Homological Theory of Representations

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.

Tropical Duality in $$(d+2)$$-Angulated Categories

Let $$\mathscr {C}$$ C be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ T and $$\mathscr {U}$$ U . A result from Jørgensen and Yakimov (Sel Math (NS) 26:71–90, 2020) and Demonet et al. (Int Math Res Not 2019:852–892, 2017) known as tropical duality says that the index with respect to $$\mathscr {T}$$ T provides an isomorphism between the split Grothendieck groups of $$\mathscr {U}$$ U and $$\mathscr {T}$$ T . We also have the notion of c-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of $$(d+2)$$ ( d + 2 ) -angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, c-vectors in the $$(d+2)$$ ( d + 2 ) -angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the c-vectors are recoverable as dimension vectors of modules in a module category.

Density of g-Vector Cones From Triangulated Surfaces

We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$ except for a closed surface with exactly one puncture, in which case it is equal to the half space of a certain explicit hyperplane in $\mathbb{R}^n$. Our main ingredients are laminations on $(S,M)$, their shear coordinates, and their asymptotic behavior under Dehn twists. As an application, if $(S,M)$ is not a closed surface with exactly one puncture, the exchange graph of cluster tilting objects in the corresponding cluster category is connected. If $(S,M)$ is a closed surface with exactly one puncture, it has precisely two connected components.

Approximations, ghosts and derived equivalences

Approximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly n-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.