Abstract representation theory of Dynkin quivers of type A

TL;DR

This paper develops a general abstract framework for the representation theory of Dynkin quivers of type A, introducing spectral bimodules, tilting modules, and higher triangulations in stable homotopy theories.

Contribution

It extends classical representation theory concepts to a broad spectral setting using stable derivators and introduces new universal tilting modules and spectral Serre duality.

Findings

Spectral tilting modules induce equivalences via spectral bimodules.

Construction of spectral Serre duality in abstract stable homotopy theories.

Canonical higher triangulations for linearly oriented A_n-quivers.

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.