Perspectives of Representation Theory of Algebras Celebrating Kunio Yamagata's 65Th Birthday

The 13th International Conference Graduate School of Mathematics, Nagoya University 第13回名古屋国際数学コンファレンス

Perspectives of Representation Theory of Algebras celebrating Kunio Yamagata's 65th birthday

November 11-15, 2013 in Nagoya University

The 13th International Conference, Graduate School of Mathematics, Nagoya University Perspectives of Representation Theory of Algebras

— celebrating Kunio Yamagata’s 65th birthday —

Period: November 11–15, 2013

Venue: Sakata–Hirata Hall, Science South Bldg. (Nov. 11–14), ES Hall, E&S Bldg. (Nov. 15), Nagoya University, Nagoya, Japan

Organizers: Hideto Asashiba (Shizuoka University), Osamu Iyama (Nagoya University), Jun-ichi Miyachi (Tokyo Gakugei University), Izuru Mori (Shizuoka University), Masahisa Sato (Yamanashi University), Andrzej Skowronski (Nicolaus Copernicus University), Morio Uematsu (Jobu University), Yuji Yoshino (Okayama University) Prof. Kunio Yamagata

TIME TABLE

Time Nov. 11 (Mon) Nov. 12 (Tue) Nov. 13 (Wed) Nov. 14 (Thu) Nov. 15 (Fri)

09:30–10:20 I. Reiten A. Skowronski K. Erdmann Y. Yoshino H. Lenzing

Coffee Break 11:00–11:30 G. Jasso M. Błaszkiewicz T. Itagaki R. Kanda K. Ueyama 11:40–12:30 T. Aihara J. Białkowski D. Zacharia R. Takahashi A. Takahashi

Lunch Break Lunch Break 13:00– 14:00–14:50 R. Kase* P. Malicki M. Yoshiwaki* Y. Kimura Y. Mizuno* H. Koga* Coffee Break Excursion Coffee Break 15:40–16:30 L. Demonet A. Skowyrski H. Minamoto M. Wemyss

16:40–17:30 O. Kerner C. Xi J. Miyachi C. M. Ringel

18:00– Banquet • Monday: 14:00–14:30 Kase, 14:35–15:05 Mizuno. • Thursday: 14:00–14:30 Yoshiwaki, 14:35–15:05 Koga. • There will be a Conference Excursion on November 13th (Wed) afternoon. • We will take a group photo just after morning session on November 14th (Thu). • A banquet is planned on November 14th (Thu) from 18:00 at Mei-dining. (See Figure C)

1 PROGRAM

November 11th (Mon) (at Sakata–Hirata Hall) 9:00– 9:30 — Registration —

9:30–10:20 Idun Reiten (Norwegian University of Science and Technology) Lattice structure of torsion classes — Coﬀee Break —

11:00–11:30 Gustavo Jasso (Nagoya University) On the simplicial complex associated with support τ-tilting modules 11:40–12:30 Takuma Aihara (Nagoya University) Tilting mutation theory — Lunch Break —

14:00–14:30 Ryoichi Kase (Osaka University) On the poset of pre-projective tilting modules over path algebras 14:35–15:05 Yuya Mizuno (Nagoya University) τ-tilting modules over preprojective algebras of Dynkin type 15:40–16:30 Laurent Demonet (Nagoya University) Ice quivers with potential associated with triangulations and Cohen-Macaulay modules

over orders (case An and Dn) 16:40–17:30 Otto Kerner (University of D¨usseldorf ) From wild hereditary to tilted algebras, and back

November 12th (Tue) (at Sakata–Hirata Hall) 9:30–10:20 Andrzej Skowronski (Nicolaus Copernicus University) Selﬁnjective algebras with deforming ideals — Coﬀee Break —

11:00–11:30 Marta Błaszkiewicz (Nicolaus Copernicus University) Selfinjective algebras of finite representation type with maximal almost split sequences 11:40–12:30 Jerzy Białkowski (Nicolaus Copernicus University) Periodicity of selfinjective algebras of polynomial growth — Lunch Break —

14:00–14:50 Piotr Malicki (Nicolaus Copernicus University) Finite cycles of indecomposable modules — Coﬀee Break —

15:40–16:30 Adam Skowyrski (Nicolaus Copernicus University) Homological problems for cycle-ﬁnite algebras

2 16:40–17:30 Changchang Xi (Capital Normal University) Higher dimensional tilting modules and recollements

November 13th (Wed) (at Sakata–Hirata Hall) 9:30–10:20 Karin Erdmann (University of Oxford) Support varieties via Hochschild cohomology: a necessary condition — Coﬀee Break —

11:00–11:30 Tomohiro Itagaki (Tokyo University of Science) Cyclic homology of truncated quiver algebras and notes on the no loops conjecture for Hochschild homology 11:40–12:30 Dan Zacharia (Syracuse University) A characterization of the graded center of a Koszul algebra 13:00– — Excursion —

November 14th (Thu) (at Sakata–Hirata Hall) 9:30–10:20 Yuji Yoshino (Okayama University) Dependence of total reﬂexivity conditions — Coﬀee Break —

11:00–11:30 Ryo Kanda (Nagoya University) Specialization orders on atom spectra of Grothendieck categories 11:40–12:30 Ryo Takahashi (Nagoya University) Existence of cohomology annihilators and strong generation of derived categories — Group Photo —

— Lunch Break —

14:00–14:30 Michio Yoshiwaki (Osaka City University) Dimensions of triangulated categories with respect to subcategories 14:35–15:05 Hirotaka Koga (University of Tsukuba) Cliﬀord extension — Coﬀee Break —

15:40–16:30 Hiroyuki Minamoto (Osaka Prefecture University) Derived Gabriel topology, localization and completion of dg-algebras 16:40–17:30 Jun-ichi Miyachi (Tokyo Gakugei University) Researches on the representation theory of algebras at University of Tsukuba 18:00– — Banquet —

3 November 15th (Fri) (at ES Hall) 9:30–10:20 Helmut Lenzing (University of Paderborn) The ubiquity of the equation x2 + y3 + z5 = 0 — Coﬀee Break —

11:00–11:30 Kenta Ueyama (Shizuoka University) Ample Group Actions on AS-regular Algebras and Noncommutative Graded Isolated Singularities 11:40–12:30 Atsushi Takahashi (Osaka University) Weyl groups and Artin groups associated to weighted projective lines — Lunch Break —

14:00–14:50 Yoshiyuki Kimura (Osaka City University) Quiver varieties and Quantum cluster algebras — Coﬀee Break —

15:40–16:30 Michael Wemyss (University of Edinburgh) Some new examples of self–injective algebras 16:40–17:30 Claus Michael Ringel (Bielefeld University) The root posets

4 ABSTRACTS

Idun Reiten (Norwegian Univ. of Sci. and Tech.)...... November 11th (Mon), 9:30–10:20 Lattice structure of torsion classes This lecture is based on work with Iyama, Thomas and Todorov. A general problem for a finite dimensional algebra A over an algebraically closed field k is to investigate when the partially ordered set of functorially finite torsion classes are a lattice. Some work by Mizuno gives a motivation for considering this problem. We give the answer when A is the path algebra kQ for a finite acyclic quiver Q.

Gustavo Jasso (Nagoya University)...... November 11th (Mon), 11:00–11:30 On the simplicial complex associated with support τ-tilting modules The class of support τ-tilting modules was introduced recently by Adachi–Iyama–Reiten so as to provide a completion of the class of tilting modules from the point of view of mutation. Let A be a ﬁnite dimensional algebra with n simple modules. In this talk, I will explain how to construct an abstract simplical complex ∆(A) whose maximal faces are in bijection with the isomorphism classes of basic support τ-tilting modules. Using established results, we will describe the combinatorial properties of ∆(A). In particular, if ∆(A) is ﬁnite, then the geometric realization of ∆(A) is homeomorphic to a (n − 1)- dimensional sphere. This work is part of a joint project with Osamu Iyama.

Takuma Aihara (Nagoya University)...... November 11th (Mon), 11:40–12:30 Tilting mutation theory In representation theory of algebras, the notion of mutation plays crucial roles. Three kinds of mutation are well-known: silting mutation [AI], quiver-mutation [FZ] and cluster tilting mutation [BMRRT, IY]. Moreover these are closely related with each other, that is, in hereditary case there are one-to-one correspondences compatible with mutation, among silting objects, clusters and cluster tilting objects [AIR]. In this talk, we focus on silting mutation, which is a generalization of tilting mutation studied first by Bernstein–Gelfand–Ponomarev and Auslander–Platzeck–Reiten, it was investigated also by Riedtmann– Schofield and Happel–Unger. We note that silting mutation is always possible, but tilting mutation is not. We mainly discuss silting quivers to observe the behavior of silting mutation. A finite dimensional algebra over a field is said to be silting-connected if its silting quiver is connected. Now we raise the following question: Question. When is a finite dimensional algebra over a field silting-connected?

Our goal of this talk is to give a partial answer of this question. Now we state the main theorem. Main Theorem. Any of the following algebras is silting-connected:

(1) local algebras [AI];

5 (2) hereditary algebras [AI]; (3) representation-ﬁnite symmetric algebras [A]; (4) Brauer graph algebras of type odd [AAC].

In particular, I will explain the strategy of the proof of (3) and (4) in this theorem. References [AAC] T. Adachi; T. Aihara; A. Chan, On tilting complexes of Brauer graph algebras I: combina- torics arising from two-term tilting complexes and mutation quivers of tilting complexes. in preparation. [AIR] T. Adachi; O. Iyama; I. Reiten, τ-tilting theory. to appear in Compos. Math. [A] T. Aihara, Tilting-connected symmetric algebras. Algebr. Represent. Theory 16 (2013), no. 3, 873–894. [AI] T. Aihara; O. Iyama, Silting mutation in triangulated categories. J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668. [BMRRT] A. B. Buan; R. Marsh; M. Reineke; I. Reiten; G. Todorov, Tilting theory and cluster combi- natorics. Adv. Math. 204 (2006), no. 2, 572–618. [FZ] S. Fomin; A. Zelevinsky, Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. [IY] O. Iyama; Y. Yoshino, Mutation in triangulated categories and rigid Cohen–Macaulay modules. Invent. Math. 172 (2008), no. 1, 117–168.

Ryoichi Kase (Osaka University)...... November 11th (Mon), 14:00–14:30 On the poset of pre-projective tilting modules over path algebras

To classify tilting modules is an important problem of the representation theory of finite dimensional algebras. Theory of tilting-mutation introduced by Riedtmann and Schofield is one of the approach to this problem. Riedtmann and Schofield defined the tilting quiver related with tilting-mutation. Happel and Unger defined the partial order on the set of (isomorphic classes of) basic tilting modules and showed that tilting quiver is coincided with Hasse quiver of this poset. In this talk we consider the poset of pre-projective tilting modules over path algebras. First we give an equivalent condition for the poset of pre-projective tilting modules to be a distributive lattice. Moreover we realize the poset of pre-projective tilting modules as an ideal-poset.

Yuya Mizuno (Nagoya University)...... November 11th (Mon), 14:35–15:05 τ-tilting modules over preprojective algebras of Dynkin type

Recently, the notion of (support) τ-tilting modules was introduced by Adachi–Iyama–Reiten. This gives a generalization of tilting modules and it also has several nice properties. In this talk, we discuss support τ-tilting modules over preprojective algebras of Dynkin type. In particular, we classify all support τ-tilting modules by giving a bijection with elements in the corresponding Weyl groups.

6 Laurent Demonet (Nagoya University)...... November 11th (Mon), 15:40–16:30 Ice quivers with potential associated with triangulations and Cohen–Macaulay modules over orders (case An and Dn) (joint with Xueyu Luo)

In this talk, we attach an ice quiver with potential (Qσ,Wσ, F) to each triangulation σ of a polygon (resp. a polygon with one puncture) where the set F of frozen vertices correspond to the sides of the polygon. This quiver extends the one introduced by Labardini–Fragoso and Cerulli Irelli. Thus, we consider the (non-completed) frozen Jacobian algebra Γσ = P(Qσ,Wσ, F). One of the main result is that

Γσ as the structure of an order over K[x], that Λ = eFΓσeF is a Gorenstein order independent of σ (eF is the sum of idempotents at the frozen vertices of σ). Moreover, the cluster tilting objects in CM(Λ) are the modules eFΓσ for all triangulations σ. More precisely, its stable category CM(Λ) is equivalent to a cluster category of type A (resp. type D).

Otto Kerner (University of Düsseldorf )...... November 11th (Mon), 16:40–17:30 From wild hereditary to tilted algebras, and back Let H be a finite dimensional basic connected wild hereditary algebra over an algebraically closed field K, and H-mod the category of finite dimensional left H-modules. The Auslander–Reiten quiver Γ(H) of H-mod consists of a unique preprojective and preinjective component and infinitely many regular components of type ZA∞. The modules at the boarder of such a regular component are called quasi- simple. The full subcategory of regular H-modules is denoted by H-reg. This category is closed under extensions and images, but not closed under kernels and cokernels. Let n be the number of pairwise nonisomorphic simple H-modules. For n ≥ 3 there always exist indecomposable regular quasi-simple modules without self-extensions. Take such a module X. Using Bongartz’s construction, one gets a squarefree tilting module T = M ⊕ X ⊥ with HomH(X, M) = 0. Thus M is in the right perpendicular category X of X, and it is a minimal ⊥ projective generator there. Hence X C-mod, where C = EndH(M), and C is connected wild hereditary ⊥ with n − 1 simple modules. We will identify X with C-mod via the functor HomH(M,−). If 0 →

τH X → Z → X → 0 is the Auslander–Reiten sequence ending in X, then Z ∈ C-reg is a quasi-simple brick. The tilted algebra B = EndH(T) C[Z] is the one-point extension of C by Z. With these notations the following holds:

Theorem (Crawley-Boevey, K.) There exists a full and dense functor

F : C-reg → H-reg

τ τ = ∈ {τi | ∈ Z} with F C H F. One has F(U) 0 if and only if U add CZ i . ∈ = τ−rτr τs τ−s ≫ τ For U C-reg one gets F(U) H T T C U, for r, s 0; here T denotes the relative Auslander– Reiten translation in the tilting torsion class T of H-modules generated by T. For the proof direct and inverse limits are used.

It follows from this theorem that any two connected wild hereditary K-algebras H and H′ are almost equivalent. This means that there exist regular H-modules U and regular H′-modules U′, such that the

7 ′ ′ factor categories of H-reg, respectively H -reg, by the τH orbit of U, respectively the τH′ -orbit of U , are equivalent.

I also will give another description of this functor, using minimal approximations instead of limits, and apply this description to deduce some consequences. For example, it follows that the module Z is orbital elementary. This means that the τC-orbit of the module Z has the following property: Each short → → ˆ → → ˆ ∈ {τi | ∈ Z} exact sequence 0 U Z V 0 with Z add CZ i splits, provided U and V are regular C- modules. Moreover, most wild hereditary algebras H have ﬁltration closed regular components. Recall that a regular component C is called ﬁltration closed, if for each short exact sequence 0 → U → A → V → 0 with A ∈ addC also U and V are in addC, provided they are regular H-modules.

Andrzej Skowronski (Nicolaus Copernicus University) . .. November 12th (Tue), 9:30–10:20 Selfinjective algebras with deforming ideals In the representation theory of selfinjective artin algebras a prominent role is played by the orbit algebras bB/G, where bB is the repetitive category of an artin algebra B and G is an admissible infinite cyclic group of automorphisms of bB. The major effort of my joint research with Kunio Yamagata during the last 20 years concerned criteria for a selfinjective artin algebra A to be isomorphic (respectively, socle b/ φν ν equivalent, stably equivalent) to an orbit algebra of the form B ( bB), where B is an artin algebra, bB the Nakayama automorphism of bB, and φ a positive automorphism of bB. This was strongly motivated by the problem of describing the structure of selfinjective artin algebras for which the Auslander–Reiten quiver admits a generalized standard component. The aim of the talk is to present the main results achieved in this direction.

Marta Błaszkiewicz (Nicolaus Copernicus University) . .. November 12th (Tue), 11:00–11:30 Selfinjective algebras of finite representation type with maximal almost split sequences This is report on joint work with Andrzej Skowroński.

Let A be a finite dimensional K-algebra (associative, with an identity) over an arbitrary field K, and mod A the category of finite dimensional right A-modules. For a nonprojective indecomposable module X in mod A, there is an almost split sequence

0 −→ τAX −→ Y −→ X −→ 0, where τAX is the Auslander–Reiten translation of X. Then we may associate to X the numerical invariant

α(X) being the number of summands in a decomposition Y = Y1 ⊕ ... ⊕ Yr of Y into a direct sum of indecomposable modules in mod A, which measures the complexity of homomorphisms in mod A with domain τAX and codomain X. It has been proved by R.Bautista and S. Brenner (1981) that, if A is of ﬁnite representation type and X is a nonprojective indecomposable module in mod A, then α(X) ≤ 4, and if α(X) = 4, then the middle Y of an almost split sequence in mod A with the right term X admits an indecomposable projective-injective direct summand. An almost split sequence in a module category mod A of an algebra A of ﬁnite representation type with the middle term being a direct sum of four indecomposable modules is called a maximal almost split sequence in mod A.

8 We will discuss the structure of basic, indecomposable, finite dimensional selfinjective K-algebras A over a field K for which mod A admits a maximal almost split sequence.

Jerzy Białkowski (Nicolaus Copernicus University).... November 12th (Tue), 11:40–12:30 Periodicity of selﬁnjective algebras of polynomial growth This is report on joint work with K. Erdmann and A. Skowro´nski

Let A be a finite dimensional K-algebra over an algebraically closed field K. Denote by ΩA the syzygy operator on the category mod A of finite dimensional right A-modules, which assigns to a module

M in mod A the kernel ΩA(M) of a minimal projective cover PA(M) → M of M in mod A. A module M Ωn ≥ in mod A is said to be periodic if A(M) M for some n 1. Then A is said to be a periodic algebra if e e op A is periodic in the module category mod A of the enveloping algebra A = A ⊗K A, that is, periodic as an A-A-bimodule. The periodic algebras A are selfinjective and their module categories mod A are periodic (all modules in mod A without projective direct summands are periodic). The periodicity of an algebra A is related with the periodicity of its Hochschild cohomology algebra HH∗(A) and is invariant under equivalences of the derived category Db(mod A) of bounded complexes over mod A. One of the exciting open problems in the representation theory of selfinjective algebras is to determine the Morita equivalence classes of periodic algebras. It has been proved by Dugas that every selfinjective algebra of finite representation type, without semisimple summands, is a periodic algebra. During the talk we will present a description of all basic, indecomposable, representation-infinite periodic algebras of polynomial growth.

Piotr Malicki (Nicolaus Copernicus University)...... November 12th (Tue), 14:00–14:50 Finite cycles of indecomposable modules This is report on joint work with J. A. de la Pe˜naand A. Skowro´nski.

Let A be a basic indecomposable artin algebra over a commutative artin ring K, mod A the category of finitely generated right A-modules and ind A the full subcategory of mod A formed by the indecom- ∞ posable modules. Denote by radA the infinite radical of mod A, being the intersection of all powers i ≥ ∞ = radA, i 1, of the radical radA of mod A. By a result of Auslander, radA 0 if and only if A is of finite ∞ 2 , representation type. On the other hand, if A is of infinite representation type then (radA ) 0, by a result due to Coelho–Marcos–Merklen–Skowronski.´ For a module M in mod A, consider a decomposition

A = PM ⊕ QM of A in mod A such that the simple summands of the semisimple module PM/rad PM are exactly the simple composition factors of M, and the ideal tA(M) in A generated by the images of all homomorphisms from QM to A in mod A. Then Supp(M) = A/tA(M) is called the support algebra of M. A cycle in ind A is a sequence

f1 fn M0 −−→ M1 → ··· → Mn−1 −−→ Mn = M0 of nonzero nonisomorphisms in ind A, and such a cycle is said to be finite if the homomorphisms f1,..., fn ∞ do not belong to radA . Following Ringel, a module M in ind A which does not lie on a cycle in ind A is called directing. A nondirecting module M in ind A is said to be cycle-finite if every cycle in ind A passing through M is finite.

9 In 1984 Ringel proved that the support algebra of a directing module is a tilted algebra. The aim of the talk is to present solution of the long standing open problem concerning the structure of the support algebras of cycle-finite indecomposable modules (being a natural extension of the problem considered by Ringel). This is achieved by a conceptual description of the support algebras of cycle-finite components in the cyclic Auslander–Reiten quiver cΓA of an algebra A. A prominent role in this description is played by the generalized multicoil algebras (defined by Malicki–Skowronski)´ and the generalized double tilted algebras (defined by Reiten–Skowronski).´

Adam Skowyrski (Nicolaus Copernicus University).... November 12th (Tue), 15:40–16:30 Homological problems for cycle-finite algebras By an algebra we mean an artin K-algebra, where K is a commutative artin ring. Given an algebra A, we denote by mod A the category of finitely generated right A-modules, by ind A the full subcategory ∞ of mod A consisting of indecomposable modules, by radA the infinite Jacobson radical of mod A, and by τA = DTr the Auslander–Reiten translation. Following Ringel, a cycle in mod A is a sequence

f1 / f2 / fr / X0 X1 ... Xr = X0 of nonzero nonisomorphisms in ind A, and such a cycle is called finite provided that all homomorphisms ,..., ∞ f1 fr do not belong to radA . There is an important and wide class of algebras introduced by Assem and Skowronski,´ called cycle-finite algebras. Recall that an algebra A is said to be a cycle-finite algebra if and only if all cycles in mod A are finite. A prominent role in the representation theory is played by quasitilted algebras and generalized double tilted algebras. We are concerned with the following three open problems formulated by Skowronski.´

(1) An algebra A is a generalized double tilted algebra or a quasitilted algebra if and only if, for all but ⩽ ⩽ ﬁnitely many isomorphism classes of modules X in ind A, we have pdA X 1 or idA X 1. (2) An algebra A is a generalized double tilted algebra or a quasitilted algebra if and only if, for all

but ﬁnitely many isomorphism classes of modules X in ind A, we have HomA(D(A), X) = 0 or

HomA(X, A) = 0. (3) An algebra A is a generalized double tilted algebra if and only if mod A admits a faithful module M, such that, for all but ﬁnitely many isomorphism classes of modules X in ind A, we have

HomA(M,τAX) = 0 or HomA(X, M) = 0.

The aim of the talk is to present solutions of these problems for cycle-ﬁnite algebras.

Changchang Xi (Capital Normal University)...... November 12th (Tue), 16:40–17:30 Higher dimensional tilting modules and recollements Infinitely generated tilting modules behave very differently from finitely generated tilting modules. In this talk, we shall present some recent results on infinitely generated tilting modules which is closely related to noncommutative localizations, homological ring epimorphisms, recollements, Mittag–Leffler conditions and coproducts of rings. Our study will be carried out under the framework of Ringel modules,

10 and focused on the derived categories of the endomorphism rings of infinitely (that is, not necessarily finitely) generated tilting modules of projective dimension at least 2. This approach allows us to deal with infinitely generated cotilting modules in a uniform way. The contents of this talk are mainly taken from a joint preprint “Higher dimensional tilting modules and homological subcategories” with H. X. Chen.

Karin Erdmann (University of Oxford)...... November 13th (Wed), 9:30–10:20 Support varieties via Hochschild cohomology: a necessary condition Let A be a finite-dimensional self-injective algebra over a field K, and let HH∗(A) be the Hochschild ∗ , ∗ cohomology algebra of A, this acts on ExtA(M M) for any A-module M. If HH (A) is noetherian and the Ext algebra of A is finitely generated over HH∗(A) then A-modules have supports defined via this action, which share many properties of supports defined via group cohomology. Unfortunately, these finite generation properties are difficult to verify. However, if the algebra has a module with complexity = 1 which is not Ω periodic (called ‘criminal’) then it follows that such support cannot exist. We show that it is very common for socle deformations of self-injective algebras to have criminals. We also show that the criminals we construct are counterexamples to the generalized Auslander–Reiten condition.

Tomohiro Itagaki (Tokyo University of Science)...... November 13th (Wed), 11:00–11:30 Cyclic homology of truncated quiver algebras and notes on the no loops conjecture for Hochschild homology (Joint work with Katsunori Sanada)

In this talk, we show the dimension formula of the cyclic homology of truncated quiver algebras over an arbitrary field, and we extend the 2-truncated cycles version of the no loops conjecture (cf. [2]) to the m-truncated cycles version for a class of finite dimesional algebras over an algebraically closed field. In [6], for a truncated quiver algebra A over a commutative ring, Skoldberg¨ gives a left Ae-projective resolution of A and computes the Hochschild homology HHn(A). By means of this result and a theorem in Loday’s book(1992), Taillefer [7] gives a dimension formula of the cyclic homology of truncated quiver algebras over a field of characteristic zero. We compute the dimension formula of the cyclic homology of truncated quiver algebras over an arbitrary field by means of chain maps in [1] and a spectral sequence. Our result generalizes the above result by Taillefer. Moreover, we show that the m-truncated cycles version of the no loops conjecture holds for a class of bound quiver algebras over an algebraically closed field as an application of the chain map from Cibils’ projective resolution (cf. [3]) to Skoldberg’s¨ projective resolution given in [1].

References [1] G. Ames, L. Cagliero, P. Tirao, Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras, J. Algebra 322(5)(2009), 1466–1497. [2] P. A. Bergh, Y. Han, D. Madsen, Hochschild homology and truncated cycles, Proc. Amer. Math. Soc. (2012), no. 4, 1133–1139.

11 [3] C. Cibils, Cohomology of incidence algebras and simplicial complexes, J. Pure Appl. Algebra 56(3) (1989), 221–232. [4] T. Itagaki, K. Sanada, The dimension formula of the cyclic homology of truncated quiver algebras over a ﬁeld of positive characteristic, submitted. [5] T. Itagaki, K. Sanada, Notes on the Hochschild homology dimension and truncated cycles, in preparation. [6] E. Skoldberg,¨ Hochschild homology of truncated and quadratic monomial algebras, J. Lond. Math. Soc. (2) 59 (1999), 76–86. [7] R. Taillefer, Cyclic homology of Hopf algebras, K-Theory 24 (2001), 69–85.

Dan Zacharia (Syracuse University)...... November 13th (Wed), 11:40–12:30 A characterization of the graded center of a Koszul algebra I will talk on joint work with Ed Green and Nicole Snashall. Let k be a ﬁeld and let S be a graded k-algebra. The graded center of S is is the graded subring Zgr(S ) generated by all the homogeneous elements u of S such that uv = (−1)|u||v|vu for every homogeneous element v of S , where |x| denotes the degree of the homogeneous element x. We will present a characterization of Zgr(S ) in the case when S is a Koszul algebra and some applications of the ideas involved.

Yuji Yoshino (Okayama University)...... November 14th (Thu), 9:30–10:20 Dependence of total reﬂexivity conditions Let R be a commutative noetherian ring. Then the following theorem is proved by using the chro- matic tower theorem of Neeman. ∈ ∈ , = L⊗ = Theorem 0.1. Let W D f g(R) and X D(R). Then, RHomR(W X) 0 if and only if W RX 0. In my talk I will show how I proved this theorem and I will present some of its application. In particular this can be presumably applied to prove the following result, but not completely veriﬁed yet.

Question 0.2. Let R be a generically Gorenstein ring, i.e. Rp is Gorenstein for all p ∈ Ass(R). If a i , = > finitely generated R-module M satisfies ExtR(M R) 0 for all i 0, then M is totally reflexive.

Ryo Kanda (Nagoya University)...... November 14th (Thu), 11:00–11:30 Specialization orders on atom spectra of Grothendieck categories The notion of the atom spectra of Grothendieck categories is a generalization of the prime spectra of commutative rings. We develop a theory of the specialization orders on the atom spectra of Grothendieck categories and introduce systematic methods to construct Grothendieck categories from colored quivers. We show that any partially ordered set is realized as the atom spectrum of some Grothendieck category, which is an analog of Hochster’s result in commutative ring theory.

Ryo Takahashi (Nagoya University)...... November 14th (Thu), 11:40–12:30 Existence of cohomology annihilators and strong generation of derived categories This talk is based on joint work with Srikanth Iyengar.

12 Let Λ be a Noether algebra, i.e., a Noetherian ring that is module-ﬁnite over its center Λc. We call a nonzerodivisor x of Λc a cohomology annihilator of Λ if there exists a positive integer n such that

n x · ExtΛ(M, N) = 0 for all finitely generated Λ-modules M and N. Studies on such cohomology annihilation were started by Dieterich [2], Popescu and Roczen [3] and Yoshino [5] in the 1980s in relation to the Brauer–Thrall conjectures for Cohen–Macaulay modules over Cohen–Macaulay rings. In the 1990s, Wang [4] proved that every complete equicharacteristic local domain with perfect residue field possesses a cohomology annihilator. Recently, Buchweitz and Flenner explored cohomology annihilation over Gorenstein orders. In this talk, we consider how to proceed in the general case. We also relate cohomology annihilation with strong generation of the bounded derived category of finitely generated Λ-modules in the sense of Bondal and Van den Bergh [1].

References [1] A. Bondal; M. Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258. [2] E. Dieterich, Reduction of isolated singularities, Comment. Math. Helv. 62 (1987), no. 4, 654–676. [3] D. Popescu; M. Roczen, Indecomposable Cohen–Macaulay modules and irreducible maps, Compos. Math. 76 (1990), no. 1-2, 277–294. [4] H.-J. Wang, On the Fitting ideals in free resolutions, Michigan Math. J. 41 (1994), no. 3, 587–608. [5] Y. Yoshino, Brauer–Thrall type theorem for maximal Cohen–Macaulay modules, J. Math. Soc. Japan 39 (1987), no. 4, 719–739.

Michio Yoshiwaki (Osaka City University)...... November 14th (Thu), 14:00–14:30 Dimensions of triangulated categories with respect to subcategories This talk is based on joint work [1] with Takuma Aihara, Tokuji Araya, Osamu Iyama and Ryo Takahashi.

The notion of dimension of a triangulated category was introduced by Rouquier [3] based on work of Bondal and van den Bergh [2] on Brown representability. It measures how many extensions are needed to build the triangulated category out of a single object, up to finite direct sum, direct summand and shift. It is still a hard problem in general to give a precise value of the dimension of a given triangulated category. Our aim is to provide new information on this problem. In this talk, we will introduce a concept of dimension of a triangulated category with respect to a fixed full subcategory. Then we will give upper bounds of our relative dimensions of derived categories in terms of global dimensions. Namely, our main result is the following. Theorem 1. Let A be an abelian category and X a contravariantly finite subcategory that generates A. Then X-tri.dimDb(A) ≤ gl.dim(modX).

Our methods not only recover some known results on the dimensions of derived categories in the sense of Rouquier, but also apply to various commutative and non-commutative noetherian rings. In fact, we obtain the following consequence of Theorem 1.

13 Corollary 2. Let Λ be a noetherian ring and T a cotilting module. Then

b XT -tri.dimD (modΛ) ≤ max{1,inj.dimT},

i where XT = {X ∈ modΛ | ExtΛ(X,T) = 0 for any i > 0}.

References [1] T. Aihara, T. Araya, O. Iyama, R. Takahashi and M. Yoshiwaki, Dimensions of triangulated categories with respect to subcategories, to appear in J. Algebra (arXiv:1204.6421). [2] A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258. [3] R. Rouquier, Dimensions of triangulated categories, J. K-Theory 1 (2008), no. 2, 193–256 and errata, 257–258.

Hirotaka Koga (University of Tsukuba)...... November 14th (Thu), 14:35–15:05 Clifford extension (Joint work with Mitsuo Hoshino and Noritsugu Kameyama) Auslander–Gorenstein rings appear in various fields of current research in mathematics (see [1], [2], [3] and [5]). However, little is known about constructions of Auslander–Gorenstein rings. We have already known that Frobenius extensions of Auslander–Gorenstein rings are Auslander–Gorenstein rings. In this note, formulating the construction of Clifford algebras (see e.g. [4]) we introduce the notion of Clifford extensions and show that Clifford extensions are Frobenius extensions. Consequently Clifford extensions of Auslander–Gorenstein rings are Auslander–Gorenstein. Let n ≥ 2 be an integer. We fix a set of integers I = {0,1,...,n − 1} and a ring R. We use the notation A/R to denote that a ring A contains R as a subring. First, we will construct a Frobenius extension Λ/R using a certain pair (σ,c) of σ ∈ Aut(R) and c ∈ R. Namely, we will define an appropriate multiplication on a free right R-module Λ with a basis {vi}i∈I. Then we restrict ourselves to the case where n = 2 in order to recover the construction of Clifford algebras. For m ≥ 1 we construct ring extensions Λm/R which we call Clifford extensions using the following data: a sequence of elements c1,c2,··· in Z(R) and signs ε(i, j) for 1 ≤ i, j ≤ m. Namely, we will define an appropriate multiplication on a free right R-module

Λm with a basis {vx}x∈Im . We show that Λm is obtained by iterating the construction above m times, that ∼ Λm/R is a Frobenius extension, and that if ci ∈ rad(R) for 1 ≤ i ≤ m then R/rad(R) −→ Λm/rad(Λm).

References [1] M. Artin, J. Tate and M. Van den Bergh, Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), no. 2, 335–388. [2] J.-E. Bjork,¨ Rings of differential operators, North-Holland Mathematical Library, 21. North-Holland Publishing Co., Amsterdam–New York, 1979. [3] J.-E. Bjork,¨ The Auslander condition on noetherian rings, in: Séminaire d′Algèbre Paul Dubreil et Marie-Paul Malliavin, 39èmeAnnée(Paris, 1987/1988), 137–173, Lecture Notes in Math., 1404, Springer, Berlin, 1989. [4] D. J. H. Garling, Clifford algebras: an introduction, London Mathematical Society Student Texts, 78. Cambridge University Press, Cambridge, 2011, viii+200pp.

14 [5] J. Tate and M. Van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), no. 1-3, 619–647.

Hiroyuki Minamoto (Osaka Prefecture University).... November 14th (Thu), 15:40–16:30 Derived Gabriel topology, localization and completion of dg-algebras Gabriel topology is a special class of linear topology on rings, which plays an important role in the theory of localization of (not necessary commutative) rings. Several evidences have suggested that there should be a corresponding notion for dg-algebras. In this talk I introduce a notion of Gabriel topology on dg-algebras, derived Gabriel topology, and show its basic properties. In the same way we give the deﬁnition of topological dg-modules over a dg-algebra equipped with derived Gabriel topology. We show that derived bi-duality module is obtained by tautological homotopy limit. From the view point of derived Gabriel topology, this is a derived version of Lambek’s theorem about localization and completion of rings. We see that this gives generalizations and conceptual proof of several results about dg-algebras.

Jun-ichi Miyachi (Tokyo Gakugei University)...... November 14th (Thu), 16:40–17:30 Researches on the representation theory of algebras at University of Tsukuba We survey researches on the representation theory of algebras at University of Tsukuba from 1980s to 1990s.

Helmut Lenzing (University of Paderborn)...... November 15th (Fri), 9:30–10:20 The ubiquity of the equation x2 + y3 + z5 = 0

This is an expository talk dealing with the ubiquity of the equation x2 + y3 + z5 = 0. Among others we will deal with a challenging interrelationship between commutative and non-commutative algebra, by considering solution sets in a commutative respectively a non-commutative environment (fields resp. matrices). The equation x2 + y3 + z5 = 0 has an interesting history relating to Plato’s classification of regular solids, via Felix Klein’s study of the symmetry group of the icosahedron and its associated invariant theory. It thus links to what is now known as McKay theory. The solutions of the equation x2 +y3 +z5 = 0 in complex 3-space form a geometric object, known as the E8-singularity, an object related to many further mathematical subjects. I will additionally consider solutions of the equation which are (composable sequences of) matrices. This immediately yields to the representation theory of the canonical algebras, as introduced by Ringel, yielding a canonical algebra Λ with arm lengths (2,3,5), and thus to the represen- e tation theory of a tame quiver of extended Dynkin type E8. Conversely, we discuss how to recover the equation from the representation theory of Λ. By using coherent sheaves and stable categories of Cohen– Macaulay modules we link the commutative and the non-commutative solutions of x2 + y3 + z5 = 0, thus showing a remarkable unity of mathematics. This fact relates to a theorem of Orlov (2009). Further appearances of the equation concern preprojective and orbit algebras, the classification of factorial rings, fundamental groups of 2-orbifolds and the invariant subspace (more precisely subflags)

15 problem for nilpotent operators, a subject treated by Simson (2007), Ringel–Schmidmeier (2008) and Kussin–Meltzer–Lenzing (2011-2013).

Kenta Ueyama (Shizuoka University)...... November 15th (Fri), 11:00–11:30 Ample Group Actions on AS-regular Algebras and Noncommutative Graded Isolated Singularities This is a report on joint work with Izuru Mori. AS-regular algebras are the most important class of algebras studied in noncommutative algebraic geometry. They are noncommutative analogues of the polynomial algebra. In this talk, we introduce a notion of ampleness of a group action on a right noetherian graded algebra, and show that, for an AS-regular algebra S , the ampleness of G on S is strongly related to the notion of S G to be a noncommutative graded isolated singularity. Moreover, we give a relationship between S G and the skew group algebra ∇S ∗G of the quantum Beilinson algebra ∇S .

Atsushi Takahashi (Osaka University)...... November 15th (Fri), 11:40–12:30 Weyl groups and Artin groups associated to weighted projective lines We report on our recent study on a correspondence among weighted projective lines, cusp singularities and cuspidal root systems. Our purpose is to study the fundamental group of the complement of the discriminant of the Frobenius manifold constructed from the Gromov–Witten theory for a weighted projective line. If the number of orbifold points is at most three, then the Frobenius manifold is isomorphic, by mirror symmetry, to the base space of a universal unfolding of a cusp singularity, which is a quotient space of a product of the complexiﬁed Tits cone and a complex line by the extended cuspidal Weyl group associated to the correponding cuspidal root system. We start from a weighted projective line and then associate to it a cuspidal Weyl group and a cuspidal Artin group by generaters and relations following Saito–Takebayashi and Yamada. We show that the cuspidal Artin group is isomorphic to the fundamental group of the regular orbit space of the complexiﬁed Tits cone under the action of the cuspidal Weyl group.

Yoshiyuki Kimura (Osaka City University)...... November 15th (Fri), 14:00–14:50 Quiver varieties and Quantum cluster algebras

Let Av(n(w)) be the quantum coordinate ring of unipotent subgroup associated with a Weyl group element w of symmetric Kac–Moody Lie algebra g. It is shown that Av(n(w)) has the dual canonical basis which is compatible with Kashiwara–Lusztig’s dual canonical basis in [2]. Geiss–Leclerc–Schroer¨

[1] have shown that Av(n(w)) has a structure of quantum cluster algebra which is induced by the preprojective algebra Λ. It is conjectured that the dual canonical basis of Av(n(w)) contains the quantum cluster monomial of that. Let Q be an acyclic quiver, cQ be the corresponding acyclic Coxeter word and = 2 ℓ = | | w cQ with (w) 2 Q . In [3], we identify the (twisted) quantum Grothendieck ring of the correspond- 2 ing graded quiver variety and Av(n(cQ)) and identify the basis of simple perverse sheaves with the dual canonical basis. Using this isomorphism, it can be shown that the set of quantum cluster monomials is 2 contained in the dual canonical basis of Av(n(cQ)). This is a joint work with Fan Qin.

16 References [1] Christof Geiß, Bernard Leclerc, and Jan Schroer.¨ Cluster structures on quantum coordinate rings. to appear in Selecta Math., 2012. e-print arxiv http://arxiv.org/abs/1104.0531. [2] Yoshiyuki Kimura. Quantum unipotent subgroup and dual canonical basis. Kyoto J. Math., 52(2):277–331, 2012. [3] Yoshiyuki Kimura and Fan Qin. Graded quiver varieties, quantum cluster algebras and dual canonical basis. e-print arxiv http://arxiv.org/abs/1205.2066v2, 2012.

Michael Wemyss (University of Edinburgh)...... November 15th (Fri), 15:40–16:30 Some new examples of self–injective algebras I will talk about my recent work with Will Donovan (arXiv:1309.0698), which gives new invariants of curves inside 3-folds X. In the minimal model program, certain surgeries called flips and flops appear, and the basic idea is that to each of these we can, using noncommutative deformation theory, associate a not-necessarily-commutative finite dimensional algebra Λcon, together with a functor

b b D (modΛcon) → D (coh X).

In the ﬂops setting, the algebra Λcon turns out to be self–injective, and control much of the homological algebra. Roughly speaking, our original curve (drawn in bold in the pictures below) deforms in our ambient variety, and by tracking how it deforms we obtain a ﬁnite dimensional algebra. The left hand picture illustrates the commutative deformations, the right hand picture the noncommutative ones.

Surprisingly (at least to us), most of the ﬁnite dimensional self-injective algebras that arise in this way turn out to be new, and so I will spend lots of time explicitly presenting these algebras, since it would be very interesting if they appeared in diﬀerent contexts. Also, there are many examples of self-injective algebras (even given by a quiver with potential) that we cannot yet explicitly present, and I will explain how the geometry helps to gain information regarding their properties. I will also discuss some open problems.

Claus Michael Ringel (Bielefeld University)...... November 15th (Fri), 16:40–17:30 The root posets Given a (ﬁnite) root system, the choice of a root basis divides it into the positive and the negative ones, it also yields an ordering on the set of positive roots. The set of positive roots with respect to this

17 ordering is called a root poset. The root posets have attracted a lot of interest in recent years: The set A of antichains in a root poset (with a suitable ordering) turns out to be a lattice, it is called lattice of (generalized) non-crossing partitions. Non-crossing partitions play an interesting role in several parts of mathematics: not only in algebra and geometry, but even in free probability theory. The maximal chains in A correspond bijectively to the factorizations of a Coxeter element in the Weyl group using reﬂections, the number of maximal chains was determined already in 1974 by Deligne–Tits–Zagier.

If R is a Dynkin algebra, then the iso classes of the indecomposable R-modules form a poset with respect to the subfactor ordering: in a joint paper with Dlab (1979) we have shown that one obtains in this way all root posets, thus one can use the representation theory of R-modules in order to study the root posets and their antichain lattices. According to Ingalls and Thomas (2009), the lattice is isomorphic to the lattice of thick subcategories of the category of R-modules, and, as observed by Krause and Hubery (2013), the maximal chains in the antichain lattice correspond bijectively to the complete exceptional sequences of R-modules.

The lecture will provide a survey of some properties of the root posets, in particular, we will focus the attention to the role of the exponents.

18 INFORMATION

Welcome to Nagoya. The following information will be useful in helping you enjoy your stay during the conference. Please feel free to get in touch with the organizers (Prof. Iyama) and the secretary (Ms. Kozaki) if any questions arise or if you encounter unexpected trouble.

Conference Venue Conference mainly takes place at Sakata–Hirata Hall in Science South Building on campus (Fig- ure A). However, please note that the talks on Friday (15th) will be held at ES Hall in E&S Building (Figure B).

Registration Desk From November 11th to 14th, the registration desk will be set in the lobby of Sakata–Hirata Hall, where the conference takes place. You can pick up your conference package. The desk operates from 9:00 to 16:30 through Monday to Thursday. If you need to make copies, please ask at the registration desk. Those who want to send messages by fax can ask at the desk as well.

Coﬀee Breaks Coﬀee will be served in the lobby of Sakata–Hirata Hall. All participants are requested not to bring food into the conference hall.

Internet Connection The Wi-Fi access is available in and around conference hall. Please ask for the detailed information at the registration desk. You can choose to connect to eduroam as well.

Restaurants For lunch and dinner, there are some restaurants in the campus, or outside of the campus. You will ﬁnd them in Figure F.

Group photo We will take a group photo of all participants in the Conference Hall. We are planning this on November 14th (Thu) just after the morning session.

Conference Excursion An English guided bus tour is planned on the afternoon of November 13th (Wed). We will visit Nagoya Castle, Noritake Garden where we will observe (and even experience) a manufacturing process of high-quality ceramic tableware. Weather permitting, we will also visit Atsuta Shrine the one of the major shrines in Japan. All the participants are welcome, and the fee is 3,500 JPY which includes a small lunch in the bus. Please sign up for it by 12:00 on November 12th (Tue) at the registration desk of the conference. We will stop accepting applications once all the seats are taken. The chartered bus will depart at no delay of 13:00 from the front of the Toyoda Auditorium. (See Figure C) We require you to be on time.

19 Banquet There will be a conference banquet on November 14th (Thu) from 18:00 at Mei-dining, which is situated west part on campus (Figure C). The banquet costs 5,000 JPY per person. If you want to join the banquet, please come to the registration desk by 13:00 on November 12th.

Library The participants may use the Mathematics Library.

Location: Science Building A, 1st Floor (Figure C) Hours: 9:00–17:00 Services: The browsing, reference are available.

Please ask the librarians for details.

Money Exchange You will find a Post Office authorized to conduct foreign money exchange on campus (Figure C). Japan Post Bank provides cash withdrawal services for credit cards and cash cards issued by overseas financial institutions. Cards bearing

VISA, VISAELECTRON, VISA PLUS, MasterCard, Maestro, Cirrus, American Express, Diners Club, JCB, China Unionpay and DISCOVER can make withdrawals at Japan Post Bank ATMs. The ATMs on campus (Figure C) and in Chunichi Building (Figure E) are most convenient for you. The cards Master, VISA and PLUS can be used at the CitiBank (Figure E) for 24 hours.

Sightseeing Please consult the brochures you received at the registration desk and the following webpages for other sight seeing spots in and around Nagoya.

http://www.ncvb.or.jp/index_e.html is a good place to start.

20 Conference Of ce

EV Conference Hall EV (ES Hall)

Entrance Hall Lounge To 2F

Conference Hall To 2F (Sakata–Hirata Hall)

Registration Desk

Restaurant Central Library of Graduate School of Engineering EV chez Jiroud

Figure A: Science South Bldg.(Nov. 11–14) Figure B: E&S Bldg. (Nov. 15)

N to Motoyama Sta.

E&S Bldg. Science Library (Conference, Nov. 15)

Subway Meijo Line Science South Bldg. (Conference, Nov. 11–14)

Post Of ce / Japan Post Bank (ATM) Graduate School of Nagoya Daigaku Sta. Mathematics

Toyoda Auditorium

Bus Stop (for Excursion)

to Kanayama Sta.

Cafeteria “Mei-dining” (Banquet)

Figure C: Campus Map

21 to Kyōto / Shin-Ōsaka

24min

Ōzone Sta.

Nagoya Sta. Fujigaoka Sta.

16–24min Motoyama Sta. Sakae Sta. Nagoya Daigaku Sta.

Meitetsu Line

Takabata Sta. JR Tokaido Shinkansen Kanayama Sta.

to Tokyō to Chubu International Airport

Subway Higashiyama Line Subway Meijō Line Subway Sakura-dori Line Subway Tsurumai Line Subway Meiko Line Subway Kamiiida Line

Figure D: Subway Map with some Railways

to Ōzone Sta. N Subway Meijo Line Hotel Trusty Nagoya Sakae

10A 9A Bus Terminal (in Oasys 21) 10B 9B CitiBank (ATM) (in Sakae Parkside Place, 1F)

Undergrdound Mall

1 2 3 4 Sakae Sta. Subway Higashiyama Line to Nagoya Sta. 7 6 to Motoyama Sta. 8

5 1 Ferris Wheel 9 10 (on Sunshine Sakae, Roof Top) 11 10 2 9 10 2 3 4 Undergrdound Mall

8 7 6 5 16 15 14 12 7 6 Bus Terminal 13

Post Ofﬁce / Japan Post Bank (ATM) (in Chunichi Bldg., 1F)

to Kanayama Sta.

Figure E: Map of Sakae (around Hotel Trusty Nagoya Sakae)

22 Masaruya Café Rest B2 (western)

SevenEleven (CVS) Sakanayama (Japanese) Ashitaba (udon, soba) N Kangaroo Pocket (western) Furaipan (western) For You (rice in omelet) Roppa (tavern) CoCo! Ichiban-ya (curry) Gaya (tavern) Bai Toong (Asian) KFC (fast food)

Tai-Sho-Ken (Chineese noodle) Bunmeikan (tavern) Bun Boo Lassi (tavern) Shoro Zushi (sushi) Motoyama55 (Chinese noodles) Mister Donuts (sweets, cafe) Machikadoya (Japanese fast food) The Don (Japanese) SevenEleven (CVS) Wakabayashi (Japanese)

Doutor (cafe) Men-ya Ryōma (Chinese noodles) Kurashiki (Japanese) Paragon (Asian) Tori-Tori-Tei (tavern) Za·Watami (tavern) SUNKS (CVS) Eegaya (curry) Bon (cafe)

Cafe Nishihara (cafe) Gusto Tori-Kin (chikin) Kakure-ga (tavern) (Japanese, western)

En (Chinese) Dorako SevenEleven (CVS) La Pallete (western) (Korean BBQ) Hamakinu (tavern) Moe (cafe) Le Sourire(French) Yorimichi (tavern) Nakau Sapporo-Tei (Chinese noodles) (Japanese fast food) McDonald’s Sukiya Marina (western) Mos Burger (hamburger) (Japanese fast food) (hamburger) CircleK (CVS) Kochab (French)

Deli Deli (CVS) Miroku (steak) Retrocalm (cafe) Ampio Fiume (Italian) Cha Raku (cafe) Kaisenkan (Chinese)

Chez Jiroud (French) SevenEleven (CVS) Hokubu Shokudō (cafeteria) Dinning Forest (cafeteria) Craig’s Cafe (cafe) Café Fronte (cafe)

Yakusou Labo Toge Palu (cafe) Nanamitei (cafeteria) Hananoki (Japanese) (cafe) Rike-Shop (CSV) Drawing (cafe & bar) Midori Sushi (sushi) I-B Cafe (cafe) Hisaya (Japanese) Starbacks (cafe) Richiru (cafe) Phonon Cafe (cafe) Ikatsu (okonomiyaku) FamilyMart (CVS) Friendly Nanbu (cafeteria) FamilyMart Nanbu Shokudō (cafeteria) (CVS) Graduate School of Mathematics Nagoya University Hoja Nasreddin (curry) Shokuin Shokudō (cafeteria) Kobalele Cafe (cafe) Miura (cafe) CircleK (CVS)

IZUMI (French) Haruten (Japanese) Universal Club (cafeteria) LAWSON100 (CVS) Haloki (hamburg, steak) Ankuru (cafe) Lawson (CVS) Pion (Korean BBQ) Bento-Man (take out lunch) Xiang Lan Lou (Chinese) Botan-tei (Chinese) Gran Piatto (Italian) Eas Salon (curry)

SevenEleven (CVS) Nanzan University

Kazukian (soba)

Cafe Downey (cafe)

Ichiemon Dining (tavern) Pastel (sweets, cafe)

0m 100 200 300 400 500m 1km

Figure F: Lunch Map