Workshop Speakers

Lars Winther Christensen (Texas Tech University, USA)

Title 1: Complexes and replacements

Abstract: The idea of resolutions, and more generally replacements, of complexes is to replace a given complex by one with isomorphic homology and

additional useful properties, typically lifting properties.  In this talk I will survey (abstract) existence theorems of replacements as well as concrete constructions.

Title 2: The derived category, invariants, equivalences, and dualities

Abstract: In this talk I will discuss homological invariants of complexes as well as dualities and equivalences of subcategories of the derived category. These technologies find applications in characterizations of Iwanaga--Gorenstein rings and regular rings.

Title 3: The derived category of a commutative noetherian ring

Abstract: Many applications of homological algebra in commutative ring theory evolve around what Kunz calls the "local--global principel in commutative algebra". I will briefly survey some instances of this principle and then move to prove characterizations of Gorenstein and regular rings in terms of properties of their complexes:

Theorem 1:  The following conditions are equivalent.

(1) R is Gorenstein.

(2) Every acyclic complex of projective R-modules is totally acyclic.

(3) Every acyclic complex of injective R-modules is totally acyclic.

(4) Every acyclic complex of flat R-modules is totally acyclic.

Theorem 2: The following conditions are equivalent.

(1) R is regular.

(2) Every complex of fintely generated projective R-modules is semi-projective.

(3) Every complex of projective R-modules is semi-projective.

(4) Every complex of injective R-modules is semi-injective.

(5) Every complex of flat R-modules is semi-flat.

(6) Every acyclic complex of projective R-modules is contractible.

(7) Every acyclic complex of injective R-modules is contractible.

(8) Every acyclic complex of flat modules is pure acyclic.

Sergio Estrada (University of Murcia, Spain)

Title: Gorenstein homological algebra over one-sided periodic cotorsion pairs

Abstract: In these lectures, we will summarize the main properties of complete cotorsion pairs in which one of the classes satisfies some periodicity property. We'll see how these pairs naturally induce Gorenstein relative global invariants, which, in particular cases, allow us to prove Gorenstein versions of well-known results in classical homological algebra by Jensen, Osofsky, and Stenström. The theory also enables us to find new classes of rings for which the class of Gorenstein projective modules is specially precovering, as well as to extend a result of Chen about the existence of an Iyengar and Krause kind of triangle equivalence between the homotopy categories of injective and projective modules over rings of finite Gorenstein global dimension.

Henning Krause (Bielefeld University, Germany)

Title: Completions of triangulated categories

Abstract: Completions arise in all parts of mathematics. For example, the real numbers are the completion of the rationals. This construction uses equivalence classes of Cauchy sequences, and there is an obvious generalisation leading to the completion of a metric space. Completions of rings and modules play an important role in algebra and geometry. For categories there is the notion of ind-completion due to Grothendieck and Verdier which provides an embedding into categories with filtered colimits. The goal of my lectures is to explain these ideas and to combine them to study completions of triangulated categories. I do not offer an elaborated theory and rather look at various examples which arise naturally in in algebra and geometry.

Liping Li (Hunan Normal University, China)

Title: Sheaf theory and its application in group representation theory

Abstract: Sheaf theory plays a central role in various areas such as algebraic geometry, differential geometry, geometric representation theory, and categorical logic. In these lectures I will give an elementary introduction on sheaf theory, and then focus on its application in investigating continuous representation of topological groups. Main topics include: Grothendieck topologies, classification of Grothendieck topologies over artinian EI categories, sheaves of modules over ringed sites, a torsion theoretic intepretation of sheaf theory, orbit categories of topological groups, Artin's theorem, classification of irreducible continuous representations of some topological groups, noetherianity of infinite polynomial rings up to symmetry.

Conference Speakers

Hongxing Chen (Capital Normal University, China)

Title: Categorical obstructions to bounded t-structures on triangulated categories

Abstract: We give a categorical obstruction (the singularity category in our sense) to the existence of bounded t-structures on general triangulated categories satisfying a finiteness condition and establish related results about the equivalence of bounded t-structures.These results aboutobstructions can be viewed as categorical generalizations of a bold conjecture by Antieau, Gepnerand Heller (2019) regarding the relationship between the regularity of finite-dimensional noetherian schemes and the existence of bounded t-structures on their derived categories of perfect complexes.The conjecture (even a major generalization of the conjecture) is now a theorem recently proved by Amnon Neeman (2022) using “good metric” techniques inthe theory of approximable triangulated categories.Our general treatment, when specialized to the case of noetherianschemes, immediately gives us Neeman's theorems an application and significantly generalizes another remarkable theorem by Neeman about the equivalence of bounded t-structures on bounded derived categories of coherent sheaves. This reports a recent joint work with Rudradip Biswas, Chris J. Parker, Kabeer Manali Rahul and Junhua Zheng.

Jianmin Chen (Xiamen University, China)

Title: Maximal almost pre-rigid representations over type D quivers

Abstract: The notion of maximal almost rigid representations was firstly introduced by Barnard, Gunawan, Meehan and Schiffler to study the representation theory of type A quivers. It is a class of important objects from algebraic, geometric and combinatorial perspective. In this talk, I will introduce an analogue of maximal almost rigid representations which we call maximal pre-rigid representations over certain type D quivers, provide a geometric realization for maximal pre-rigid representations, and then show their general form as well as their application in the tilting theory.

Xiao-Wu Chen (University of Science and Technology of China, China)

Title: Preprojective algebras, skew group algebras and Morita equivalences

Abstract: The folding process is classic in Lie theory and palys a role in the representation theory of quivers. By the work of Buan-Iyama-Reiten-Scott, Mizuno, and Fu-Geng,  there are remarkable bijections between certain ideal monoids of preprojective algebras and Weyl groups. We use the folding process to compare these bijections. This is joint with Ren Wang.

Ioannis Emmanouil (University of Athens, Greece)

Title: Strong fp-injectivity and duality

Abstract: We plan to present a list of results, which suggest that the appropriate dual of the class of flat modules is the class of strongly fp-injective modules. These results involve the purity of resolutions, the Ext1-orthogonality in the module category and certain global invariants appearing in Gorenstein homological algebra.

Changjian Fu (Sichuan University, China)

Title: Intersection vectors over tilings with application to gentle algebras and cluster algebras

Abstract: A tiling is a marked surface with a partial triangulation, which provides a geometric model for gentle algebras and cluster algebras.The class of  permissible arcs play a central role, which corresponds to indecomposable tau-rigid modules over the corresponding gentle algebra and non-initial cluster variables of the corresponding cluster algebras. We observe a local-global criterion for multiset of pairwise compatible permissible arcs, and prove that a multiset of permissible arcs are uniquely determined by its intersection vector under a mild conditon on the partial triangulation.  As an appliction, we confirm the denominator conjecture for cluster algebras of type ABC. This is based on a joint work with Shengfei Geng.

Xianhui Fu (Northeast Normal University, China)

Title: Ghosts, phantoms and Cartan-Eilenberg DG-modules for a DG-ring 

Abstract: We firstly investigate the ghost ideal and the phantom ideal in the (derived) category of DG-modules of a DG-ring. This allows us to introduce and investigate the notions of a Cartan-Eilenberg projective module, a Cartan-Eilenberg injective module, and a Cartan-Eilenberg flat module for a DG-ring. An immediate application is that we can give an affirmative answer to a conjecture of Minamoto. Also we may investigate notherian DG-ring from our approach, and parallel to classical theory of rings and modules, introduce and investigate the notions of a coherent DG-ring, and a perfect DG-ring. We also investigate the global dimension and weakly global dimension of a DG-ring in the sense of Hovey and Lockridge. This talk is based on an ongoing project with Xiaoyan Yang.

Sira Gratz (Aarhus University, Denmark)

Title: Metric completions of discrete cluster categories 

Abstract: Methods for generating new triangulated categories from old are notoriously few and far between. Neeman’s recent innovation  allows one to complete with respect to suitable metrics on a triangulated category to construct a new triangulated category. This promises the opportunity to construct a plethora of new examples of triangulated categories. However, up to now, explicit computations have all taken place within an existing ambient triangulated category. In this talk, based on joint work with Charley Cummings, we present a cluster-flavoured example where the computation can be done without this crutch. More specifically, we investigate discrete cluster categories of type A and show that, with a suitable choice of metric, the metric completion of such a category mirrors a topological completion of its combinatorial model.

Jiwei He (Hangzhou Normal University, China)

Title: An introduction to Shephard-Todd-Chevalley Theorem for finite group actions on AS-regular algebras 

Abstract: The classical Shephard-Todd-Chevalley Theorem says that for a finite group G acting faithfully on a polynomial algebra A, then the invariant subalgebra A^G is of finite global dimension if and only if G is generated by pseudo-reflections. In this talk, I will report some progresses on noncommutative version of Shephard-Todd-Chevalley Theorem.

Wei Hu (Beijing Normal University, China)

Title: Derived  invariance of the contravariantly finiteness of Gorenstein projectives 

Abstract: There is an open problem that, for an arbitrary ring, whether the subcategory consisting of all Gorenstein projectives is contravariantly finite in the module category. In this talk, we show that the stable functor may play a role in the study of this problem. Particularly, we show that the contravariantly finiteness of  the subcategory consisting of all Gorenstein projectives is preserved by derived equivalences. This is based on an ongoing work with Xianhui Fu.

Zhaoyong Huang (Nanjing University, China)

Title: Auslander-type conditions and weakly Gorenstein algebras

Abstract: Let R be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if R is left quasi Auslander, then R is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if R satisfies the Auslander condition, then R is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander--Reiten's conjecture, which states that R is Gorenstein if R satisfies the Auslander condition.

Yajun Ma (Lanzhou Jiaotong University, China)

Title:  Model structures and Q-shaped derived category

Abstract: In this talk, we first give an informal introduction to model structures. We then construct a flat model structure on the category of additive functors from a preadditive category satisfying certain conditions to the module category, whose homotopy category is the Q-shaped derived category introduced by Holm and Jorgensen. Finally, we describe Gorenstein object on the functor category, and hence improve a result by Dell'Ambrogio, Stevenson and Št'ovíček. This is the joint work with Zhenxing Di, Liping Li and Li Liang.

Lixin Mao (Nanjing Institute of Technology, China)

Title:  A class of special formal triangular matrix rings

Abstract: Let R and S be rings and C an (S, R)-bimodule. Then we can construct a formal triangular matrix ring T with usual matrix addition and multiplication. We consider a class of special formal triangular matrix ring, i.e., C is a semidualizing (S, R)-bimodule. We will exhibit the connections between C-reflexive (resp.C-Gorenstein projective, C-tilting) left S-modules and reflexive (resp. Gorenstein projective, tilting) left T-modules.

Greg Stevenson (Aarhus University, Denmark)

Title: Proxy-smallness and presentations

Abstract: Proxy-smallness, introduced by Dwyer, Greenlees, and Iyengar, is a rather mysterious condition for objects of a compactly generated triangulated category. An object P is proxy-small if the localizing subcategory P generates and the localizing subcategory generated by the compacts in thick(P) coincide. Every compact object is proxy-small, but there are many more examples. For instance, the residue field of any commutative noetherian local ring is proxy-small in the derived category. I’ll give an introduction to this condition and why it is so interesting. Then I’ll reinterpret it in a more philosophical way: proxy-small objects are presentations of compactly generated subcategories. This leads to a slick characterization of proxy-smallness. The results in this talk are based on joint work with Benjamin Briggs and Srikanth Iyengar.

Peder Thompson (Mälardalen University, Sweden)

Title: Dimension and depth inequalities over complete intersections

Abstract: For a pair of modules M and N over a complete intersection ring R, there is a natural inequality whenever the tensor product of M and N has finite length: dim(M)+dim(N) is at most dim(R)+codim(R). This is an extension of Serre’s fundamental dimension inequality for regular local rings, which was motivated by the geometric intersection of varieties. We will discuss an extension of Hochster’s theta invariant to complete intersections and consider when the non-vanishing of this invariant detects equality. We will also explore versions of this inequality involving depth and complexity, and their relation to some open questions in homological algebra. This is joint work with Petter Andreas Bergh and David Jorgensen.

Jiaqun Wei (Zhejiang Normal University, China)

Title: An extended version of Zhang’s question

Abstract: Zhang [X. Zhang, Self-orthogonal τ-tilting modules and tilting modules. J. Pure Appl. Algebra 226 (2022),106860.] asked if self-orthogonal τ-tilting modules are tilting. In this paper, we extend his question to derived category and provide a partial answer to it.

Guodong Zhou (East China Normal University, China)

Title: Categorical properties and homological conjectures for bounded extensions of algebras

Abstract: An extension $B\subset A$ of finite dimensional algebras is bounded if the $B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is finite and $\Tor_i^B(A/B, (A/B)^{\otimes_B j})=0$ for all $i, j\geq 1$.  We show that for a bounded extension $B\subset A$, the algebras $A$ and $B$ are singularly equivalent of Morita type with level. Additively, under some conditions, their stable categories of Gorenstein projective modules and Gorenstein defect categories are equivalent, respectively. Applications to trivial extensions, triangular matrix algebras and Morita context algebras are given.  Some homological conjectures are also investigated for bounded extensions, including Auslander-Reiten conjecture, finististic dimension conjecture, Han's conjecture, and Keller's conjecture.

Yu Zhou (Tsinghua University, China)

Title: Connectedness of mutation graph of support tau-tilting modules for skew-gentle algebras

Abstract: In the tau-tilting theory of finite-dimensional algebras, an important problem is whether the mutation graph of support tau-tilting modules is connected. In this talk, I will provide a proof of the connectedness of this graph for skew-gentle algebras using the cluster category and its geometric model.

Bin Zhu (Tsinghua University, China)

Title: Support τ-tilting subcategories in exact categories

Abstract: Let E=(A, S) be an exact category with enough projectives P. We introduce the notion of support τ-tilting subcategories of E. It is compatible with the existing definitions of support τ-tilting modules (subcategories) in various context. It is also a generalization of tilting subcategories of exact categories. We show that there is a bijection between support τ-tilting subcategories and certain τ-cotorsion pairs. Given a support τ-tilting subcategory T, we find a subcategory E_T of E which is an exact category and T is a tilting subcategory of E_T. If E is Krull-Schmidt, we prove the cardinal |T| is equal to the number of isomorphism classes of indecomposable projectives Q such that Hom_E(Q, T) \neq 0. We also show a functorial version of Brenner-Butler’s theorem. This is a joint work with Jixin Pan and Yaohua Zhang.