Abstract representation theory of Dynkin quivers of type A

Abstract

We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields. These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result. Finally, using abstract representation theory of linearly oriented $A_n$-quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable $\infty$-categories.

References in corpus (5)

• Higher Topos Theory
• On triangulated orbit categories
• The linearity of traces in monoidal categories and bicategories
• Tilting theory for trees via stable homotopy theory
• Universal derived equivalences of posets

Cited by in corpus (11)

• Simplicial structures in higher Auslander-Reiten theory
• Tilting theory for trees via stable homotopy theory
• Generalised BGP reflection functors via the Grothendieck construction
• Characterizations of abstract stable homotopy theories
• On good morphisms of exact triangles
• Abstract cubical homotopy theory
• Generalized stability for abstract homotopy theories
• Derivator Six Functor Formalisms -- Definition and Construction I
• The bivariant parasimplicial ${\mathsf{S}}_{\bullet }$-construction
• Higher symmetries in abstract stable homotopy theories
• Higher Auslander algebras of type $\mathbb{A}$ and the higher Waldhausen $S$-constructions