Separable extensions preserve finiteness of global dimension, Gorensteinness and regularity in compactly generated triangulated categories while relating their singularity categories up to retracts.
Amiot , Cluster categories for algebras of global dimension 2 and quivers with potential, Ann.\ Inst.\ Fourier (Grenoble) 59 (2009), no.\ 6, 2525–2590
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Finiteness of global dimension relative to compact silting objects is intrinsic to the triangulated category and independent of the silting object chosen.
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Homological Aspects of Separable Extensions of Triangulated Categories
Separable extensions preserve finiteness of global dimension, Gorensteinness and regularity in compactly generated triangulated categories while relating their singularity categories up to retracts.
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Global dimension of dg algebras via compact silting objects
Finiteness of global dimension relative to compact silting objects is intrinsic to the triangulated category and independent of the silting object chosen.