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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.)