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Auslander-Reiten sequences, Brown-Comenetz duality, and the \(K(n)\)-local generating hypothesis. (English) Zbl 1380.55008
Freyd’s generating hypothesis is a prominent open problem in stable homotopy theory. It asks whether a map between finite spectra is null-homotopic if it induces the zero map on homotopy groups. An analogous question can be asked in more general triangulated categories, and it has for example been explored under which conditions the generating hypothesis holds in the derived category of a commutative ring [K. H. Lockridge, J. Pure Appl. Algebra 208, No. 2, 485–495 (2007; Zbl 1112.55011)] or in the stable module category of a finite group [J. F. Carlson et al., Proc. Am. Math. Soc. 137, No. 8, 2575–2580 (2009; Zbl 1188.20006)].
One main result of the paper under review is that the statement of the generating hypothesis does not hold in the \(K(n)\)-localization of the stable homotopy category as soon as \(n > 0\). Here \(K(n)\) is the Morava \(K\)-theory spectrum of height \(n\) at a prime \(p\). The \(K(n)\)-localizations are important because they provide the layers in the chromatic filtration of the stable homotopy category.
The strategy of the paper is to employ the notion of Auslander-Reiten sequences in compactly generated triangulated categories that has for example been studied by H. Krause [\(K\)-Theory 20, No. 4, 331–344 (2000; Zbl 0970.18012)]. The author shows that the simultaneous presence of Auslander-Reiten sequences and of indecomposables which are not a sum of shifted copies of the generator violates the generating hypothesis. This criterion is then verified for the \(K(n)\)-localization of the stable homotopy category when \(n > 0\). The Auslander-Reiten sequences for this category determine an Auslander-Reiten translation functor \(T\), and it is shown that \(T\) is the dual of the Brown-Comenetz duality functor.

MSC:
55P42 Stable homotopy theory, spectra
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
18E30 Derived categories, triangulated categories (MSC2010)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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