q is the (in general non-symmetric) bilinear
form on ZQ0 given by ⟨d, e⟩ =
∑
i ∈ Q0 diei −
∑
α:i→j diej for d, e ∈ ZQ0.
We define the formal quantum affine space Q(q1/2)q[[Q0]] as the skew formal power
series ring with topological basis td for d ∈ NQ0 and multiplication twisted by the anti-
symmetrized Euler form
td · te =
(
−q1/2
)⟨d,e⟩−⟨e,d⟩
td+e.
We fix linear functions Θ, κ ∈ (ZQ0)∗ on Q such that κ(d) > 0 for d ∈ NQ0 \ 0, and
consider the associated slope function µ(d) = Θ(d)κ(d) for d ∈ NQ0 \ 0. We denote by Λa the
set of all d ∈ NQ0 \ 0 of slope a ∈ Q.
We define the Grothendieck ring of varieties K0(VarC) as the free abelian group in
isomorphism classes of complex algebraic varieties X modulo the “cut-and-paste” relation
[X] = [A] + [U ] if A ⊂ X closed, U = X \ A, with product given by [X] · [Y ] = [X × Y ].
We abbreviate the Lefschetz motive [A1] =: L.
We consider the localization R = K0(VarC)[L±1/2, (1 − Ln)−1 : n ≥ 1], and define the
formal motivic affine space RL[[Q0]] as above, with R replacing the coefficient ring Q(q1/2)
and L replacing q. In fact, all our computations will happen in the smaller coefficient
ring of motives which are rational functions in L1/2. Note that the existence of motivic
measures such as the virtual Hodge polynomial shows that this subring is isomorphic to
a subring of Q(q1/2), so by a slight abuse of notation, we can and will identify q and L
from now on.
Given a dimension vector d ∈ NQ0, we fix C-vector spaces Vi of dimension di for i ∈ Q0.
We consider the base change action∏
i ∈ Q0
GL(Vi) = Gd ↷ Rd(Q) =
⊕
α:i → j
Hom(Vi, Vj)
given by
(gi)i · (fα)α =
(
gjfαg
−1
i
)
α:i → j
,
whose orbits, by definition, correspond bijectively to the isomorphism classes of complex
representations of Q of dimension vector d.
We denote by Rµ−sstd (Q) ⊂ Rd(Q) the open subset of µ-semistable points. Using this
notation, we can formulate the motivic wall-crossing formula, which is a formal conse-
quence of the existence of the Harder–Narasimhan filtration (see [8] for a version involving
counts over finite fields; the proof of [11, Theorem 3.5] shows that the same formula holds
for motives):
Ann. Repr. Th. 1 (2024), 3, p. 385–391 https://doi.org/10.5802/art.12
Wild quantum dilogarithm identities 389
Theorem 3.1 (Motivic wall-crossing formula). In Rq[[Q0]], we have the identity
Φ(t1) · . . . · Φ(tn) =
∑
d
(
−q1/2
)⟨d,d⟩ [Rd(Q)]
[Gd]
td
=
→∏
a ∈Q decreasing
1 + ∑
d ∈ Λa
(
−q1/2
)⟨d,d⟩ [Rµ−sstd (Q)]
[Gd]
td
 .
Next, we can define motivic Donaldson–Thomas invariants. We generalize the shorthand
notation of the previous section to
Φ(x)◦P (q) =
∏
k
Φ
(
qk/2x
)(−1)kck
for P (q) =
∑
k ck(−q1/2)k.
Definition 3.2 ([5]). Assume that ⟨_, _⟩ is symmetric on Λa. Define DTµd(q) ∈ Q[q±1/2]
by factorization in Rq[[Q0]]:
1 +
∑
d ∈ Λa
(−q1/2)⟨d,d⟩
[Rµ−sstd (Q)]
[Gd]
td =
∏
d ∈ Λa
Φ(td)◦DT
µ
d(q).
The DTµd(q) are called the motivic Donaldson–Thomas invariants of the quiver Q with
stability µ.
We remark that the more common definition in terms of the plethystic exponential Exp
is equivalent to this one since, by definition, Φ(x) = Exp( x
q−1/2−q1/2 ).
The motivic Donaldson–Thomas invariants admit a geometric interpretation in terms
of intersection homology of moduli spaces. Namely, we consider the GIT quotient
Rµ−sstd (Q)//Gd = M
µ−sst
d (Q),
the moduli space of µ-semistable representations of Q of dimension vector d.
It is an irreducible projective normal (typically singular) complex algebraic variety. If
d is µ-stable, that is, if there exists a µ-stable representation of dimension vector d, then
dim Mµ−sstd (Q) = 1−⟨d, d⟩. In terms of this moduli space, we have the following geometric
interpretation of the motivic Donaldson–Thomas invariants [6]:
Theorem 3.3 (Geometricity of DT invariants). We have
DTµd(q) = (−q
1/2)⟨d,d⟩−1
∑
k ≥ 0
dim IHk(Mµ−sstd (Q),Q)
(
−q1/2
)k
if d is µ-stable, and DTµd(q) = 0 otherwise.
4. Quantum dilogarithm identity for quivers
To combine the methods prepared in the previous section, we consider quivers and
stabilities such that the restriction of ⟨_, _⟩ is symmetric on all Λa. In particular, this
holds if the antisymmetrized Euler form of Q is determined by the stability function, in
the sense that
⟨d, e⟩ − ⟨e, d⟩ = κ(d)Θ(e)− κ(e)Θ(d) for all d, e ∈ ZQ0,
Ann. Repr. Th. 1 (2024), 3, p. 385–391 https://doi.org/10.5802/art.12
390 Markus Reineke
compare [10, Proposition 5.2]. Besides generalized Kronecker quivers, this property holds,
for example, for complete bipartite quivers.
Theorem 4.1. Assume that ⟨_, _⟩ is symmetric on all Λa. Then we have a factorization
Φ(t1) · . . . · Φ(tn) =
→∏
µ(d) decreasing
Φ
(
td
)◦(−q1/2)⟨d,d⟩−1·Pd(q)
in Rq[[Q0]], for polynomials Pd(q) with the following properties:
(1) Non-vanishing: We have Pd(q) ̸= 0 if and only if d is µ-stable,
(2) Positivity: We have Pd ∈ N[q], of degree deg Pd(q) = 1− ⟨d, d⟩,
(3) Unimodality: Pd(q) is palindromic and unimodal,
(4) Simplicity: We have Pd(q) = 1 if, additionally, ⟨d, d⟩ = 1.
This is now readily proved: combining Theorem 3.1, Definition 3.2 and Theorem 3.3, we
see that Pd(q) is precisely the Poincaré polynomial in intersection homology of Mµ−sstd (Q)
in case d is µ-stable, proving the non-vanishing property. Positivity and unimodality are
proven in [6, Corollary 1.2]. The degree statement follows from the dimension formula for
the moduli space, and consequently the simplicity statement follows.
To derive Theorem 2.7, we consider the m-arrow Kronecker quiver Km = 1
(m)← 2, with
stability function given by Θ(d) = m · (d2 − d1) and κ(d) = d1 + d2.
We identify the variables t1 = x and t2 = y, leading to an identification of Q(q1/2)q
[[(Km)0]] with Q(q1/2)qm [[x, y]], such that td = (−q1/2)md1d2xd1yd2 and
Φ
(
td
)◦(−q1/2)⟨d,d⟩−1·P (q)
= Φ◦P (q)(a,b)
for d = (a, b).
Theorem 4.1 then provides a factorization in Q(q1/2)qm [[x, y]] of the form
Φ(1,0)Φ(0,1) =
→∏
a/b increasing
Φ◦P(a,b)(a,b) ,
such that
P(a,b)(q) =
∑
k ≥ 0
dim IH2k(K(a,b),Q)qk,
where
K(a,b) = Ma×b(C)msst//GLa(C)×GLb(C)
are the Kronecker moduli of [1].
All remaining properties in Theorem 2.7, as well as Theorems 2.4 and 2.5, now follow
from properties of these Kronecker moduli spaces. Using [13, Theorem 4.1], we see that
all roots, that is, dimension vectors (a, b) such that ⟨(a, b), (a, b)⟩ ≤ 1, are already Schur
roots, with the exception of proper multiples of (1, 1) for m = 2. By [13, Theorem 6.1]
(and the discussion following the theorem) Schur roots are already µ-stable. In case
⟨(a, b), (a, b)⟩ = 1, the dimension vector is a real root, and the moduli space K(a,b) reduces
to a point, thus P(a,b) = 1. This already (re-)proves Theorem 2.4 and, together with
K(1,1) ≃ Pm−1, also proves Theorem 2.5. Otherwise, we have ⟨(a, b), (a, b)⟩ ≤ 0, which
translates to µ− ≤ a/b ≤ µ+. This proves the completeness statement of the theorem. By
linear duality, we have K(a,b) ≃ K(b,a). Moreover, reflection functors induce isomorphisms
K(a,b) ≃ K(b,mb−a) by [15, Proposition 4.3]. This establishes the dihedral symmetry.
Ann. Repr. Th. 1 (2024), 3, p. 385–391 https://doi.org/10.5802/art.12
Wild quantum dilogarithm identities 391
The Kronecker moduli K(1,k) identify with the Grassmannians Grmk , determining the lowest
order terms in the factorization. Finally, the special value P(k,k)(1) is computed in [9,
Theorem 5.2].
Acknowledgments
The author would like to thank Daping Weng for several discussions on these wild
identities, Bernhard Keller for the opportunity to present them in his seminar, Vladimir
Fock as well as Sergey Mozgovoy for valuable comments, and Timm Peerenboom for
carefully reading a draft of this text and suggesting several improvements.
References
[1] Jean-Marc Drezet, Cohomologie des variétés de modules de hauteur nulle, Math. Ann. 281 (1988),
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no. 5, 427–434.
[3] Mark Gross and Rahul Pandharipande, Quivers, curves, and the tropical vertex, Port. Math. 67
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— Markus Reineke —
Faculty of Mathematics, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum,
Germany
E-mail address: markus.reineke@ruhr-uni-bochum.de
E-mail address: markus.reineke@rub.de
URL: http://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Algebra/reineke.html.de
Ann. Repr. Th. 1 (2024), 3, p. 385–391 https://doi.org/10.5802/art.12


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Luke Kershaw & Jeremy Rickard
A finite dimensional algebra with infinite delooping level
Volume 1, issue 1 (2024), p. 61-65
https://doi.org/10.5802/art.3
Communicated by Henning Krause.
© The authors, 2024
This article is licensed under the
Creative Commons Attribution (CC-BY) 4.0 License.
http://creativecommons.org/licenses/by/4.0/
Annals of Representation Theory is published by the
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and a member of the
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C EN T R E
MER S ENN E

Annals of Representation Theory
Volume 1, issue 1 (2024), p. 61–65
https://doi.org/10.5802/art.3
A finite dimensional algebra with infinite delooping
level
Luke Kershaw and Jeremy Rickard ∗
Abstract. We give an example of a finite dimensional algebra with infinite delooping level, based
on an example of a semi-Gorenstein-projective module due to Ringel and Zhang.
1. Introduction
One of the most celebrated open problems in the representation theory of finite dimen-
sional algebras is the finitistic dimension conjecture, publicized by Bass [2] in 1960, and
subsequently proven to imply a host of other homological conjectures.
In recent years other, potentially stronger, conditions that would imply the finitistic
dimension conjecture have been considered. One is the notion of “injective generation”
[8], that we consider briefly in Section 5. But our main focus is the finiteness of delooping
level, introduced by Gélinas [5]. He works in greater generality, but since the main example
in this paper is a finite dimensional algebra, we shall describe the delooping level in that
context.
Given a finite dimensional algebra A and a finitely generated right A-module M , the
delooping level of M is the smallest nonnegative integer n such that ΩnM is a direct
summand (in the stable module category) of Ωn+1N for some other finitely generated
module N , or is infinite if no such n exists. The delooping level is denoted by dell M .
Then the delooping level dell A is defined to be the largest delooping level of a simple
A-module.
Manuscript received 2023-06-20, revised 2023-11-05 and accepted 2023-11-27.
Keywords. delooping level, semi-Gorenstein-projective modules, finitistic dimension conjecture.
2010 Mathematics Subject Classification. 16E10, 16G10.
The first author was supported by Engineering and Physical Sciences Research Council Doctoral Training
Partnership award EP/R513179/1.
∗ Corresponding author.
61
62 Luke Kershaw & Jeremy Rickard
One of the main properties of the delooping level is that dell A is an upper bound for
the (big) finitistic dimension of the opposite algebra Aop and so if dell A < ∞ then the big
finitistic dimension conjecture holds for Aop [5, Proposition 1.3].
In this paper we shall give what we believe to be the first known example of a finite
dimensional algebra with infinite delooping level. This does not give a counterexample to
the finitistic dimension conjecture.
Although the delooping level of an algebra A is defined in terms of the delooping levels
of simple modules, we show in Section 2 that if there is any finitely generated A-module
M with dell M = ∞, then there is another finite dimensional algebra B with dell B = ∞.
Our main result, Proposition 4.1, is that a particular “semi-Gorenstein-projective” mod-
ule studied by Ringel and Zhang [9] has infinite delooping level. Gorenstein projective
modules are easily seen to have delooping level equal to zero, but the same argument
does not apply to semi-Gorenstein-projective modules. What inspired us to look at this
example is that Gélinas showed in [5, Theorem 1.10] that the module N in the definition
of delooping level can always be taken to be Σn+1ΩnM (where Σ denotes the left adjoint
of Ω), which in the case of a semi-Gorenstein-projective module simplifies to ΣM , so that
proving dell M = ∞ reduces to proving that ΩnM is never a summand of Ωn(ΩΣ)M .
There are other examples known of modules that are semi-Gorenstein-projective, but not
Gorenstein projective. The first was given by Jorgensen and Şega [6], and Marczinzik [7]
gave others. It might be interesting to study the delooping level of these examples.
2. Delooping levels of one point extensions
The proof of the following proposition uses the construction of a one-point extension al-
gebra, a particular kind of triangular matrix algebra. More details about this construction
can be found in, for example, [1, Section III.2].
Proposition 2.1. If M is a finitely generated module with infinite delooping level for a
finite dimensional k-algebra A, then there is a finite dimensional k-algebra B with infinite
delooping level.
Proof. Let B be the one point extension algebra
B = A[M ] =
(
k M
0 A
)
,
and let S be the simple B-module
(
k M
)
/
(
0 M
)
Then for n ≥ 1, ΩnS =
(
0 Ωn−1M
)
. If S has finite delooping level, then for all
sufficiently large n, ΩnS is a summand of Ωn+1X for some B-module X, and ΩX =
(
0 N
)
for some A-module N , and Ωn+1X =
(
0 ΩnN
)
.
Therefore Ωn−1M is a summand of ΩnN , and so M has finite delooping level. □
3. Ringel and Zhang’s example
In [9], Ringel and Zhang exhibited an example of a three-dimensional module M for
a six-dimensional local finite dimensional algebra Λ that is semi-Gorenstein-projective
(meaning that Exti(M, Λ) = 0 for all i ≥ 1), but is not torsionless (i.e., is not a submodule
of a projective Λ-module), and hence not Gorenstein-projective.
In this section, we gather the information that we will need about this example. All of
the results in this section are taken from Ringel and Zhang’s paper, except for the remarks
about the subalgebra Γ. Note that they work with left modules, whereas we are using
right modules, so the algebra we define is the opposite of theirs.
Ann. Repr. Th. 1 (2024), 1, p. 61–65 https://doi.org/10.5802/art.3
63
3.1. The algebra Λ = Λ(q) and the subalgebra Γ [9, §6.1]. Let k be a field, and q ∈ k
an element with infinite multiplicative order. Then Λ = Λ(q) is the algebra
k⟨x, y, z⟩/
(
x2, y2, z2, zy, yx + qxy, zx − xz, yz − xz
)
,
which is easily seen to be six-dimensional, with basis 1, x, y, z, yx, zx.
Let Γ be the two-dimensional subalgebra of Λ generated by x. Note that Λ is a free
right Γ-module.
3.2. The modules M(α) [9, §6.1, §6.3]. For each α ∈ k, M(α) is the three-dimensional
Λ-module with basis v, v′, v′′, where vx = αv′, vy = v′, vz = v′′, with v′ and v′′ annihilated
by x, y and z.
Next, we gather the results that we need regarding the modules M(α). We shall state
them in terms of the effect of Ω and Σ on the modules, but apart from the brief remark
about the subalgebra Γ, everything in the following proposition is a restatement of results
of Ringel and Zhang [9].
Proposition 3.1. Let α ∈ k.
(1) If α ̸= 1 then ΩM(α) = M(qα).
(2) If α ̸= q then ΣM(α) = M(q−1α).
(3) ΩΣM(q) is a two-dimensional module that is free as a Γ-module.
Proof. Both (1) and (2) follow immediately from [9, Lemma 6.4 and Lemma 3.2]. Note
that Ringel and Zhang use the notation ℧M(α) where we use ΣM(α).
For any module M , ΩΣM is the maximal torsionless quotient of M . For M = M(q),
this is shown in [9, Lemma 6.2] to be M(q)/M(q)z, which is freely generated as a Γ-module
by v. □
4. The delooping level of M(q)
Proposition 4.1. The Λ-module M(q) has infinite delooping level.
Proof. Gélinas showed that if a module M has delooping level n < ∞, then ΩnM is a
stable direct summand of Ωn+1Σn+1ΩnM [5, Theorem 1.10]. We shall show that this is
not the case for M = M(q).
By Proposition 3.1(1), ΩnM(q) ∼= M(qn+1) is three-dimensional.
By Proposition 3.1(2), ΣnΩnM(q) ∼= M(q), so
Ωn+1Σn+1ΩnM ∼= Ωn+1ΣM(q).
But since ΩΣM(q) and Λ are both free as Γ-modules,
Ωn+1ΣM(q) = Ωn (ΩΣM(q))
is also free as a Γ-module and therefore every direct summand has even dimension. □
Corollary 4.2. The one-point extension algebra Λ[M(q)] has infinite delooping level.
Proof. Proposition 3.1 and Proposition 4.1. □
Ann. Repr. Th. 1 (2024), 1, p. 61–65 https://doi.org/10.5802/art.3
64 Luke Kershaw & Jeremy Rickard
5. Other properties of Λ[M(q)]
In this final section, we consider some other related properties of the algebra Λ[M(q)].
One of the main reasons that the delooping level was introduced was its connection
with the finitistic dimension conjecture: for Artinian rings A, Gélinas proved that the
delooping level dell A of A is an upper bound for the big finitistic dimension Findim Aop of
its opposite algebra, so that finiteness of the delooping level of A implies the big finitistic
dimension conjecture for Aop [5, Proposition 1.3].
It is therefore natural to wonder whether Λ[M(q)]op might have infinite finitistic dimen-
sion. This is not the case.
Proposition 5.1. Findim Λ[M(q)]op = 1
Proof. Since
Λ[M(q)]op =
(
k 0
M(q) Λop
)
there is an obvious module with projective dimension equal to one – namely,
(
0 Λop
)
=(
M(q) Λop
)
/
(
M(q) 0
)
, and so 1 ≤ Findim Λ[M(q)]op.
But since Λ[M(q)]op is a triangular matrix ring, it is well-known [4, Corollary 4.21] that
Findim Λ[M(q)]op ≤ 1 + Findim k + Findim Λop,
and since Λop is a local finite dimensional algebra, Findim Λop = 0. So Findim Λ[M(q)]op
≤ 1. □
Another property that has recently been shown to imply the big finitistic conjecture for a
finite dimensional algebra A is “injective generation”. If the unbounded derived category
D(A) is equal to Loc(Inj A), the localizing subcategory of D(A) generated by injective
modules, then the big finitistic dimension conjecture holds for A [8, Theorem 4.3].
We shall end by showing that the algebra Λ[M(q)]op is not a counterexample to injective
generation.
Proposition 5.2. Injectives generate for Λ[M(q)]op.
Proof. Since Λ[M(q)]op is a triangular matrix ring, by a result of Cummings it is sufficient
to show that injectives generate for Λop [3, Example 6.11].
Since M(0) = ΩM(0), M(0) has finite syzygy type as a Λ-module, and so, taking the
vector space dual, DM(0) has finite cosyzygy type as a Λop-module, which implies that
DM(0) is in Loc(Inj Λop) [8, Proposition 7.2].
Since DM(0) has radical length two, there is a short exact sequence
0 → S1 → DM(0) → S2 → 0
where S1 and S2 are nonzero semisimple Λop-modules. Taking the coproduct of infinitely
many copies of this sequence, we get a short exact sequence
0 → S →
⊕
DM(0) → S → 0
where S is an infinitely generated semisimple module. Splicing together copies of this
sequence gives a resolution
· · · →
⊕
DM(0) →
⊕
DM(0) → S → 0
by coproducts of copies of DM(0), and so S is in Loc(Inj Λop) [8, Proposition 2.1(h)].
Therefore the unique simple Λop-module is in Loc(Inj Λop) and so D(Λop) = Loc(Inj Λop) [8,
Proposition 2.1(e) and Lemma 6.1]. □
Ann. Repr. Th. 1 (2024), 1, p. 61–65 https://doi.org/10.5802/art.3
65
5.1. The opposite algebra. The reader might wonder about corresponding properties
of the opposite algebra. In that case two things are easier, because ( k 00 0 ) and ( 0 00 U ), where
U is any one-dimensional subspace of rad2(Λ), are simple left submodules of Λ[M(q)],
so that every simple module for Λ[M(q)]op is isomorphic to a submodule of a projective
module, a property which is well known ([2, Lemma 6.2]) to be equivalent to the condition
Findim Λ[M(q)] = 0,
and is trivially equivalent to the condition
dell Λ[M(q)]op = 0.
It is also the case that injectives generate for Λ[M(q)]. It follows from the results of [9,
Section 6.5] that M(0)∗ = HomΛ(M(0), Λ) is a three-dimensional Λop-module isomorphic
to its syzygy, and from there the proof follows that of Proposition 5.2: D(M(0)∗) is in
Loc(Inj Λ), and so the unique simple Λ-module is in Loc(Inj Λ), and so injectives generate
for Λ, and so injectives generate for Λ[M(q)].
References
[1] Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cam-
bridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, 1995.
[2] Hyman Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Am.
Math. Soc. 95 (1960), 466–488.
[3] Charley Cummings, Ring constructions and generation of the unbounded derived module category, Al-
gebr. Represent. Theory 26 (2023), no. 1, 281–315.
[4] Robert M. Fossum, Phillip A. Griffith, and Idun Reiten, Trivial extensions of abelian categories. Homo-
logical algebra of trivial extensions of abelian categories with applications to ring theory, Lecture Notes
in Mathematics, vol. 456, Springer, 1975.
[5] Vincent Gélinas, The depth, the delooping level and the finitistic dimension, Adv. Math. 394 (2022),
Paper no. 108052.
[6] Liana M. Jorgensen, David A. andŞega, Independence of the total reflexivity conditions for modules,
Algebr. Represent. Theory 9 (2006), no. 2, 217–226.
[7] René Marczinzik, Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra,
2021, https://arxiv.org/abs/1608.04212v2.
[8] Jeremy Rickard, Unbounded derived categories and the finitistic dimension conjecture, Adv. Math. 354
(2019), Paper no. 106735 (21 pages).
[9] Claus Michael Ringel and Pu Zhang, Gorenstein-projective and semi-Gorenstein-projective modules,
Algebra Number Theory 14 (2020), no. 1, 1–36.
— Luke Kershaw —
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8
1UG, United Kingdom
E-mail address: l.kershaw@bristol.ac.uk
— Jeremy Rickard —
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8
1UG, United Kingdom
E-mail address: j.rickard@bristol.ac.uk
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