A remark on the additivity of traces in triangulated categories

The additivity of traces in certain tensor triangulated categories for endomorphisms of finite order of distinguished triangles is investigated. For the identity endomorphism this has been fully established by J. P. May (“The additivity of traces in triangulated categories”, Adv. Math., 2001, 163, 34-73). By imposing extra conditions on the coefficients we show how May’s result implies a stronger additivity.

💡 Research Summary

The paper investigates the additivity of traces in tensor‑triangulated categories for endomorphisms of finite order acting on distinguished triangles. The starting point is J. P. May’s 2001 theorem, which proves that for a distinguished triangle (X\to Y\to Z\to \Sigma X) consisting of strongly dualizable objects, the trace of the identity satisfies (\operatorname{tr}(\mathrm{id}_X)+\operatorname{tr}(\mathrm{id}_Z)=\operatorname{tr}(\mathrm{id}_Y)). May’s result, however, is limited to the identity endomorphism; it does not directly address non‑trivial finite‑order automorphisms.

The author introduces additional hypotheses on the coefficient ring (R). Specifically, (R) is assumed to be split‑semisimple and to contain all (n)‑th roots of unity, where (n) is the order of the endomorphism under consideration. Under these conditions any endomorphism (f) with (f^{,n}= \mathrm{id}) can be diagonalised over (R); its eigenvalues are (n)-th roots of unity and the associated idempotent projectors commute with the morphisms of the triangle. This diagonalisation makes the trace (\operatorname{tr}(f)) a linear combination of the eigenvalues, allowing a direct comparison of traces on the three vertices of the triangle.

The main theorem states: let (X\to Y\to Z\to \Sigma X) be a distinguished triangle in a tensor‑triangulated category (\mathcal{T}) and let (f_X,f_Y,f_Z) be endomorphisms of finite order (n) compatible with the triangle maps. If the coefficient ring satisfies the split‑semisimple and root‑of‑unity conditions, then
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