Hereditary triangulated categories
The authors introduce the notion of hereditary triangulated categories as triangulated categories equivalent to the bounded derived category of a hereditary abelian category and the equivalence commutes with the translation functors. This notion is equivalent to the existence of a hereditary \(t\)-structure on the triangulated category such that its heart coincides with its full additive subcategory. In Section 3, it is shown that if the triangulated category is hereditary and algebraical, i.e. triangle equivalent to the stable category of a Frobenius category, then it is triangle equivalent to a bounded derived category of a hereditary abelian category. Any hereditary triangulated category is algebraical [\textit{A. Hubery}, ``Characterising the bounded derived category of an hereditary abelian category'', Preprint, \url{arXiv:1612.06674}] Next section is devoted to the study of paths in triangulated categories. The existence of paths for block triangulated categories is shown. In the last section, hereditary block triangulated categories are characterized as those having two indecomposable objects \(X,Y\) with no path from \(Y\) to \(X\), equivalently having an indecomposable object \(X\) with no path from \(X[1]\) to \(X\). Applications to piecewise hereditary algebras are given.
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