Relative homological dimensions in recollements of triangulated categories

Abstract

Let \mathcal{E} be a proper class of triangles in a triangulated category \mathcal{C}, and let (\mathcal{A},\mathcal{B},\mathcal{C}) be a recollement of triangulated categories. Based on Beligiannis's work, we prove that \mathcal{A} and \mathcal{C} have enough \mathcal{E} -projective objects whenever \mathcal{B} does. Moreover, in this paper, we give the bounds for the \mathcal{E} -global dimension of \mathcal{B} in a recollement (\mathcal{A},\mathcal{B},\mathcal{C}) by controlling the behavior of the \mathcal{E} -global dimensions of the triangulated categories A and \mathcal{C} : In particular, we show that the niteness of the \mathcal{E} -global dimensions of triangulated categories is invariant with respect to the recollements of triangulated categories.

Keywords

Triangulated category / proper class of triangles / recollement / \mathcal{E} - global dimension / derived category

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