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The \(n\)-order of algebraic triangulated categories. (English) Zbl 1285.18018
Let \({\mathcal T}\) be a triangulated category and let \(n\) be a positive integer. The \(n\)-order of \({\mathcal T}\) is an invariant whose value is a positive integer or infinity. It measures the strength of the relation \(n\cdot(Y/n)\) for objects \(Y\) in \({\mathcal T}\), where \(Y/n\) is the cone of multiplication by \(n\) in \(Y\). If \({\mathcal T}\) is obtained from chain complexes in an additive category (that is, if \({\mathcal T}\) is an algebraic triangulated category), then the \(n\)-order of \({\mathcal T}\) is infinite for all \(n\). But the \(p\)-local stable homotopy category for a prime number \(p\) has \(p\)-order \(p-1\) and is therefore not algebraic.

18E30 Derived categories, triangulated categories (MSC2010)
55P42 Stable homotopy theory, spectra
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