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Let [Formula: see text] be an artin algebra, [Formula: see text] be a [Formula: see text]-Gorenstein [Formula: see text]-module and [Formula: see text], then [Formula: see text] is a [Formula: see text]-[Formula: see text]-bimodule. We use the restricted flat dimension of [Formula: see text] and the finitistic [Formula: see text]-dimension of [Formula: see text] to characterize the finitistic dimension of [Formula: see text], and obtain the following main result: if [Formula: see text] is [Formula: see text]-finite with [Formula: see text], then we have: (1) If [Formula: see text] or [Formula: see text], then [Formula: see text] (2) If [Formula: see text], then [Formula: see text]

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Let [Formula: see text] be an artin algebra and [Formula: see text] be an idempotent ideal of [Formula: see text]. Then there is a recollement of [Formula: see text] relative to [Formula: see text] and [Formula: see text]. In this paper, we study some properties of Lat-syzygy-finite classes, special Lat-syzygy-finite classes and Lat-Igusa–Todorov classes. Then, we compare the algebras [Formula: see text], [Formula: see text] and [Formula: see text] with respect to Lat-syzygy-finite properties, special Lat-syzygy-finite properties and Lat-Igusa–Todorov properties under suitable conditions.

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