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2019 Noncommutative Algebraic Geometry

Shanghai Workshop

Nov. 11-15th

Shanghai Center for Mathematical Sciences

Agenda

Monday，Nov. 11

09:00-09:10 Opening speech

09:10-10:00 Speaker: Changchang Xi, Capital Normal University

10:00-10:30 Tea break

10:30-11:20 Speaker: Osamu Iyama, Nagoya University

11:30-12:00 Speaker: Ruipeng Zhu, Fudan University

12:00-14:00 Lunch

14:00-14:50 Speaker: Xingting Wang, Howard University

14:50-15:20 Tea break

15:20-16:10 Speaker: Sei-Qwon Oh, Chungnam National University

16:10-17:00 Speaker: Theo Raedschelders, University of Glasgow

Tuesday, Nov. 12

09:00-09:50 Speaker: Milen Yakimov, Louisiana State University

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09:50-10:40 Speaker: Yu Zhou, Tsinghua University

10:40-11:10 Tea break

11:10-12:00 Speaker: Zheng Hua, The University of Hong Kong

12:00-14:00 Lunch

14:00-14:50 Speaker: Pu Zhang, Shanghai Jiaotong University

14:50-15:20 Tea break

15:20-16:10 Speaker: Guisong Zhou, Ningbo University

16:10-17:00 Speaker: Junwu Tu, Shanghai Tech University.

18:00-20:00 Banquet

Wednesday, Nov. 13

09:00-09:50 Speaker: Hiroyuki Minamoto, Osaka Prefecture University

09:50-10:40 Speaker: Kenta Ueyama, Hirosaki University

10:40-11:10 Tea break

11:10-12:00 Speaker: Masahisa Sato, Yamanashi University

12:00-14:00 Lunch

Free afternoon

Thursday, Nov. 14

09:00-09:50 Speaker: Guodong Zhou, East China Normal University

09:50-10:40 Speaker: Hongxing Chen, Capital Normal University

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10:40-11:10 Tea break

11:10-12:00 Speaker: Jianmin Chen, Xiamen University

12:00-14:00 Lunch

14:00-14:50 Speaker: Manuel Lionel Reyes, Bowdoin College

14:50-15:20 Tea break

15:20-16:10 Speaker: Liyu Liu, Yangzhou University

16:10-17:00 Speaker: Ryo Kanda, Osaka University

Friday, Nov. 15

09:00-09:50 Speaker: Will Donovan, Tsinghua University

09:50-10:40 Speaker: Jiwei He, Hangzhou Normal University

10:40-11:10 Tea break

11:10-12:00 Speaker: Daniel Rogalski, University of California San Diego

Closing Remark

12:00-14:00 Lunch

Free afternoon

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Abstracts

Speaker: Changchang Xi, Capital Normal University

Title: Good tilting modules and recollements of derived module categories

Abstract: Tilting modules are one of the interesting topics in the representation theory

of algebras and rings. In this talk, we shall consider when an infinitely generated tilting

module induces a recollement of derived module categories. A sufficient and necessary

condition is presented in terms of cohomologies of a complex related only to the given

tilting module. Examples of such tilting modules over noncommutative rings are

constructed from the ones over commutative rings. The talk reports parts of a joint work

with H. X. Chen:

[1] H.X. Chen; C.C. Xi, Good tilting modules and recollements of derived module

categories, II. J. Math. Soc. Japan 71(2019), No.2, 515-554.

Speaker: Osamu Iyama, Nagoya University

Title: Tilting theory of contracted preprojective algebras and cDV singularities

Abstract: A preprojective algebra of non-Dynkin type has a family of tilting modules

associated with the elements in the corresponding Coxeter group W (Buan-I-Reiten-

Scott). This family plays an important role to understand the representation theory of

the preprojective algebra. In this talk, I will discuss tilting theory of a contracted

preprojective algebra, which is a subalgebra eAe of a preprojective algebra A given by

an idempotent e of A. It has a family of tilting modules associated with the double

cosets in W modulo certain parabolic subgroups. I will apply our results to classify

certain family of Cohen-Macaulay modules over cDV singularities. This is a joint work

with Michael Wemyss.

Speaker: Ruipeng Zhu, Fudan University

Title: Nakayama automorphisms of ltered quantizations

Abstract: By using homological determinants of Hopf actions on skew Calabi-Yau

algebras, we describe relations between the Nakayama automorphisms of skew Calabi-

Yau algebras and the modular derivations of Poisson algebras under ltered deformation

quantization. As an application, we prove that the rings of differential operators over

smooth affine varieties are Calabi-Yau algebras. This is a joint work with Quanshui Wu.

Speaker: Xingting Wang, Howard University

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Title: Topological criterion for Poisson Dixmier-Moeglin equivalence

Abstract: In this work, we provide some topological criteria for the Poisson Dixmier-

Moeglin equivalence for a complex affine Poisson algebra A in terms of the poset of its

Poisson prime spectrum and the symplectic leaf/core stratification on its maximum

spectrum. In particular, we prove that the Zariski topology of the Poisson prime

spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin

equivalence for A. This is a joint work with Quanshui Wu and Juan Luo.

Speaker: Sei-Qwon Oh, Chungnam National University

Title: Endomorphisms of quantized algebras and their semiclassical limits

Abstract: Poisson algebras appear in classical mechanical system and their quantized

algebras appear in quantum mechanical system. Let F be a commutative ring and let A

be an F-algebra. Suppose that a central element h ∈ A is a nonzero, nonunit, non-

zero-divisor such that A := A/hA is commutative. Then A becomes a Poisson algebra

with Poisson bracket

In such case, A is called a semiclassical limit of A and A is called a quantization of A.

Here we discuss relationships between quantized algebras and their semiclassical limits.

In particular, it is shown that there exists a natural homomorphism from semigroup

induced by endomorphisms of quantized algebras into semigroup of Poisson

endomorphisms of their semiclassical limits.

Speaker: Theo Raedschelders, University of Glasgow

Title: Deformations of P-functors

Abstract: I will discuss how (generalised) deformations of a smooth projective variety

interact with arbitrary Fourier-Mukai functors (based on work by Toda). This

machinery can be applied to P-functors, allowing one to obtain a criterion for a P-

functor to become spherical on the total space of a deformation of the target,

generalising a result by Huybrechts and Thomas. Finally, I will explain how this

abstract criterion can be checked in the case of Hilbert schemes of points on a K3

surface, thus providing new examples of derived autoequivalences for certain

deformations of these Hilbert schemes. This talk contains joint work with Ciaran

Meachan.

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Speaker: Milen Yakimov, Louisiana State University

Title: Derived actions of groupoids of 2-Calabi-Yau categories

Abstract: Starting with work of Seidel and Thomas, there has been a great interest in

the construction of faithful actions of various classes of groups on derived categories

(braid groups, fundamental groups of hyperplane arrangements, mapping class groups).

We will describe a general construction of such actions in the setting of algebraic 2-

Calabi-Yau triangulated categories. It is applicable to categories coming from algebraic

geometry, cluster algebras and topology. To each algebraic 2-Calabi-Yau category, we

associate a groupoid, defined in an intrinsic homological way, and then construct a

representation of it by derived equivalences. In a certain general situation we prove that

this action is faithful and that the green green groupoid is isomorphic to the Deligne

groupoid of a hyperplane arrangement. This applies to the 2-Calabi-Yau categories

arising from algebraic geometry. We will also illustrate this construction for categories

coming from cluster algebras, where one gets categorical actions of braid groups. This

is a joint work with Peter Jorgensen (Newcastle University).

Speaker: Yu Zhou, Tsinghua University

Title: Realization functors and derived equivalences

Abstract: For the heart H of a bounded t-structure in the bounded derived category

D^b(A) of an abelian category A, there exists a triangle functor from D^b(H) to D^b(A),

which is so-called a realization functor. I will give necessary and sufficient conditions

on a realization funtor to be an equivalence. This is based on joint work with Xiao-Wu

Chen and Zhe Han.

Speaker: Zheng Hua, The University of Hong Kong

Title: Feigin-Odesskii Poisson structures via derived geometry

Abstract: The Feigin-Odesskii Poisson structures are semiclassical limits of the Feigin-

Odesskii elliptic algebras. These noncommutative algebras are vast generalization of

Sklyanin algebras. It is an open problem that how to classify the symplectic leaves of

these Poisson structures. With Sasha Polischuk, we construct a (1-d) shifted Poisson

structure on the moduli stack of bounded complexes of vector bundles on projective

Calabi-Yau d-folds. When d=1, our Poisson structure descends to Feigin-Odesskii’s

Poisson structure on certain components of the moduli stack. The derived geometry of

the moduli stack leads to a geometric description of the symplectic leaves. Using

algebraic geometry, we give an explicit classification of symplectic leaves for those

Poisson structures of “endomorphism” type. This is a joint work with Sasha Polishchuk.

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Speaker: Pu Zhang, Shanghai Jiaotong University

Title: Exceptional cycles in triangulated categories

Abstract: We will recall basic properties of exceptional cycles in a Hom-finite Krull-

Schmidt triangulated category with Serre functor, recently introduced by

N.BroomheadD.Pauksztello-D.Ploog. As an example, we will give all the exceptional

cycles in the bounded derived category Db (kQ) of finite acyclic quivers Q. Main

attention will be focused on the bounded homotopy category Kb (A-proj) of perfect

complexes over gentle algebras A. We classify all the exceptional cycles in Kb (A-proj),

and determine the actions of the twist functors induced by exceptional cycles. Namely,

the mouth of each characteristic component of Kb (A-proj) forms an exceptional cycle;

if the quiver of A is not of type A3, this gives all the exceptional n-cycle in Kb (A-proj)

with n ≥ 2, up to shift and rotation; and a string complex is an exceptional 1-cycle (a

spherical object) iff it is at the mouth of a characteristic component with AG-invariant

(1, m). A band complex which is an exceptional 1- cycle is also at the mouth (of a

homogeneous tube); however, a band complex which is at the mouth is not necessarily

an exceptional 1-cycle. This is a joint work with Peng Guo.

Speaker: Guisong Zhou, Ningbo University

Title: The structure of connected (graded) Hopf algebras

Abstract: Connected Hopf algebras of finite Gelfand-Kirillov dimension over a field

of characteristic 0 can be viewed as generalizations of enveloping algebras of finite

dimensional Lie algebras, as deformations of polynomial algebras in finitely many

variables, and as noncommutative counterpart of connected unipotent algebraic groups.

In this talk, we will show that connected graded Hopf algebras of finite Gelfand-

Kirillov dimension over a field of characteristic 0 are iterated Hopf Ore extensions, that

is, they can be obtained from the base field by Hopf Ore extensions in finitely many

times. The approach is based on the combinatorial properties of Lyndon words and the

standard bracketing on words. This is a Joint work with Di-Ming Lu and Yuan Shen.

Speaker: Junwu Tu, Shanghai Tech University.

Title: Enumerative invariants from Calabi-Yau categories

Abstract: In this talk, we present a detailed construction of categorical enumerative

invariants defined by Costello back in 2004/2005.

These invariants conjecturally generalize Gromov-Witten invariants/FJRW invariants,

and BCOV invariants simultaneously.

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Speaker: Hiroyuki Minamoto, Osaka Prefecture University

Title: On a Heisenberg analog of the preprojective algebra

Abstract: This is a joint work with Martin Herschend. We introduce a Heisenberg

analog of the preprojectve algebra of a quiver Q.

If we look the preprojective algebra as a quiver version of the polynomial algebra in

two variables, then our algebra can be looked as a quiver version of the Heisenberg

algebra in two variables.

We note that our algebra is a special case of algebras introduced by Etingof-Rains,

which is a special case of algebras introduced by Cachazo-Katz-Vafa.

We show that our algebra of very special case is a proper one dimnsional higher version

of the preprojective algebra which is closely related to Auslander-Reiten theory of KQ.

We also discuss the invariant subalgebra of Heisenberg algebra in two variables by a

finite subgroup of SL(2).

Speaker: Kenta Ueyama, Hirosaki University

Title: Stable categories of graded Cohen-Macaulay modules over skew quadric

hypersurfaces

Abstract: In this talk, we study the stable categories of graded maximal Cohen-

Macaulay modules over $S/(f)$ where $S$ is a ($\pm 1$)-skew polynomial algebra

generated in degree one, and $f \in S$ is the sum of all squared variables. Our method

is to use a certain graph associated with $S$. We present four graphical operations

called mutation, relative mutation, Kn\"orrer reduction, and two points reduction, and

show that the above stable categories can be completely computed by using these

graphical operations. This talk is based on joint work with Izuru Mori, and on joint

work with Akihiro Higashitani.

Speaker: Masahisa Sato, Yamanashi University

Title: Generalized Nakayama-Azumaya Lemma and Ware's problem

Abstract: In this talk, we have two main topics which include basic but important

results in noncommutative ring theory. The first topic is about a generalization of the

Nakayama-Azumaya Lemma. The second topic is about an affirmative answer for

Ware’s problem. Also we give more general result and an interesting example relating

to Ware’s problem.

The original Nakayama-Azumaya Lemma asserts that MJ(R) = M implies M = 0 for a

finitely generated R-module M, here J(R) is the Jacobson radical of a ring R. Also this

lemma holds for any projective modules by H.Bass [1]. We unify and generalize these

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two lemmas for an R-module M isomorphic to a direct summand of a direct sum of

finitely generated modules in the following theorem.

Theorem 1 (Generalized Nakayama-Azumaya Lemma). Let M be an R-module

isomorphic to a direct summand of a direct sum of finitely generated modules. Then

MJ(R) = M implies M = 0. R. Ware [5] proposed the following problem.

Problem 1. If a projective right R-module P has unique maximal submodule

L, then L is the largest maximal submodule of P.

We give an affirmative answer of this problem and give more general result.

Theorem 2. Let M be an R-module isomorphic to a direct summand of a direct sum of

finitely generated modules. If M has unique maximal submodule L, then L is the largest

maximal submodule of M. We need Generalized Nakayama-Azumaya Lemma to prove

the above theorem. Also in the proof of the above theorem, we show that M is

indecomposable. As a consequence, M is countably generated. (See [3, 4].)

Relating to Ware’s problem, we give the following examples.

For a uniserial module U, the paper [2] asserts K is an infinitely generated projective

module with unique maximal submodule, but we will know that this assertion does not

seem to be true as an application of above results of our example.

References

[1] H. Bass, Finitistic dimension and a homological generalization of semiprimary rings,

Trans. American Math. Soc. Vol.95 (1960), 466-488.

[2] A. Facchini, D. Herbera, I. Sakhajev, Finitely Generated Flat Modules and a

Characterization of Semiperfect Rings, Comm. in Algebra, Vol.31 No.9 (2003), 4195–

214.

[3] F.W. Anderson, K.R. Fuller, Rings and Categories of Modules, GTM 13,

SpringerVerlag (1992).

[4] I. Kaplansky, Projective modules, Ann. of Math. Vol.68 (1958), 372―377.

[5] R. Ware, Endomorphism rings of projective modules, Trans. Amer. Math. Soc. 155

(1971), 233-256.

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Speaker: Guodong Zhou, East China Normal University

Title: Computing Hochschild cohomology via weak self-homotopies and algebraic

Morse theory

Abstract: This talk is a survey about computing methods for Hochschild cohomology.

After introducing this cohomology theory, I will introduce weak self-homotopies and

algebraic Morse theory, which enable us to construct projective resolutions, comparison

morphisms (or homotopy equivalences) between resolutions as well as homotopies

realizing them. I will illustrate these methods by various examples.

Speaker: Hongxing Chen, Capital Normal University

Title: Homological theory of self-orthogonal modules and Tachikawa's second

conjecture

Abstract: In 1970, Hiroyuki Tachikawa proposed two homological conjectures arising

from Nakayama's conjecture. Tachikawa's second conjecture says that if a finitely

generated module over a self-injective Artin algebra is self-orthogonal, then it is

projective. In this talk, we first discuss some homological properties of self-orthogonal

generators over self-injective algebras in terms of the stable categories of Gorenstein

projective modules over their endomorphism algebras, and then provide several

equivalent characterizations of Tachikawa's second conjecture. It turns out that a class

of generalized symmetric algebras (that is, endomorphism algebras of generators over

symmetric algebras) is shown to satisfy Nakayama's conjecture. This is based on an

ongoing work with Professor Changchang Xi.

Speaker: Jianmin Chen, Xiamen University

Title: Frobenius-Perron theory of endofunctors

Abstract: The spectral radius (also called the Frobenius-Perron dimension) of a matrix

is an elementary and extremely useful invariant in linear algebra, combinatorics,

topology, probability and statistics. The Frobenius-Perron dimension has become a

crucial concept in the study of fusion categories and representations of semisimple

weak Hopf algebras since it was introduced by Etingof-Nikshych-Ostrik in early 2000.

In this talk, I will generalize the Frobenius-Perron dimension of an object in a fusion

category, introduce the Frobenius-Perron dimension and several Frobenius-Perron type

invariants of an endofunctor of a category, give some basic properties of them and apply

them to study the derived category of coherent sheaves on projective schemes and

modules over finite dimensional algebras. The talk is based on joint works with Zhibin

Gao, Elizabeth Wicks, James Zhang, Xiaohong Zhang and Hong Zhu.

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Speaker: Manuel Lionel Reyes, Bowdoin College

Title: An invitation to twisted Calabi-Yau algebras

Abstract: This talk will be a survey of some recent results on twisted Calabi-Yau (CY)

algebras (joint work with Daniel Rogalski). These algebras include many "smooth"

algebras that arise in noncommutative algebraic geometry, especially Artin-Schelter

(AS) regular algebras and Calabi-Yau algebras. We will discuss the precise way in

which $\mathbb{N}$-graded twisted CY algebras can be viewed as AS regular

algebras that are not necessarily connected, as well as the structure of these algebras in

dimension $\leq 3$.

Speaker: Liyu Liu, Yangzhou University

Title: Nakayama automorphisms of Ore extensions over polynomial algebras

Abstract: Let $R$ be a skew Calabi--Yau algebra. It was proved that every Ore

extension $E=R[x; \sigma, \delta]$ is also skew Calabi--Yau when $\sigma$ is an

automorphism. However, the Nakayama automorphism $\nu$ of $E$ has not been

completely determined so far. In this talk, I will present the explicit formula of $\nu$ in

the case that $R$ is a polynomial algebra in $n$ variables for an arbitrary integer $n\geq

1$. This is joint work with Wen Ma.

Speaker: Ryo Kanda, Osaka University

Title: Feigin-Odesskii's elliptic algebras

Abstract: This is based on ongoing joint work with Alex Chirvasitu and S. Paul Smith.

Feigin and Odesskii introduced a family of noncommutative graded algebras, which are

parametrized by an elliptic curve and some other data, and claimed a number of

remarkable results in their series of papers. The family contains all higher dimensional

Sklyanin algebras, which have been widely studied and recognized as important

examples of Artin-Schelter regular algebras. In this talk, I will explain some properties

of Feigin-Odesskii's algebras, including the nature of their point schemes and algebraic

properties obtained by using the quantum Yang-Baxter equation.

Speaker: Will Donovan, Tsinghua University

Title: Stringy Kaehler moduli, mutation and monodromy

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Abstract: The derived symmetries associated to a 3-fold admitting an Atiyah flop may

be organised into an action of the fundamental group of a sphere with three punctures,

thought of as a stringy Kaehler moduli space. I extend this to general flops of irreducible

curves on 3-folds in joint work with M Wemyss, using a novel helix of sheaves

supported on the flopping curve, and relative spherical objects over noncommutative

base rings.

Speaker: Jiwei He, Hangzhou Normal University

Title: Maximal Cohen-Macaulay modules of noncommutative hypersurfaces and

Clifford deformations of Frobenius algebras

Abstract: Let $E$ be a Koszul Frobenius algebra. A Clifford deformation of $E$ is a

finite dimensional $\mathbb Z_2$-graded algebra $E(\theta)$, which corresponds to a

noncommutative quadric hypersurface $E^!/(z)$, for some central regular element

$z\in E^!_2$. It turns out that the bounded derived category $D^b(\gr_{\mathbb

Z_2}E(\theta))$ is equivalent to the stable category of the maximal Cohen-Macaulay

modules over $E^!/(z)$ provided that $E^!$ is noetherian. As a consequence,

$E^!/(z)$ is a noncommutative isolated singularity if and only if the corresponding

Clifford deformation $E(\theta)$ is a semisimple $\mathbb Z_2$-graded algebra. The

preceding equivalence of triangulated categories also indicates that Clifford

deformations of trivial extensions of a Koszul Frobenius algebra are related to the

Kn\"{o}rrer Periodicity Theorem for quadric hypersurfaces.

Speaker: Daniel Rogalski, University of California San Diego

Title: The Brown-Goodearl conjecture for weak Hopf algebras

Abstract: Brown and Goodearl conjectured that any Noetherian Hopf algebra should

have finite injective dimension. The conjecture is known to be true in some cases, in

particular for affine polynomial identity Hopf algebras. Weak Hopf algebras are an

important generalization of Hopf algebras in which the axioms on the unit and counit

are weakened. Just as for Hopf algebras, the category of modules over a weak Hopf

algebra has a monoidal structure, and this has important consequences for the

homological properties of the algebra. We study the extension of the Brown-Goodearl

conjecture to the case of weak Hopf algebras, and show that a weak Hopf algebra which

is finite over an affine center has finite injective dimension, and is a direct sum of AS

Gorenstein algebras. (Joint with Rob Won and James Zhang.)