Auslander-Reiten sequences, Brown-Comenetz duality, and the $K(n)$-local generating hypothesis

TL;DR

This paper develops Auslander-Reiten sequences in the $K(n)$-local stable homotopy category using Brown-Comenetz duality, providing counterexamples to the $K(n)$-local generating hypothesis and extending methods to other triangulated categories.

Contribution

It introduces a $K(n)$-local Auslander-Reiten sequence framework with Brown-Comenetz duality, and constructs counterexamples to the generating hypothesis across all heights and primes.

Findings

Counterexamples to the $K(n)$-local generating hypothesis for all $n>0$ and primes.

Establishment of Auslander-Reiten sequences via Brown-Comenetz duality in the $K(n)$-local category.

Applicability of methods to other triangulated categories like derived categories of sheaves.

Abstract

In this paper, we construct a version of Auslander-Reiten sequences for the $K(n)$-local stable homotopy category. In particular, the role of the Auslander-Reiten translation is played by the local Brown-Comenetz duality functor. As an application, we produce counterexamples to the $K(n)$-local generating hypothesis for all heights $n>0$ and all primes. Furthermore, our methods apply to other triangulated categories, as for example the derived category of quasi-coherent sheaves on a smooth projective scheme.