Resolving resolution dimension of recollements of abelian categories

Abstract

In this paper, we investigate the resolving resolution dimension with respect to the recollements of abelian categories. Let [Formula: see text] be a recollement of abelian categories such that [Formula: see text] and [Formula: see text] have enough projective objects and let [Formula: see text], [Formula: see text], [Formula: see text] be resolving subcategories of [Formula: see text], [Formula: see text] and [Formula: see text], respectively. We establish some upper and lower bounds of [Formula: see text]-resolution dimension of [Formula: see text] in terms of the [Formula: see text]-resolution dimension of [Formula: see text] and [Formula: see text]-resolution dimension of [Formula: see text]. Based on these upper and lower bounds, we study the Gorensteinness of abelian categories involved in [Formula: see text]. Under some suitable assumptions, we show that if [Formula: see text] and [Formula: see text] are Gorenstein, then [Formula: see text] is Gorenstein. As applications, we apply our results to ring theory and the triangular matrix artin algebras, we study the quasi-Frobenius and Gorenstein hereditary properties of the ring [Formula: see text] and [Formula: see text], where [Formula: see text] is an idempotent element of [Formula: see text]. We also investigate Gorensteinness of the triangular matrix artin algebras, some known results are obtained as corollaries. At the end of this paper, we give two examples to illustrate our results.