Bounded complexes of objects of finite flat dimensions

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematics and Computer Science, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

10.48308/cgasa.2024.184656.1499

Abstract

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.

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References

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Categories and General Algebraic Structures with Applications

Articles in Press, Accepted Manuscript
Available Online from 25 February 2025

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