Bounded Complexes of Objects of Finite Flat Dimensions

Let $(\mathcal{R},\otimes)$ be a symmetric monoidal closed Grothendieck category which has enough flat objects. It is shown that a given object ${\mathcal{G}}$ in $\mathcal{R}$ has finite flat dimension if and only if it is quasi-isomorphic to a bounded complex of objects of finite flat dimension. In the case in which $\mathcal{R}$ has enough projective objects, we prove that finite flat dimension in $\mathcal{R}$ implies finite projective dimension if and only if any object in $\mathcal{R}$ that is quasi-isomorphic to a bounded complex of objects of finite flat dimension has finite projective dimension. This leads to a generalization of [4, Proposition 2.3] and [15, Theorem]. Moreover, we present a wide class of $n$-perfect rings.