The 51st Symposium on Ring Theory
and Representation Theory

ABSTRACT

Okayama University of Science, Okayama
September 19 – 22, 2018

Program
September 19 (Wednesday)
9:40–10:10

Mitsuo Hoshino (University of Tsukuba), Hirotaka Koga (Tokyo Denki University), Noritsugu Kameyama (Salesian Polytechnic)
Frobenius ring homomorphisms

10:20–10:50 Yoshihiro Otokita (Chiba University)
Centers of modular group algebras
11:00–12:00 Kohji Yanagawa (Kansai University)
Homological methods in combinatorial commutative algebra I
13:10–13:40 Yusuke Nakajima (The University of Tokyo)
Finite dimensional algebras arising from dimer models and their derived equivalences
13:50–14:20 Toshitaka Aoki (Nagoya University)
Torsion classes for algebras with radical square zero
14:30–15:00 Norihiro Hanihara (Nagoya University)
Auslander correspondence for triangulated categories
15:20–15:50 Sota Asai (Nagoya University)
The chamber structures of the Grothendieck groups coming from bricks
16:00–17:00 Yu Qiu (Tsinghua University)
Decorated marked surfaces

September 20 (Thursday)
9:40–10:10 Hiroki Matsui (Nagoya University)
Connectedness of the Balmer spectrum of the right bounded derived category of a commutative noetherian ring
10:20–10:50 Tokuji Araya (Okayama University of Science), Kei-ichiro Iima (Nara College), Maiko Ono(Okayama University), Ryo Takahashi (Nagoya University)
Dimensions of singular categories of hypersurfaces of countable representation type
11:00–12:00 Kohji Yanagawa (Kansai University)
Homological methods in combinatorial commutative algebra II
13:10–13:40 Hiroyuki Minamoto (Osaka Prefecture University)
Resolution of DG-modules and their applications for commutative DG-algebras
13:50–14:20 Haruhisa Enomoto (Nagoya University)
Relations for Grothendieck groups and representation-finiteness
14:30–15:00 Yuya Mizuno (Shizuoka University)
Sortable elements and torsion pairs for quivers
15:20–15:50 Takahide Adachi (Osaka Prefecture University), Aaron Chan (Nagoya University)
A geometric model of Brauer graph algebras–i–16:00–16:30 Kaoru Motose
Cyclotomic polynomials
17:30– Conference dinner

September 21 (Friday)
9:40–10:10 Tomohiro Itagaki (Tokyo University of Science)
Batalin-Vilkovisky algebra structures on the Hochschild cohomology of self-injective Nakayama
algebras
10:20–10:50 Kazunori Nakamoto (University of Yamanashi), Takeshi Torii (Okayama University)
An application of Hochschild cohomology to the moduli of subalgebras of the full matrixring
11:00–12:00 Hiroyuki Nakaoka (Kagoshima University)
Some remarks on Avella-Alaminos-Geiss invariants of gentle algebras
13:10–13:40 Masahisa Sato (Aichi University & Yamanashi University)
On projective modules with unique maximal submodule
13:50–14:20 Gangyong Lee, Mauricio Medina-Bárcenas (Chungnam National University)
On (finite) Σ-Rickart modules: On a module theoretic setting of the (semi-)hereditaryproperty of rings
14:30–15:00 Mayu Tsukamoto (Osaka City University)
A strongly quasi-hereditary structure on Auslander–Dlab–Ringel algebras
15:20–15:50 Kengo Miyamoto (Osaka University)
Components of the stable Auslander-Reiten quiver for a symmetric order over a complete
discrete valuation ring
16:00–17:00 Yu Qiu (Tsinghua University)
X-stability conditions on Calabi-Yau-X categories

September 22 (Saturday)
9:40–10:10 Shigeo Koshitani, Taro Sakurai (Chiba University)
On certain Morita invariants involving commutator subspace and radical powers
10:20–10:50 Ayako Itaba (Tokyo University of Science), Masaki Matsuno (Shizuoka University)
The defining relations of geometric algebras of Type EC
11:00–11:30 Akira Masuoka (University of Tsukuba)
Quotients G/H is super-symmetry–ii–

Frobenius ring homomorphisms
Mitsuo Hoshino, Noritsugu Kameyama and Hirotaka Koga
Let R be a left and right noetherian ring. We call a ring homomorphism ϕ : R → A Frobenius if A
is a finitely generated right R-module with ExtiR (A, R) = 0 for all i ≥ 1 and if HomR (A, R) is a finitely
generated projective generator for right A-modules (cf. [1, 7, 8]). Our aim is to show that if a Frobenius
ring homomorphism ϕ : R → A is given then A inherits various homological properties from R.
Assume that a Frobenius ring homomorphism ϕ : R → A is given. Then A also is left and rightnoetherian (cf. [5]). Furthermore, if R is Auslander-Gorenstein then so is A with inj dim A ≤ inj dim R(cf. [3, 4, 6, 9]). Also, in connection with the generalized Nakayama conjecture ([2]), we notice that
ExtiA (X, R) = 0 if and only if ExtiR (X, R) = 0 for any right A-module X and i ≥ 0, and that a ringhomomorphism ψ : A → Γ is Frobenius if and only if so is ψ ◦ ϕ. On the other hand, it seems in generalthat one can not expect R to inherit any homological property from A. For instance, if R is an arbitrary
finite dimensional algebra over a field k, then Endk (R) is a Frobenius algebra and the canonical ringhomomorphism R → Endk (R), x 7→ (y 7→ yx), is Frobenius.
References[1] H. Abe and M. Hoshino, Frobenius extensions and tilting complexes, Algebras and Representation Theory 11(3) (2008),
215–232.
[2] M. Auslander and I. Reiten, On a generalized version of the Nakayama conjecture, Proc. Am. Math. Soc. 52 (1975),
69–74.
[3] J. -E. Björk, Rings of diﬀerential operators, North-Holland Mathematical Library, 21. North-Holland Publishing Co.,
Amsterdam-New York, 1979.
[4] J. -E. Björk, The Auslander condition on noetherian rings, in: Séminaire d′ Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988), 137-173, Lecture Notes in Math., 1404, Springer, Berlin, 1989.
[5] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473.
[6] R. M. Fossum, Ph. A. Griﬃth and I. Reiten, Trivial extensions of abelian categories, Lecture Notes in Math., 456,
Springer, Berlin, 1976.
[7] T. Nakayama and T. Tsuzuku, On Frobenius extensions I, Nagoya Math. J. 17 (1960), 89–110.
[8] T. Nakayama and T. Tsuzuku, On Frobenius extensions II, Nagoya Math. J. 19 (1961), 127–148.
[9] A. Zaks, Injective dimension of semi-primary rings, J. Algebra 13 (1969), 73–86.

Institute of Mathematics
University of Tsukuba
Ibaraki, 305-8571, Japan
Email: hoshino@math.tsukuba.ac.jp
Department of General Education
Salesian Polytechnic
Tokyo, 194-0212, Japan
Email: n-kameyama@salesio-sp.ac.jp
Department of Mathematics
Tokyo Denki University
Tokyo, 120-8551, Japan
Email: koga@mail.dendai.ac.jp–1–

Centers of modular group algebras
Yoshihiro Otokita
This talk deals with the Loewy structure of the center of a modular group algebra.
Let G be a finite group and F an algebraically closed field of characteristic p > 0. We denote by Z thecenter of the group algebra F G. Then the dimension of Z is equal to the number of conjugacy classes of
G. For a primitive idempotent b in Z, Zb is the center of a block ideal of F G and it is a local algebra in
the sense that the Jacobson radical J(Zb) has codimension 1. Our studies focus on the relation between
Zb and defect group D. Then it is known that D = {1} if and only if J(Zb) = 0 (i.e. dim Zb = 1). In [1]Héthelyi and Külshammer conjectured that :√2 p − 1 ≤ dim Zb if D ̸= {1} ?
In order to solve this problem we examine the algebraic structure of Zb. Our main theorem in this talkgives a lower bound for its Loewy length LL(Zb):pm + p − 2≤ LL(Zb) where pm is the exponent of the center of D.
p−1
As a corollary, we conclude that p + 2 ≤ dim Zb provided m ≥ 2.
References[1] L. Héthelyi and B. Külshammer, On the number of conjugacy classes of a finite solvable group, Bull. London Math.
Soc. 32 (2000), 668–672.

Department of Mathematics and Informatics
Graduate School of Science
Chiba University
JAPAN
Email: otokita@chiba-u.jp–2–

HOMOLOGICAL METHODS INCOMBINATORIAL COMMUTATIVE ALGEBRA
Kohji Yanagawa
Let S = K[x1 , . . . , xn ] be a polynomial ring over a field K.

∏
Definition 1. For a simplicial complex ∆ ⊂ 2{1,...,n} , consider the monomial ideal I∆ := ( i∈F xi | F ⊂{1, . . . , n}, F ̸∈ ∆) ⊂ S. We call K[∆] := S/I∆ the Stanley-Reisner ring of ∆.
Using this concept, R. Stanley proved Upper Bound Conjecture for simplicial spheres. Since then,this has been one of the central tools in the combinatorial study of finite simplicial complexes (c.f. [2]).
Moreover, the algebraic study of Stanley-Reisner rings is still very active. It is a classical result that the
Cohen–Macaulay-ness of K[∆] only depends on the topology of the underlying space |∆| (and char(K)).
For example, if |∆| is homeomorphic to a sphere, then K[∆] is Cohen–Macaulay for all K (moreover, it
is Gorenstein).
In [3], I introduced the following notion, to use homological methods more systematically.
Definition 2. We say a finitely generated Zn -graded S-module M is squarefree, if it is Nn graded(i.e., Ma = 0 for a ̸∈ Nn ), and the multiplication map Ma ∋ y 7−→ xi y ∈ Ma+ei is bijective for alla = (a1 , . . . , an ) ∈ Nn and all i with ai > 0.
Many modules appearing in the Stanley-Reisner ring theory are squarefree. At the same time, theclass of squraefree modules is quite small. In fact, the category Sq S of squarefree S-modiles is equivalent
to the category of finitely generated left modules over the incidence K-algebra of the poset 2{1,...,n} .
The following are selected results on this notion.
• There is a duality functor A : Sq S → Sq S such that A(K[∆]) ∼= I∆∨ , where ∆∨ is the Alexander
dual of ∆.
• R HomS (−, ωS ) gives a duality functor D from Db (Sq S) to itself, where ωS is the Zn -gradedcanonical module (it is just S itself as an underlying module). Moreover, we have (A ◦ D)3 ∼=T−2n , where T stands for the translation of Db (Sq S).
• For M ∈ Sq S, we have a constructible sheaf M † on the (n − 1)-simplex B. For example, K[∆]†is the K-constant sheaf on |∆| ⊂ B (more precisely, its direct image to B). In this context, theabove mentioned duality D corresponds to Poincaré-Verdier duality on B.
On the other hand, I have begun to recognized that some “ring theoretic” arguments (e.g., Lefschetz property of artinian graded algebras, see [1]) are indispensable for further development of the
Stanley-Reisner ring theory. Unfortunately, the compatibility between squarefree modules and such aring theoretic argument is quite bad now. If time allows, I will discuss these problems.
References[1] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, and J. Watanabe, The Lefschetz Properties, Springer Lecture
Notes 2080, Springer-Verlag, 2013. 19[2] R. P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Progress in Mathematics 41, Birkhäuser, 1996.
[3] K. Yanagawa, Alexander duality for Stanley-Reisner rings and squarefree Nn -graded modules, J.Algebra 225 (2000),
630–645.
, Derived category of squarefree modules and local cohomology with monomial ideal support,, J. Math. Soc. Japan[4]
56 (2004), 289–308.
[5]
Stanley-Reisner rings, sheaves, and Poincaré-Verdier duality, Math. Res. Lett. 110 (2003) 635–650.

Department of Mathematics
Kansai University
Suita, Osaka 564-8680, JAPAN
Email: yanagawa@kansai-u.ac.jp–3–

Finite dimensional algebras arising from dimer models and their derived equivalences
Yusuke Nakajima
A dimer model Γ is a bipartite graph described on the torus, and we can define the quiver Q as thedual of a dimer model. Precisely, we assign a vertex of Q dual to each face of Γ, an arrow of Q dual toeach edge of Γ. For the quiver Q obtained as the dual of a “consistent” dimer model, the path algebrawith certain relations, which is called the Jacobian algebra, has nice properties, for example it gives a
non-commutative crepant resolution of a 3-dimensional Gorenstein toric singularity.
In the theory of dimer models, the notion of perfect matchings is so important. Here, we say that asubset D of edges of a dimer model Γ is a perfect matching if for any node n of Γ there is a unique edgein D containing n as the endpoint. We can also define a perfect matching of Q associated with Γ as thedual of D. By using a perfect matching D of Q, we define the degree dD on each arrow a ∈ Q1 of Q as{
1 if a ∈ D
dD (a) =0 otherwise
and this makes the Jacobian algebra a graded algebra. Then, it is known that this graded Jacobian algebra
arising from a consistent dimer model is a bimodule 3-Calabi-Yau algebra of Gorenstein parameter 1 [1, 2].
Moreover, if the degree zero part of such an algebra is finite dimensional, then it is a 2-representation
infinite algebra (or quasi 2-Fano algebra in the context of noncommutative algebraic geometry) which isa generalization of a representation infinite hereditary algebra (see [3, 4]).
On the other hand, by using perfect matchings we can assign a lattice polygon ∆Γ , which is called
the perfect matching polygon, to each consistent dimer model Γ. In this talk, we first show that internal
perfect matchings which correspond to interior lattice points of ∆Γ give 2-representation infinite algebras.
Theorem 1 ([3]). Let Q be the quiver obtained as the dual of a consistent dimer model Γ, and Λ be thegraded Jacobian algebra whose degree is induced by a perfect matching D. Then, we see that D is aninternal perfect matching of Q if and only if the degree zero part Λ0 is a finite dimensional algebra, inwhich case Λ0 is a 2-representation infinite algebra.
After that, I also discuss the relationship between internal perfect matchings by using the mutationsof perfect matchings.
Theorem 2 ([3]). For a consistent dimer model Γ, internal perfect matchings of Γ are transformed into
each other by the mutations if and only if they correspond to the same interior lattice point of ∆Γ . When
this is the case, 2-representation infinite algebras arising from these internal perfect matchings are derived
equivalent.
References[1] N. Broomhead, Dimer model and Calabi-Yau algebras, Mem. Amer. Math. Soc., 215 no. 1011, (2012).
[2] C. Amiot, O. Iyama and I. Reiten, Stable categories of Cohen-Macaulay modules and cluster categories, Amer. J. Math.
137 (2015), no. 3, 813–857.
[3] M. Herschend, O. Iyama and S. Oppermann, n-representation infinite algebras, Adv. Math. 252 (2014), 292–342.
[4] H. Minamoto and I. Mori, The structure of AS-Gorenstein algebras, Adv. Math. 226 (2011), no. 5, 4061–4095.
[5] Y. Nakajima, On 2-representation infinite algebras arising from dimer models, arXiv:1806.05331.

Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS,The University of Tokyo
5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan
Email: yusuke.nakajima@ipmu.jp–4–

Torsion classes for algebras with radical square zero
Toshitaka Aoki
Let A be a finite dimensional algebra over a field k. A torsion class of the category mod A of finitely generated right A-modules is a full subcategory which is closed under factor modules, extensions and isomorphisms. We denote by tors A the set of torsion classes of mod A, then it forms a complete lattice with respect to inclusion [3]. In general, tors A can be decomposed into subsets torsϵ A (:= [Filt (top ϵA), Fac ϵA]),intervals associated with idempotents ϵ in A. We denote by faithful-tors A the set of faithful torsion classesof mod A, that is, containing all injective modules. From the point of view of tilting theory, torsion classes
have been extensively studied. Recently, it is shown in τ -tilting theory that a class of modules, called
support τ -tilting modules, corresponds to functorially finite torsion classes. Note that tilting modules
are precisely faithful support τ -tilting modules.
In this talk, we give a classification of torsion classes for an arbitrary algebra with radical squarezero in terms of faithful torsion classes for hereditary algebras. A connection between the two classes ofalgebras were first studied by Gabriel [2]. Let A be a finite dimensional k-algebra with radical squarezero. To a given idempotent ϵ in A we can attach a factor algebra Aϵ of A, which is a hereditary algebrawith radical square zero. Then we have the following result.
Theorem 1. Let A be a finite dimensional k-algebra with radical square zero. For each idempotent ϵ inA, we have the following commutative diagram of partially ordered sets:∼

torsϵ A −−−−→ torsϵ Aϵ
x
x




∼

∼

−−−−→ faithful-tors Aop
ϵ
x


∼

sτ -tiltϵ A −−−−→ sτ -tiltϵ Aϵ −−−−→

tilt Aopϵwhere sτ -tiltϵ A (respectively, tilt A) is the set of isomorphism classes of support τ -tilting modules corresponding to functorially finite torsion classes in torsϵ A (respectively, tilting modules over A).
A key observation is that there is a reflection functor mod Aϵ → mod Aop
ϵ . We can get numerical
data on tors A through these isomorphisms. This result is a refinement of the previous work [1] from acategorical perspective.
References[1] T. Aoki, Two-term silting complexes over algebras with radical square zero, arXiv:1903.03795v1.
[2] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103; correction, ibid. 6 (1972), 309.
[3] O. Iyama, I. Reiten, H. Thomas, G. Todorov, Lattice structure of torsion classes for path algebras, Bull. LMS 47 (2015),
no. 4, 639-650.

Graduate School of Mathematics
Nagoya University
Frocho, Chikusaku, Nagoya 464-8602 Japan
Email: m15001d@math.nagoya-u.ac.jp–5–Auslander correspondence for triangulated categories
Norihiro Hanihara
The classical Auslander correspondence [2] ‘for abelian categories’ states a bijection between abeliancategories with finitely many indecomposables and algebras with certain homological invariants, which
provides an important viewpoint in representation theory. The aim of this talk is to give a triangulated
analogue of this theorem. Namely, we give a homological characterization of triangulated categories
satisfying some finiteness conditions. Let k be a field and T be a k-linear, Hom-finite, idempotentcomplete triangulated category.
We consider two finiteness conditions. The first one is that T is finite, that is, T has an additive generator M . In this case, we call EndT (M ) the Auslander algebra of T . We have the following homologicalcharacterization of Auslander algebras of finite triangulated categories:
Theorem 1. Let k be a perfect field. Then, the following are equivalent for a basic finite dimensional
algebra A.
(1) A is the Auslander algebra of a finite triangulated category.
(2) A is self-injective and there exists an automorphism α of A such that Ω3 ≃ (−)α on mod A.
Our proof depends on Amiot’s result [1] as well as Green-Snashall-Solberg’s [3], which plays an essential
role in introducing a triangle strucure on certain additive categories.
The second finiteness condition is the following:• There exists M ∈ T such that T = add{M [n] | n ∈ Z}.
• For any X, Y ∈ T , HomT (X, Y [n]) = 0 for almost all n ∈ Z.
In this case, we say T is [1]-finite. For example, the bounded derived categories of representationfinite hereditary algerbas are [1]-finite. We have the following Auslander correspondence for [1]-finite
triangulated categories:
Theorem 2. Let k be an algebraically closed field. Then, there exists a bijection between the following.
(1) The set of triangle equivalence classes of [1]-finite algebraic triangulated categories.
(2) The set of graded Morita equivalence classes of finite dimensional graded self-injective algebras
such that Ω3 ≃ (−1).
(3) A disjoint union of Dynkin diagrams of type A, D, and E.
A key step toward this bijection is, as is stated in (3), the classfication of [1]-finite algebraic triangulated
categories, which is base on the tilting theorem due to Keller [5].
Applying this classification, we obtain the following result, which partially recovers an equivalene
between the stable categories of simple singularities and the derived categories of Dynkin quivers.
⊕
Corollary 3. Let k be an algebraically closed field and Λ = n≥0 Λn be a positively graded CM-finite
Iwanaga-Gorenstein algebra such that each Λn is finite dimensional over k and Λ0 has finite global dimension. Then, there exists a triangle equivalence CMZ Λ ≃ Db (mod kQ) for a disjoint union Q of Dynkinquivers.
References[1] C. Amiot, On the structure of triangulated categories with finitely many indecomposables, Bull. Soc. math. France 135(3), 2007, 435-474.
[2] M. Auslander, Representation dimension of artin algebras, Queen Mary College Mathematics notes, London, 1971.
[3] E. L. Green, N. Snashall, and Ø. Solberg, The Hochschild cohomology ring of a selfinjective algebra of finite representation type, Proc. Amer. Math. Soc. 131 (11) (2003), 3387-3393.
[4] N. Hanihara, Auslander correspondence for triangulated categories, arXiv:1805.07585.
[5] B. Keller, Deriving DG categories, Ann. scient. Éc. Norm. Sup. (4) 27 (1) (1994) 63-102.

Graduate School of Mathematics
Nagoya University
Furocho, Chikusa-ku, Nagoya, 464-8602 Japan
Email: m17034e@math.nagoya-u.ac.jp–6–THE CHAMBER STRUCTURES OF THE GROTHENDIECK GROUPS COMINGFROM BRICKS
Sota Asai
Let A be a finite-dimensional algebra over a field K, and mod A be the category of finite-dimensional
A-modules, and proj A ⊂ mod A be the subcategory consisting of the projective A-modules.
For the Grothendieck groups K0 (proj A) and K0 (mod A), there exists a non-degenerate Z-bilinear form
K0 (proj A) × K0 (mod A) → Z called Euler form. The indecomposable projective A-modules and thesimple A-modules give dual bases of K0 (proj A) and K0 (mod A) with respect to Euler form.
By [4, 6], there is a bijection from the set 2-silt A of 2-term silting complexes of the homotopy category
Kb (proj A) to the set 2-smc A of 2-term simple-minded collections of the ⊕derived category Db (mod A).
m
Let n be the number of isomorphic classes of simple A-modules. If T = i=1 Ti ∈ 2-silt A with all Ti
indecomposable is sent to X = {Xi }m
i=1 ∈ 2-smc A, then the g-vectors [T1 ], . . . , [Tn ] ∈ K0 (proj A) and thec-vectors [X1 ], . . . , [Xn ] ∈ K0 (mod A) form dual bases [1].
Now we consider the real-valued Grothendieck group K0 (proj A)R := K0 (proj A)⊗Z R. Then K0 (proj A)R
is identified with the Euclidean space Rn . We define the cone C(T ) ⊂ K0 (proj A)R for each T ∈ 2-silt A
by C(T ) := {a1 [T1 ] + · · · + an [Tn ] | a1 , . . . , an ∈ R≥0 }. The cone C(T ) has n walls, and each wall corresponds to a mutation of T . The wall for the mutation of T at Ti is contained in the orthogonal space of[Xi ] ∈ K0 (mod A) with respect to Euler form [2].
For θ ∈ K0 (proj A)R , we can define the numerical torsion class Tθ ⊂ mod A as in [3]. By [7], if θ isan element in the interior of the cone C(T ) for T ∈ 2-silt A, then Tθ coincides with the functorially finite
torsion class TT := Fac H 0 (T ) corresponding to T in [1]. In particular, the set of θ satisfying Tθ = TTcoincides with C(T ) up to boundaries, and we can say that the numerical torsion class Tθ changes in eachtime θ ∈ K0 (proj A)R leaps a wall of cones.
Moreover, θ ∈ K0 (proj A)R also gives the semistable subcategory Wθ of mod A defined in [5]. It is anabelian subcategory of mod A, so every simple object of Wθ is a brick. Thus, for each brick S, we set
Θ(S) := {θ ∈ K0 (proj A)R | S ∈ Wθ } and consider the chamber structure of K0 (proj A)R with the walls
given by Θ(S) for all bricks S. If A is τ -tilting finite, then each chamber is nothing but the cone for a2-term silting complex. However, if A is not τ -tilting finite, then this observation is incomplete, because
the Euclidean space K0 (proj A)R is not covered by the cones for the 2-term silting complexes.
To solve this problem, we define an equivalence relation ∼T called T -equivalence by θ ∼T θ′ : ⇐⇒Tθ = Tθ′ . I have obtained the following two important results on the chamber structure of K0 (proj A)R .
Theorem 1. The walls ΘS determine the T -equivalence classes.
Theorem 2. Let θ ∈ K0 (proj A)R , then the interior of the T -equivalence class of θ is not empty if andonly if Tθ is functorially finite.
References[1] T. Adachi, O. Iyama, I. Reiten, τ -tilting theory, Compos. Math. 150 (2014), no. 3, 415–452.
[2] S. Asai, Semibricks, arXiv:1610.05860, to appear in Int. Math. Res. Not.
[3] P. Baumann, J. Kamnitzer, P. Tingley, Aﬃne Mirković-Vilonen polytopes, Publ. Math. Inst. Hautes Études Sci. 120(2014), Issue 1, 113–205.
[4] T. Brüstle, D. Yang, Ordered exchange graphs, Advances in representation theory of algebras, 135–193, EMS Ser. Congr.
Rep., Eur. Math. Soc., Zürich, 2013.
[5] A. King, Moduli of representations of finite dimensional algebras, Quart. J. Math. Oxford 45 (1994), 515–530.
[6] S. Koenig, D. Yang, Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional
algebras, Doc. Math. 19 (2014), 403–438.
[7] T. Yurikusa, Wide Subcategories are Semistable, Doc. Math. 23 (2018), 35–47.

Graduate School of Mathematics
Nagoya University
Nagoya, Aichi 464-8602 JAPAN
Email: m14001v@math.nagoya-u.ac.jp–7–Decorated Marked Surfaces
Qiu, Yu
Abstract: We introduce the decorated marked surface S△ to study the associated Calabi-Yau-3 category D(S) of an unpunctured marked surface S. In particular, we prove the following:• [DMS I] There is a bijection X from the set CA(S△ ) of closed arcs on S△ to the set Sph D(S)/[Z]of shifts orbits of (reachable) spherical objects in D(S).
• [DMS I] X induces an isomorphism between the braid twist group BT(S△ ), which is a subgroupof the mapping class group of S△ , to the spherical twist group ST D(S).
• [DMS I(B)] There is a bijection from the set of open arcs on S△ to a class of rigid indecomposable
objects in the corresponding perfect category per S.
• [DMS II] The intersection number between arcs equals the dimension of Hom• between objects
under the bijections above.
• [DMS III] The composition of Keller-Yang equivalences for a sequence of mutations in the surface
is path-independent.
• [DMS IV] We give finite presentations of the group BT(S△ ) ∼= ST D(S) w.r.t. triangulations orquivers with potentials.
• [DMS V] The corresponding space Stab◦ D(S) of stability conditions is simply connected.
References[DMS I] Y. Qiu, Decorated marked surfaces: Spherical twists versus braid twists, Math. Ann. 365 (2016) 595–633.
(arXiv:1407.0806)[DMS I(B)] Y. Qiu. Decorated marked surfaces (part B): topological realizations. Math. Z. 288 (2018) 39–53..
[DMS II] Y. Qiu and Y. Zhou, Decorated marked surfaces II: Intersection numbers and dimensions of Homs, to appear in
Tans. Amer. Math. Soc. (arXiv:1411.4003)[DMS III] T. Buan, Y. Qiu and Y. Zhou, Decorated marked surfaces III: The derived category of a decorated marked
surface, arXiv:1804.00094[DMS IV] Y. Qiu and Y. Zhou, Finite presentations for spherical/braid twist groups from decorated marked surfaces,
arXiv:1703.10053.
[DMS V] A. King and Y. Qiu, Cluster exchange groupoids and framed quadratic diﬀerentials, arXiv:1805.00030.

–8–

Connectedness of the Balmer spectrum of the right bounded derived category of acommutative noetherian ring
Hiroki Matsui
This talk is based on the paper [2]. Tensor triangulated geonetry is a theory introduced by Balmer[3] to study tensor triangulated categories by algebro-geometric methods. Let (T , ⊗, 1) be a tensor
triangulated category (i.e., a triangulated category equipped with a symmetric monoidal tensor product⊗ which is compatible with the triangulated structure). Then we can define the notion of ideals, radical
ideals, and prime ideals as in the case of commutative rings when we regard ⊗ as multiplication. For
instance, a prime ideal of T is a proper thick subcategory P of T such that
(1) (ideal) ∀X ∈ T , ∀Y ∈ P, one has X ⊗ Y ∈ P, and(2) (prime) if X ⊗ Y ∈ T , then X ∈ P or X ∈ P.
Then Balmer defines a topological space SpcT as the set of all prime ideals of T with Zariski topology.
Balmer’s celebrated result says that
Theorem 1 (Balmer). There is an order-preserving one-to-one correspondence between(1) the set of radical thick tensor ideals of T and(2) the set of Thomason subsets of SpcT .
From this result, if we want to classify the radical thick tensor ideals of T , we have only to understandthe topological structure of SpcT .
In this talk, we consider the right bounded derived category D− (modR) of a commutative noetherianring R. This triangulated category is a tensor triangulated category with respect to derived tensor product⊗LR , and we can consider its Balmer spectrum SpcD− (modR). We discuss some topological structures ofthis topological space SpcD− (modR). The following theorem is one of the main result in this talk.
Theorem 2. The Balmer spectrum SpcD− (modR) is connected (irreducible) if and only if so is the
Zariski spectrum SpecR.
The key to prove our main theorem is the classification of thick tensor ideals of D− (modR) generated
by bounded complexes given in [3].
References[1] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168.
[2] H. Matsui, Connectedness of the Balmer spectra of right bounded derived categories, Connectedness of the Balmer
spectra of right bounded derived categories, to appear in J. Pure Appl. Algebra.
[3] H. Matsui; R. Takahashi, Thick tensor ideals of right bounded derived categories, Algebra Number Theory 11 (2017),
no.7, 1677–1738.

Graduate School of Mathematics
Nagoya University
Furo-cho, Chikusa-ku, Nagoya 400-8511 JAPAN
Email: m14037f@math.nagoya-u.ac.jp–9–Dimensions of singular categories of hypersurfaces of countable representation type
Tokuji Araya, Kei-ichiro Iima, Maiko Ono, Ryo Takahashi
Throughout this talk, let k be an algebraically closed field of characteristic zero, and let R be acomplete local hypersurface over k with countable representation type. Denote by Dsg (R) the singularity
category of R, and let Dosg (R) be the full subcategory of Dsg (R) consisting of objects locally zero on the
punctured spectrum.
Let T be a triangulated category. For a subcategory X of T we denote by ⟨X ⟩ the smallest subcategory of T containing X which is closed under isomorphisms, shifts, finite sums and summands. For
subcategories X , Y of T we denote by X ∗ Y the subcategory consisting of objects M ∈ T such that there
is an exact triangle X → M → Y → X[1] with X ∈ X and Y ∈ Y. Set X ⋄ Y := ⟨⟨X ⟩ ∗ ⟨Y⟩⟩. For asubcategory X of T we put ⟨X ⟩0 := 0, ⟨X ⟩1 := ⟨X ⟩, and inductively define ⟨X ⟩n := X ⋄ ⟨X ⟩n−1 for n ≥ 2.
The dimension of T with respect to X , denote by dimX T , is the infimum of {n ≥ 0 | T = ⟨X ⟩n+1 }.
The main results in this talk are the following two theorems.
D (R)Theorem 1. For all nonzero objects M of Dsg (R), the residue field k belongs to ⟨{M }⟩2 sg.

Theorem 2. Let T be a nonzero thick subcategory of Dsg (R), and let X be a full subcategory of T . Thenthe following statements hold.
(1) T coincides with either Dsg (R) or Dosg (R).
(2) (a) If T = Dsg (R), then(⟨X ⟩ = T ),
0dimX T = 1(⟨X ⟩ =
̸ T , ⟨X ⟩ ⊈ Dosg (R)),
∞ (⟨X ⟩ ⊆ Dosg (R)).
(b) If T = Dosg (R), then0dimX T = 1
∞(⟨X ⟩ = T ),(⟨X ⟩ =
̸ T , #ind⟨X ⟩ = ∞),(#ind⟨X ⟩ < ∞).
References[1] T. Aihara; T. Araya; O. Iyama; R. Takahashi; M. Yoshiwaki, Dimensions of triangulated categories with respect to
subcategories, J. Algebra 399 (2014), 205–219.
[2] M. Ballard; D. Favero; L. Katzarkov, Orlov spectra: bounds and gaps, Invent. Math. 189 (2012), 359–430.

Department of Applied Science, Faculty of Science, Okayama University of Science,
Ridaicho, Kitaku, Okayama 700-0005, Japan.
Email: araya@das.ous.ac.jp
Department of Liberal Studies, National Institute of Technology, Nara College, 22
Yata-cho, Yamatokoriyama, Nara 639-1080, Japan.
Email: iima@libe.nara-k.ac.jp
Graduate School of Natural Science and Technology, Okayama University,
Okayama 700-8530, Japan
Email: onomaiko@s.okayama-u.ac.jp
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya,
Aichi 464-8602, Japan
Email: takahashi@math.nagoya-u.ac.jp–10–Resolution of DG-modules and their applications for commutative DG-algebras
Hiroyuki MinamotoDiﬀerential graded (DG) algebra lies in the center of homological algebra and allows us to use techniques of homological algebra of ordinary algebras in much wider context. Projective resolutions andinjective resolutions which are the fundamental tools of homological algebra already have their DGversions, which are called a DG-projective resolution and a DG-injective resolution. The aim of this talkis to introduce a diﬀerent DG-versions of projective resolution (sup-projective resolution) and injective
resolutions (inf-injective resolutions) for DG-modules over a connective DG-algebra.
Recall that a (cohomological) DG-algebra R is called connective if the vanishing condition H>0 (R) = 0
of the cohomology groups is satisfied. There are rich sources of connective DG-algebras: the Koszul
algebra KR (x1 , · · · , xd ) in commutative ring theory, and an endomorphism DG-algebra RHom(S, S) of asilting object S. We would like to point out that a commutative connective DG-algebras are regarded as
the coordinate algebras of derived aﬃne schemes in derived algebraic geometry (see e.g. [1]).
The motivation for this work came from the projective dimensions and the injective dimensions for
DG-modules introduced by Yekutieli. In the paper [3] he introduced projective dimension and injective dimension of DG-modules by generalizing the characterization of projective dimension and injective
dimension of ordinary modules by vanishing of Ext-group. An important feature of newly introduced
resolutions is that, roughly speaking, the “length” of these resolutions give projective or injective dimensions.
Theorem 1. Let R be a connective DG-algebra. Then a DG-R-module M is of pdR M = d if and only
if it has a sup-projective resolution P• of length e such thatd = e + sup M − sup Pe .
Other conditions for pdM = d will be given in the talk or the poster.
We show that these resolutions allows us to investigate basic properties of projective and injective
dimensions of DG-modules. As an application we introduce the global dimension of a connective DGalgebra and show that finiteness of global dimension is derived invariant.
If time permits, we will discuss a commutative DG-algebra. We observe that a DG-counter partER (R/p) of the class of indecomposable injective modules are parametrized by prime ideals p ∈ SpecH0 (R)of the 0-th cohomology algebra. This fact is compatible with the view point of derived algebraic geometrythat the base aﬃne scheme of the derived aﬃne scheme SpecR associated to a CDGA R is the aﬃnescheme SpecH0 (R). We give a structure theorem of minimal injective resolution of dualizng complex.
This talk is a report of my paper [2].
References[1] Gaitsgory, Dennis; Rozenblyum, Nick, A study in derived algebraic geometry. Vol. I. Correspondences and duality.
Mathematical Surveys and Monographs, 221. American Mathematical Society, Providence, RI, 2017.
[2] Minamoto. H, Resolutions of DG-modules and their applications to commutative DG-algebras, arXiv: 180201994[3] Yekutieli, A. Duality and Tilting for Commutative DG Rings, arXiv:1312.6411

Hiroyuki Minamoto
Department of Mathematics and Information Sciences Osaka Prefecture University
Sakai Nakamozu, Osaka 599-8531 JAPAN
Email: minamoto@mi.s.osakafu-u.ac.jp–11–Relations for Grothendieck groups
and representation-finiteness
Haruhisa Enomoto
A Grothendieck group is the abelian group associated with an exact category. It was known that thedefining relations of the Grothendieck group is closely related to the representation-finiteness. My aimof this talk is to unify and generalize several known results about this.
Let E be a Krull-Schmidt exact category. A Grothendieck group K0 (E) of E is defined to be the quotientgroup K0 (E, 0)/Ex(E), where K0 (E, 0) is the free abelian group with basis indE (the set of isomorphismclasses of indecomposables in E) and Ex(E) is the subgroup of K0 (E) generated by{[X] − [Y ] + [Z] | there exists a short exact sequence 0 → X → Y → Z → 0 in E}.
Among short exact sequences, AR sequences are minimal in some sense, and have played an essential
role in the representation theory of algebras and commutative rings. We denote by AR(E) the subgroupof Ex(E) generated by AR sequences.
For an artin algebra Λ, it was proved by Butler [Bu] and Auslander [Au] that AR(mod Λ) = Ex(mod Λ)holds if and only if mod Λ has finitely many indecomposables. Similar results were obtained for certain subcategories of mod Λ, e.g. [MMP, PR]. Our result about artin algebras is the following, which
generalizes all of these results.
Theorem 1. Let Λ be an artin algebra and E a contravariantly finite resolving subcategory of mod Λ.
Then E has finitely many indecomposables if and only if AR(E) = Ex(E) holds.
Next I will discuss what happens if we drop the assumption of artin-ness. Similar equivalences wasproved for the category CM Λ of Λ-lattices over an order Λ under some restrictions: [AR, Hi], and wehave a partial result [Ko]. I will give a partial result on this which generalizes these results:
Theorem 2. Let R be a complete Cohen-Macaulay local ring and Λ an R-order. Then the following
holds.
(1) If CM Λ has finitely many indecomposables, then AR(CM Λ) = Ex(CM Λ) holds.
Suppose that AR(CM Λ) = Ex(CM Λ) holds. Then we have the following.
(2) ΩCM Λ, the category of syzygies of modules in CM Λ, has finitely many indecomposables.
(3) Assume that Λ is a Gorenstein order or has finite global dimension. Then CM Λ has finitely many
indecomposables.
We use the functorial method to prove these results. If time permits, I will give a general relations,which is used to prove above, between the conditions: (a) having finitely many indecomposables, (b)some functorial condition and (c) AR=Ex.
References[Au] M. Auslander, Relations for Grothendieck groups of Artin algebras, Proc. Amer. Math. Soc. 91 (1984), no. 3, 336–340.
[AR] M. Auslander, I. Reiten, Grothendieck groups of algebras and orders, J. Pure Appl. Algebra 39 (1986), no. 1–2, 1–51.
[Bu] M. C. R. Butler, Grothendieck groups and almost split sequences, Lecture Notes in Math., 882, Springer, Berlin-New
York, 1981.
[Hi] N. Hiramatsu, Relations for Grothendieck groups of Gorenstein rings, Proc. Amer. Math. Soc. 145 (2017), no. 2,
559–562.
[Ko] T. Kobayashi, Syzygies of Cohen-Macaulay modules and Grothendieck groups, J. Algebra 490 (2017), 372–379.
[MMP] E. N. Marcos, H. A. Merklen, M. I. Platzeck, The Grothendieck group of the category of modules of finite projective
dimension over certain weakly triangular algebras, Comm. Algebra 28 (2000), no. 3, 1387–1404.
[PR] M. I. Platzeck, I. Reiten, Modules of finite projective dimension for standardly stratified algebras, Comm. Algebra 29(2001), no. 3, 973–986.

Graduate School of Mathematics
Nagoya University
Chikusa-ku, Nagoya. 464-8602, Japan
Email: m16009t@math.nagoya-u.ac.jp–12–Sortable elements and torsion pairs for quivers
Yuya Mizuno
Path algebras are one of the most fundamental and important classes of algebras. In this talk, we
discuss torsion pairs for the module category of a path algebra, and explain a close relationship between
torsion pairs and the elements of the Coxeter group of the quiver.
Let Q be acyclic quiver and W the Coxeter group of Q. Then the result of [ORT] asserts that there
exists a bijection between the elements of W and the set of cofinite quotient-closed subcategories ofmodkQ. Thus, for a given w ∈ W , we can give a cofinite quotient-closed category Cw . Then we pose thefollowing natural questions.
Question: (1) When is Cw a torsion class of modkQ for w ∈ W ?
(2) When Cw is a torsion class, how can we describe the corresponding torsion free class ?
In this talk, we will give an answer for the above question. Our method is the theory of preprojective
algebra Π of Q. The result [BIRS] allows a connection between the representation theory of Π and W .
Using this connection, we parametrize torsion pairs of modkQ by some elements of W , called sortableelements, and we explain a conjecture by [ORT].
This is based on our work [MT].
References[BIRS] A. B. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups
Compositio Mathematica, Volume 145, Issue 4 (2009) pp. 1035–1079.
[MT] Y. Mizuno, H. Thomas, Torsion pairs for quivers and the Weyl groups, arXiv:1804.02148.
[ORT] S. Oppermann, I. Reiten, H. Thomas, Quotient closed subcategories of quiver representations, Compositio Mathematica, Volume 151, Issue 3 (2015) pp. 568–602.

Faculty of Science Shizuoka University
836 Ohya, Suruga-ku, Shizuoka, 422-8529
Email: yuya.mizuno@shizuoka.ac.jp–13–

A geometric model of Brauer graph algebras
Takahide Adachi and Aaron Chan
Given a surface, we can construct various triangulated categories which are usually motivated by thehomological mirror symmetry conjecture and related Calabi-Yau algebras/categories. In the constructions, curves on a surface frequently give rise to an important class of objects. Since a Brauer graphcan be embedded into a certain surface with marked points and boundary, we can also ask if curves ofthe surface give us any interesting objects in the triangulated categories associated to a Brauer graphalgebra. Generalizing ideas from works by Khovanov-Seidel [1], Seidel-Thomas [3] and Marsh-Schroll [2],we associate curves on the surface with complexes in the bounded homotopy category. In this talk, wegive details on this construction and show how one can interpret some homological phenomenons andproblems of a Brauer graph algebra using the combinatorics of the associated surface.
References[1] M. Khovanov, P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002), no. 1,
203–271.
[2] R. Marsh, S. Schroll, The geometry of Brauer graph algebras and cluster mutations, J. Algebra 419 (2014), 141–166.
[3] P. Seidel, R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), no. 1,
37–108.

Takahide Adachi
Graduate school of Science
Osaka Prefecture University
1-1 Gakuen-cho, Nakaku, Sakai, Osaka, 599-8531, JAPAN
Email: adachi@mi.s.osakafu-u.ac.jp
Aaron Chan
Graduate School of Mathematics
Nagoya University
Furocho, Chikusaku, Nagoya 464-8602 JAPAN
Email: aaron.kychan@gmail.com–14–Cyclotomic polynomials
Kaoru Motose
In this talk, I present about relationship between the fundamental results about cyclotomic polynomials
and the next mathematical items, 1. Order, 2. Decomposition of cyclotomic polynomials over fields,
3. Feit Thomphson conjecture, and 4. Rational primes.
References[1] M. Aigner and G. M. Ziegler, Proofs from the book, 2009, Springer.
[2] K. Ireland and M. Rosen, A classical introduction to modern number theory,2nd ed. 1990, G. T. M. 84, Springer.
[3] K. Motose, On values of cyclotomic polynomials, Math. J. Okayama Univ. 35 (1993), 35-40.
[4]II.,, 37 (1995), 27-36.
[5]V.,, 45 (2003), 29-36.
[6] T. Ono, An introduction to algebraic number theory, 1990, Plenum Press.

036-8171 Toriage, 5-13-5, Hirosaki, JAPAN–15–

Batalin-Vilkovisky algebra structures on the Hochschild cohomology of self-injective
Nakayama algebras
Tomohiro Itagaki
In this talk, we give Batalin-Vilkovisky (BV) algebra structures on the Hochschild cohomology ofself-injective Nakayama algebras over an algebraically closed field.
Tradler [2] discovered that Hochschild cohomology of arbitrary symmetric algebra has a BV algebra
structure given by a symmetric bilinear form. Later, Lambre, Zhou and Zimmermann [1] discovered that
Hochschild cohomology of Frobenius algebras with diagonalizable Nakayama automorphism has a BV
algebra structure. However, it is not known that Hochschild cohomology of Frobenius algebras has a BV
algebra structure in general, and there are few examples of complete calculation of BV diﬀerentials on
Hochschild cohomology of Frobenius algebras which are not symmetric.
Recently, for any Frobenius algebra A, Volkov [3] defined the cohomology HH∗ (A)ν↑ of Hochschild complex related to Nakayama automorphism ν, which induces Gerstenhaber algebra (HH∗ (A)ν↑ , ⌣, [ , ]).
Moreover, Volkov also found a BV algebra structure on (HH∗ (A)ν↑ , ⌣, [ , ]). In particular, if the
Nakayama automorphism ν is diagonalizable, then HH∗ (A)ν↑ ∼= HH∗ (A) and the BV diﬀerential on∗(HH (A)ν↑ , ⌣, [ , ]) induces the one on the Gerstenhaber algebra (HH∗ (A), ⌣, [ , ]).
We give BV diﬀerentials on the Hochschild cohomology of self-injective Nakayama algebras by dividing
the computation into two cases: (a) the characteristic of the ground field does not divide the order of the
Nakayama automorphism; (b) the characteristic of the ground field divides the order of the Nakayama
automorphism. For a self-injective Nakayama algebra Λ in case (b), by computing HH∗ (Λ)ν↑ and BV
diﬀerentials on (HH∗ (Λ)ν↑ , ⌣, [ , ]), we have HH∗ (Λ)ν↑ ∼= HH∗ (Λ) and [ , ] = 0. On the other hand,[ , ] ̸= 0 on Hochschild cohomology of Λ in case (a) in general.
References[1] T. Lambre, G. Zhou and A. Zimmermann, The Hochschild cohomology ring of a Frobenius algebra with semisimple
Nakayama automorphism is a Batalin-Vilkovisky algebra, J. Algebra 446 (2016), 103–131.
[2] T. Tradler, The Batalin-Vilkovisky algebra on Hochschild cohomology induced by infinity inner products, Ann. Inst.
Fourier 58 (7) (2008), 2351–2379.
[3] Yu. V. Volkov, BV diﬀerential on Hochschild cohomology of Frobenius algebras, J. Pure. Appl. Algebra 220 (2016),
3384–3402.

Department of Mathematics
Faculty of Science Division 1
Tokyo University of Science
1-3, Kagurazaka, Shinjuku, Tokyo, 162-8601, JAPAN
Email: titagaki@rs.tus.ac.jp–16–

An application of Hochschild cohomology to the moduli of subalgebrasof the full matrix ring
Kazunori Nakamoto and Takeshi Torii
Let us begin with the definition of the moduli of subalgebras of the full matrix ring.
Definition 1. We say that a subsheaf A of OX -algebras of Mn (OX ) is a mold of degree n on a scheme
X if Mn (OX )/A is a locally free sheaf. We denote by rankA the rank of A as a locally free sheaf.
Proposition 2. The following cotravariant functor is representable by a closed subscheme of the Grassmann scheme Grass(d, n2 ):Moldn,d:(Sch)opX→ (Sets){}A A is a rank d mold of degree n on X
7→.

Let A be the universal mold on Moldn,d . For x ∈ Moldn,d , set A(x) := A ⊗OMoldn,d k(x), where k(x)is the residue field of x. We describe the dimension of the Zariski tangent space Tx Moldn,d and thesmoothness of Moldn,d → Z at x by using Hochschild cohomology H i (A(x), Mn (k(x))/A(x)).
Theorem 3. Set N (A(x)) := {Y ∈ Mn (k(x)) | [X, Y ] ∈ A(x) for X ∈ A(x)}. The dimension of the
Zariski tangent space Tx Moldn,d is given by
dim Tx Moldn,d = dim H 1 (A(x), Mn (k(x))/A(x)) + n2 − dim N (A(x)).
By using cohomology classes of H 2 (A(x), Mn (k(x))/A(x)), we can describe the smoothness of Moldn,d →Z at x. In particular, we have
Theorem 4. If H 2 (A(x), Mn (k(x))/A(x)) = 0, then Moldn,d → Z is smooth at x.
Let A be an R-subalgebra of Mn (R) over a commutative ring R. Assume that Mn (R)/A is a projective
R-module. We introduce several results on H i (A, Mn (R)/A).
Let Q = (Q0 , Q1 ) be an ordered quiver. For a, b ∈ Q0 , we say that a ≥ b if a = b or there exists
an oriented path from b to a. The incidence algebra RQ/I can be written by Λ = ⊕a≥b Reba . Fix
a numbering Q0 = {1, 2, . . . , n}. By regarding eba as Eba ∈ Mn (R), Λ can be identified with the Rsubalgebra ⊕a≥b REba of Mn (R). In a similar way as [1], we have the following theorem:
Theorem 5 ([2]). Let Λ be as above. Then H i (Λ, Mn (R)/Λ) = 0 for i ≥ 0.

∑s
Theorem 6 ([2]). For n1 +· · · +nr = n, we set Pn1 ,n2 ,...,nr (R) = {(aij ) ∈ Mn (R) | aij = 0 if
k=1 nk <∑s∑s+1
ii ≤ k=1 nk and j ≤ k=1 nk }. Then H (Pn1 ,n2 ,...,nr (R), Mn (R)/Pn1 ,n2 ,...,nr (R)) = 0 for i ≥ 0.
References[1] C. Cibils, Cohomology of incidence algebras and simplicial complexes, Journal of Pure and Applied Algebra 56 (1989)
221–232.
[2] K. Nakamoto and T. Torii, An application of Hochschild cohomology to the moduli of subalgebras of the full matrix ring,in preparation.

Center for Medical Education and Sciences
Faculty of Medicine
University of Yamanashi
Chuo, Yamanashi 409-3898, Japan
Email: nakamoto@yamanashi.ac.jp
Department of Mathematics
Okayama University
Okayama 700-8530, Japan
Email: torii@math.okayama-u.ac.jp–17–Some remarks on Avella-Alaminos-Geiss invariants of gentle algebras
Hiroyuki Nakaoka
In [1], Avella-Alaminos and Geiss have introduced derived invariants ϕA : N×N → N for gentle algebras
A = KQ/I. Since these invariants can be calculated combinatorially from the bound quiver (Q, I), they
are eﬀectively used in the classifiation of gentle algebras with some conditions, e.g. in [2]. They also haverelation with the dimensions of the Hochschild cohomologies HH n (A), as shown in [3] and [4].
In this talk, I would like to introduce its definition and properties from related literatures, and possibly
give an experimental construction of algebras related to these invariants.
References[1] D. Avella-Alaminos, C. Geiss, Combinatorial derived invariants for gentle algebras, J. Pure Appl. Algebra 212 (2008),
no. 1, 228–243.
[2] G. Bobiński, Derived equivalence classification of the gentle two-cycle algebras, Algebr. Represent. Theory 20 (2017),
no. 4, 857–869.
[3] S. Ladkani, Hochschild cohomology of gentle algebras, Algebr. Represent. arXiv:1208.2230.
[4] M.J. Redondo, L. Román, Gerstenhaber algebra structure on the Hochschild cohomology of quadratic string algebras,
Algebr. Represent. Theory 21 (2018), no. 1, 61–86.

Research and Education Assembly
Science and Engineering Area
Research Field in Science
Kagoshima University
1-21-35 Korimoto, Kagoshima, 890-0065 JAPAN
Email: nakaoka@sci.kagoshima-u.ac.jp–18–

On projective modules with unique maximal submodule
唯一の極大部分加群を持つ射影加群について
Masahisa Sato
Aichi University & Yamanashi University
INTRODUCTION
It passed 47 years after R.Ware gave the following problem in Endomorphism rings of projective
modules, Trans. Amer. Math. Soc. 155 (1971), 233-256.;
Let R be a ring and P a projective right R-module with unique maximal submodule L, then L is thelargest submodule of P .
In this talk, we report aﬃrmative answer for this problem.
Before we solve this problem, we show the following fact;
Any projective module has a maximal submodule.
This implies the following fact which generalizes Nakaya-Azumaya Lemma for projective modules.
If P J(R) = P for a projective module P , then P = 0.
ON PRIMITIVE RINGS AND IDEALS
We review the definition of primitive rings and primitive ideals and their basic properties.
Definition 1. A ring R is called a right primitive ring if there is a faithful simple right R-module. A
two sided ideal T of R is called a primitive right ideal if the factor algebra R/T is a primitive ring.
The Jacobson radical J(R) of R is the intersection of all maximal right ideals of R. This is same as
the intersection of all primitive right ideals of R.
Let M be a right R-module and S a subset of M . An (right) annihilator of S is defined by AnnR (S) ={r ∈ R |Sr = 0}.
Remark 2. A primitive right ideal T with a faithful simple right R/T -module R/J is given by the form
T = AnnR (R/J).
∩ T is maximal between two sided ideals included in J.
∩ HenceI, here Γ is the set consisting of a maximal right ideal I with T ⊂ I and ∆ isI =Also T =I∈ΓI∈∆

the set consisting of a maximal right ideal I with R/J ∼= R/I.
STRUCTURE THEOREM
The following is a key theorem to solve R.Ware’s problem.
Proposition 3. Let R be a ring and P a projective right R-module with unique maximal submodule L,then P is indecomposable or there are direct summands P1 and P2 such that P = P1 ⊕ P2 , P1 has uniquemaximal submodule and P2 does not have any maximal submodules.
EXAMPLES
We give the following examples.
(1) Non-projective indecomposable module with unique maximal submodule but not largest submodule.
i.e., Ware’s problem is not ture for non-projective modules in general.
(2) A non-projective module M with the properties M J ̸= M for only one maximal right ideal J and
M I = M for any maximal right ideal I ̸= J.
(3) An infinitely generated projective module P with the properties P J ̸= P for only one maximal rightideal J and P I = P for any maximal right ideal I ̸= J.

–19–

On (finite) Σ-Rickart modules:
On a module theoretic setting of the (semi-)hereditary property of rings
Gangyong Lee∗ and Mauricio Medina-Bárcenas(Chungnam National University∗ , Chungnam National University)
After Kaplansky introduced hereditary rings in the earliest 50’s, they have been extensively investigatedin the literature. Hereditary rings have been characterized in diﬀerent ways, the most common of themis that given in [1, Ch.I, Theorem 5.4]: a ring R is right hereditary if and only if every submodule of any
projective right R-module is projective if and only if every factor module of any injective right R-module
is injective. Semi-hereditary rings also ware characterized as a similar way as hereditary rings.
In this talk, we introduce the notion of (finite) Σ-Rickart modules by utilizing the endomorphism ringof a module and by using the recent notion of Rickart modules [2] as a module theoretic analogue of aright (semi-)hereditary ring. A module M is called (finite) Σ-Rickart if every (finite) direct sum of copies
of M is Rickart ([3, Definition 2.21]). It is shown that any direct summand and any (finite) direct sum of
copies of a (finite) Σ-Rickart module are Σ-Rickart modules. Also, we provide several characterizations of(finite) Σ-Rickart modules which are including generalizations of the most common results (see the above
results for rings) of (semi-)hereditary rings in a module theoretic setting. Also, we have a characterizationof a finitely generated (finite) Σ-Rickart module in terms of its endomorphism ring.
This talk is based on a joint work with Mauricio-Bárcenas.
References[1] H. Cartan; S. Eilenberg, Homological Algebra, Princeton University Press (1956)[2] G. Lee; S.T. Rizvi; C.S. Roman, Rickart modules, Comm. Algebra, 2010 38(11), 4005–4027[3] G. Lee; S.T. Rizvi; C.S. Roman, Direct sums of Rickart modules, J. Algebra, 2012 353, 62–78–20–

A strongly quasi-hereditary structure on Auslander–Dlab–Ringel algebras
Mayu Tsukamoto
Ringel introduced special classes of quasi-hereditary algebras called left-strongly quasi-hereditary algebras and strongly quasi-hereditary algebras [5]. It is known that left-strongly quasi-hereditary algebras
have better upper bound for global dimension than that of arbitrary quasi-hereditary algebras.
Left-strongly (resp. strongly) quasi-hereditary algebras are strongly related to rejective chains defined
below. Let C be a Krull–Schmidt category. A chain C = C0 ⊃ C1 ⊃ · · · ⊃ Cn = 0 of subcategories of C iscalled a total left rejective chain if the following conditions hold for 1 ≤ i ≤ n:(a) Ci is a left rejective subcategory of C (i.e., for any X ∈ C, there exists an epic left Ci -approximationf X ∈ C (X, Y ) of X);(b) the Jacobson radical of the factor category Ci−1 /[Ci ] is zero.
Dually, a total right rejective chain is defined. A rejective chain is defined as a total left rejective chain
and a total right rejective chain.
Proposition 1 ([6, Theorem 3.22]). Let A be an artin algebra. Then A is left-strongly (resp. strongly)
quasi-hereditary if and only if the category projA has a total left rejective (resp. rejective) chain.
Let A be an artin algebra with Loewy length⊕m m and J the Jacobson radical of A. Auslander provedthat the endomorphism algebra B := EndA ( i=1 A/J i ) has finite global dimension [1]. Moreover, Dlab
and Ringel showed that B is quasi-hereditary [3]. Hence B is called an Auslander–Dlab–Ringel (ADR)
algebra. Recently, Conde gave a left-strongly quasi-hereditary algebra structure on ADR algebras [2].
In this talk, we study ADR algebras of semilocal modules introduced by Lin and Xi [4]. Recall thata module M is said to be semilocal if M is a direct sum of modules which have a simple top. Since any
artin algebra is a semilocal module, the ADR algebras of semilocal modules are a generalization of theoriginal ADR algebras. In [4], they proved that ADR algebras of semilocal modules are quasi-hereditary.
We refine this result by using Proposition 1.
Theorem 2 ([7, Theorem 2.2]). Let A be an artin algebra,
a semilocal A-module with Loewy length⊕Mmim and J the Jacobson radical of A. Thenthecategory
add(
i=1 M/M J ) has a total left rejective chain.
⊕mi
In particular, the ADR algebra EndA ( i=1 M/M J ) is left-strongly quasi-hereditary.
As an application, we give a tightly upper bound for global dimension of an ADR algebra.
It is known that the global dimension of any strongly quasi-hereditary algebra is at most two [5,Proposition A.2]. We note that algebras with global dimension at most two are not always strongly
quasi-hereditary. We prove that the converse holds if B is an original ADR algebra.
Theorem 3 ([7, Theorem 3.1]).
⊕mLet A be an artin algebra with Loewy length m ≥ 2 and J the Jacobsonradical of A. Let B := EndA ( i=1 A/J i ) be the ADR algebra of A. Then B is strongly quasi-hereditary
if and only if gl.dimB = 2 holds.
References[1] M. Auslander, Representation dimension of Artin algebras, Queen Mary College, 1971.
[2] T. Conde, The quasihereditary structure of the Auslander–Dlab–Ringel algebra, J. Algebra 460 (2016), 181–202.
[3] V. Dlab and C. M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1989), no. 1, 1–5.
[4] Y. N. Lin and C. C. Xi, Semilocal modules and quasi-hereditary algebras, Arch. Math. (Basel) 60 (1993), no. 6, 512–516.
[5] C. M. Ringel, Iyama’s finiteness theorem via strongly quasi-hereditary algebras, J. Pure Appl. Algebra 214 (2010), no.
9, 1687–1692.
[6] M. Tsukamoto, Strongly quasi-hereditary algebras and rejective subcategories, arXiv:1705.03279, to appear in Nagoya
Math. J.
[7]
, On upper bound for global dimension of Auslander–Dlab–Ringel algebras, arXiv:1805.08085.

Advanced Mathematical Institute
Osaka City University
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 JAPAN
Email: tsukamoto@sci.osaka-cu.ac.jp–21–Components of the stable Auslander-Reiten quiver for a symmetric order over a complete
discrete valuation ring
Kengo Miyamoto
In representation theory of algebras, we often use Auslander–Reiten theory to analysis various additive
categories arising in representation theory because we may prove many important combinatorial andhomological properties with the help of the theory. For important classes of finite dimensional algebras,
there exist strong restrictions on stable Auslander–Reiten quivers, for example see [2, 6, 7, 8]. However, if
the base ring is not a field but a complete regular local ring, then the shape of (stable) Auslander–Reiten
components for algebras are mostly unknown.
Let O be a complete discrete valuation ring, κ is the residue field, K the quotient field and A an
O-order. We denote by latt-A the category consisting of A-lattices. Then, the category latt-A admits
almost split sequences if and only if A is an isolated singularity. In this case, one can find some results on
the shape of Auslander–Reiten components, for example [3]. Otherwise, we have to consider a suitable
full subcategory of latt-A which admits almost split sequences. In [1], we considered the full subcategoryof latt-A, say latt(♮) -A, consisting of A-lattices M such that M ⊗O K is projective as an A ⊗O K-module,
and we defined the concept of the stable Auslander–Reiten quiver for latt(♮) -A, for a symmetric O-order
A. The following is the main results.
Theorem 1 ([1, 4]). Let A be a symmetric O-order and C be a stable AR component for latt(♮) -A with
infinitely many vertices. Then, the following statements hold.
(1) Suppose that C is periodic. Then, one of the following statements holds:(i) If C has no loops, then the tree class is one of infinite Dynkin diagrams.
(ii) If C has loops, then C \ {loops} = ZA∞ /⟨τ ⟩, and the loops appear on the boundary of C.
(2) Suppose that C is non-periodic. Then, C has no loops. Moreover, if either C does not containHeller lattices or A ⊗O κ has finite representation type, then the tree class of C is one of infinite
Dynkin diagrams or Euclidean diagrams.
By applying this restriction to the symmetric Kronecker algebra A, we have the following.
Theorem 2 ([4, 5]). Let A be the symmetric Kronecker algebra over O, and let C be a stable ARcomponent for latt(♮) -A that contains a Heller lattice. If C is non-periodic, then C ≃ ZA∞ . Otherwise, C
is isomorphic to ZA∞ /⟨τ ⟩ if the Heller lattice is given from strings, else ZA∞ /⟨τ 2 ⟩.
References[1] S. Ariki, R. Kase and K. Miyamoto, On Components of stable Auslander -Reiten quivers that contains Heller lattices:
the case of truncated polynomial rings, Nagoya math. J., 228 (2017), 72–113.
[2] M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge studies in advanced mathematics 36, Cambridge University Press, 1995.
[3] X. Luo, 0-Calabi–Yau configurations and finite Auslander–Reiten quivers of Gorenstein orders, J. Pure and Applied
Algebra Volume 219, Issue 12, 5590–5630.
[4] K. Miyamoto, On the non-periodic stable Auslander–Reiten Heller component for the Kronecker algebra over a complete
discrete valuation ring, arXiv:1601.06256.
[5] K. Miyamoto, On periodic stable Auslander–Reiten component containing Heller lattices over the symmetric Kronecker
algebra, preprint.
[6] C. Riedtmann, Representation-finite selfinjective algebras of class An , Lecture Notes in Mathematics 832 (Springer,
Berlin, 1980), 449–520.
[7] G. Todorov, Almost split sequences for DT r-periodic modules, Lecture Notes in Mathematics 832 (Springer, Berlin,1980), 600–631.
[8] P. J. Webb, The Auslander-Reiten quiver of a finite group, Math. Z. 179 (1982), 97–121.

Pure and Applied Mathematics
Graduate School of Information Science and Technology
Osaka University
Suita, Osaka 565-0871, JAPAN
Email: k-miyamoto@ist.osaka-u.ac.jp–22–X-Stability Conditions on Calabi-Yau-X Categories
Qiu, Yu
Abstract: We introduce the notion of X-stability condition, consisting of a Bridgeland stability condition and a complex number s, on a Calabi-Yau-X category DX satisfying the equation
X(σ) = s · σ.

(1)Here, the Grothendieck group of DX isK(DX ) ∼= R⊕n ,

R = Z[q ±1 ]and the R-structure on K(DX ) is provided by the auto-equivalence X. This gives the q-deformationof Bridgeland’s stability condition, in the sense that when fixing s, the space XStabs DX of X-stability
conditions (σ, s) is a complex manifold with dimension n.
We also introduce the R-generalization of global dimension for algebras, the global dimension function
on a stability condition σ = (Z, P):(2)

gldim σ = sup{ϕ2 − ϕ1 | Hom(P(ϕ1 ), P(ϕ2 )) ̸= 0},

where P is the slicing (a R-collection of t-structures). We give a criterion
gldim σb + 1 ≤ Re(s)of constructing an X-stability condition on DX from an usual stability condition σb on the X-heart D∞ of
DX .
We discuss motivation/application that relates to Saito-Frobenius structure, mirror symmetry andcluster theory.
References[CY-X I] A. Ikeda and Y. Qiu, X-Stability conditions on Calabi-Yau-X categories and twisted periods, arXiv:1807.00469[CY-X II] A. Ikeda and Y. Qiu, X-Stability conditions on Calabi-Yau-X categories II, in preparation.
[CY-X III] A. Ikeda, Y. Qiu and Y. Zhou, X-Stability conditions on Calabi-Yau-X categories III, in preparation.
[CY-X 0] Y. Qiu, Global dimension function, Gepner equations and X-Stability conditions, arXiv:1807.00010–23–

On certain Morita invariants involving
commutator subspace and radical powers
Shigeo Koshitani and Taro Sakurai
In 1941, Brauer-Nesbitt [2] established a characterization of a block of a finite group with trivial defectgroup as a block B with k(B) = 1 where k(B) is the number of irreducible ordinary characters of B. In
1982, Brandt [1] established a characterization of a block of a finite group with defect group of order two
as a block B with k(B) = 2. These correspond to the cases when the block is Morita equivalent to the
one-dimensional algebra and to the non-semisimple two-dimensional algebra, respectively [5].
In this talk, we redefine k(A) to be the codimension of the commutator subspace K(A) of a finitedimensional algebra A and prove analogous statements for arbitrary (not necessarily symmetric) finitedimensional algebras.
Theorem 1 (Chlebowitz [3], Koshitani-Sakurai [4]). Suppose F is a splitting field for A. Then thefollowing holds.
(i) k(A) = 1 ⇐⇒ mod A ≃ mod F.
(ii) k(A) = 2 and ℓ(A) = 1 ⇐⇒ mod A ≃ mod F [X]/(X 2 ).
This is achieved by extending the Okuyama refinement [6] of the Brandt result to this setting (part(ii) of Theorem 2; see also Shimizu [8]).
To this end, we study the codimension of the sum of the commutator subspace K(A) and nth Jacobsonradical Radn (A), which is denoted by KRn (A) = K(A) + Radn (A).
Theorem 2 (Koshitani-Sakurai [4]). Let { Si | 1 ≤ i ≤ ℓ(A) } be a complete set of pairwise nonisomorphic simple right A-modules and let CA be the Cartan matrix of A. Suppose F is a splitting field
for A. Then the following holds.
(i) codim KR1 (A) = ℓ(A).
∑ℓ(A)(ii) codim KR2 (A) = ℓ(A) + i=1 dim Ext1A (Si , Si ).
∑ℓ(A)(iii) ℓ(A) + i=1 dim Ext1A (Si , Si ) ≤ k(A) ≤ tr CA .
We prove that the codimension is Morita invariant and give an upper bound as well.
References[1] J. Brandt, A lower bound for the number of irreducible characters in a block, J. Algebra 74 (1982) 509–515.
doi:10.1016/0021-8693(82)90036-9.
[2] R. Brauer and C. Nesbitt, On the modular characters of groups, Ann. of Math. 42 (1941) 556–590. doi:10.2307/1968918.
[3] M. Chlebowitz, Über Abschätzungen von Algebreninvarianten (German), PhD Thesis, Universität Augsburg, 1991.
[4] S. Koshitani and T. Sakurai, On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras,
preprint (2018) 9pp. arXiv:1803.00025v2.
[5] P. Landrock, Finite Group Algebras and Their Modules, London Mathematical Society Lecture Note Series. 84 (Cambridge University Press, Cambridge, 1983). doi:10.1017/CBO9781107325524.
[6] T. Okuyama, Ext1 (S, S) for a simple kG-module S (Japanese), in Proceedings of the Symposium “Representations ofGroups and Rings and Its Applications,” ed. S. Endo (1981), pp. 238–249.
[7] Y. Otokita, On diagonal entries of Cartan matrices of p-blocks, preprint (2016) 4pp. arXiv:1605.07937v2.
[8] K. Shimizu, Further results on the structure of (co)ends in finite tensor categories, preprint (2018) 50pp.
arXiv:1801.02493v2.

Center for Frontier Science
Chiba University
1-33, Yayoi-cho, Inage-ku, Chiba-shi, Chiba, 263-8522 JAPAN
Email: koshitan@math.s.chiba-u.ac.jp
Department of Mathematics and Informatics
Graduate School of Science
Chiba University
1-33, Yayoi-cho, Inage-ku, Chiba-shi, Chiba, 263-8522 JAPAN
Email: tsakurai@math.s.chiba-u.ac.jp–24–

The defining relations of geometric algebras of Type EC
Ayako Itaba and Masaki Matsuno
This talk is a report of our paper [2]. Let k be an algebraically closed field of characteristic 0.
An algebra A means a graded k-algebra finitely generated in degree 1. For a quadratic algebra A =k⟨x1 , · · · , xn ⟩/I where I = (f1 , · · · , fm ) and fj are homogeneous elements of degree 2, V(A) := {(p, q) ∈Pn−1 × Pn−1 | fj (p, q) = 0, j = 1, · · · , m} is called the point scheme of A.
Definition 1 ([3]). A quadratic algebra A = k⟨x1 , · · · , xn ⟩/I is a geometric algebra if there exists ageometric pair (E, σ) where E is a projective variety in Pn−1 and σ is an automorphism of E such that
V(A) = {(p, σ(p)) | p ∈ E} and I2 = {f ∈ k⟨x1 , · · · , xn ⟩2 | f (p, σ(p)) = 0, ∀p ∈ E}.
If A is a geometric algebra, then A determines and is determined by the geometric pair (E, σ), so wewrite A = A(E, σ). A geometric algebra is related to an AS-regular algebra which is one of the maininterests in noncommutative algebraic geometry. Artin-Tate-Van den Bergh [1] showed that there is a
one-to-one correspondence between the set of 3-dimensional quadratic AS-regular algebras and the set of
regular geometric pairs (E, σ) where E is P2 or a cubic curve in P2 .

(elliptic curve)
In this talk, we study geometric algebras of Type EC, i.e., the algebras whose point schemes are elliptic
curves in P2 . We calculate defining relations of all geometric algebras of Type EC up to isomorphismof graded algebras by using the defining relations of 3-dimensional Sklyanin algebras. For p ∈ E, theautomorphism σp defined by σp (q) := p + q is called the translation by a point p. We will choose asuitable τ ∈ Autk E such that Autk E = {σp τ i | p ∈ E, i ∈ Z|τ | } where |τ | is the order of τ . We denoteby E[3] the set of 3-torsion points of E. The following is our main result.
Theorem 2 ([2]). Let p, q ∈ E \ E[3] and i, j ∈ Z|τ | .
(1) Geometric algebras A(E, σp τ i ) and A(E, σq τ j ) of Type EC are isomorphic as graded algebras if
and only if there exist r ∈ E[3] and l ∈ Z|τ | such that i = j and q = τ l (p) + r − τ i (r).
(2) Geometric algebras A(E, σp τ i ) and A(E, σq τ j ) of Type EC are graded Morita equivalent if andonly if there exist r ∈ E[3] and l ∈ Z|τ | such that p − τ j−i (p) ∈ E[3] and q = τ l (p) + r.
References[1] M. Artin, J. Tate and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck
Festschrift, vol. 1, Progress in Mathematics vol. 86 (Brikhäuser, Basel, 1990) 33–85.
[2] A. Itaba and M. Matsuno, Defining relations of 3-dimensional quadratic AS-regular algebras, (2018) submitted(arXiv:1806.04940).
[3] I. Mori, Non commutative projective schemes and point schemes, Algebras, Rings and Their Representations, World
Sci. Hackensack, N. J., (2006), 215–239.

Ayako Itaba
Department of Mathematics
Faculty of Science
Tokyo University of Science
1-3 Kagurazaka, Shinjyuku, Tokyo, 162-8601, Japan
Email: itaba@rs.tus.ac.jp
Masaki Matsuno
Graduate School of Integrated Science and Technology
Shizuoka University
Ohya 836, Shizuoka 422-8529, Japan
Email: matsuno.masaki.14@shizuoka.ac.jp–25–Quotients G/H is super-symmetry
Akira Masuoka
This is a report of my joint work with Yuta Takahashi, a PhD student at Tsukuba. Throughout wework over a field k of characteristic ̸= 2.
Background. The trivial symmetry v ⊗ w 7→ w ⊗ v, V ⊗ W → W ⊗ V on vector spaces is generalized
to the super-symmetry{−w ⊗ v if v and w are odd,v ⊗ w 7→ (−1)|v||w| w ⊗ v =w ⊗ v otherwise.
For the latter V and W are supposed to be super-vector spaces, or namely, vector spaces V0 ⊕V1 , W0 ⊕W1graded by Z/(2) = {0, 1}, and v, w homogeneous elements of degree |v|, |w|. The super-vector spaces
form a k-linear abelian, symmetric tensor category with respect to the obvious tensor product and thesuper-symmetry above. Ordinary objects, such as Lie algebra or Hopf algebra, are generalized to superobjects, such as Lie super-algebra or Hopf super-algebra, in the category. My interest in the super-world,
especially in algebraic super-groups, arose from the following striking result by P. Deligne (2002): if k is
algebraically closed and char k = 0, a k-linear abelian, rigid symmetric tensor category satisfying somemild assumption is realized as the category G-modules of finite-dimensional super-modules over a certain
aﬃne algebraic super-group G. Of course, before the result, there were already produced a huge numberof fruitful results in super-geometry both by mathematicians and by physicists. But it was only around2008 when the following basic question was explicitly posed by J. Brundan: Given an aﬃne algebraic
super-group G and its algebraic super-subgroup H, is the quotient (fppf-)sheaf G/H a super-scheme?
The results. In fact, the question was solved positively by Alexandr Zubkov and myself [J. Algebra
348 (2011), 135–170]. But this time Takahashi and I found a more acceptable proof of the result in
the generalized situation that G may not be aﬃne; the main point is to give an explicit description of
G/H, see (∗) below. The description could be hopefully extended to Lie super-groups in an analytic
context. The super-Grassmannians are typical, important examples of such quotients both in algebraic
and analytic contexts.
Recall that schemes (over k) are defined, from two viewpoints, as a sort of (i) topological spaces given
structure sheaf of commutative algebras over k, as well as, of (ii) functors defined on the category ofcommutative algebras over k. It is not diﬃcult to define super-schemes, replacing commutative algebras
with super-commutative super-algebras. Algebraic super-groups are the group objects of algebraic superschemes.
To show our result more explicitly, we let H ⊂ G be as in Brundan’s question above, concentratingon the aﬃne case for simplicity. Then there are naturally associated (a) algebraic groups H ⊂ G and(b) Lie super-algebras Lie(H) ⊂ Lie(G). (Here G is, from the viewpoint (ii), a group-valued functor,
and G is the restricted functor defined on the category of the trivially graded super-algebras.) Note that(b) is an inclusion of left H-modules, where H acts by adjoint. Let VG/H = Lie(G)1 / Lie(H)1 be theassociated quotient restricted to the odd component. A classical result tells us that the quotient sheaf
G/H is a Noetherian scheme, and the quotient morphism π : G → G/H is aﬃne, faithfully flat andfinitely presented.
Theorem. The quotient sheaf G/H is a Noetherian super-scheme, whose underlying topological space|G/H| is the same as |G/H| of G/H. The quotient morphism G → G/H has the same, desirable properties
as π. Given an aﬃne open subset U ⊂ |G/H|, we have a non-canonical isomorphism of super-algebras(∗)

OG/H (U ) ≃ ∧OG/H (U ) (HomH (VG/H , π∗ OG (U ))), the exterior algebra over OG/H (U ).

Institute of Mathematics
University of Tsukuba
Tsukuba, Ibaraki 305-8571 JAPAN
Email: akira@math.tsukuba.ac.jp–26–