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arXiv · 2306.09140

Little finitistic dimensions and generalized derived categories

Xiaolei Zhang·Tiwei Zhao·Dingguo Wang
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Abstract

In this paper, we introduced a generalization of the derived category, which is called the $n$-derived category and denoted by $\D_{n}(R)$, of a given ring $R$ for each $n\in\mathbb{N}\cup\{\infty\}$. The $n$-derived category of a ring is proved to be very closely connected with its left little finitistic dimension. We also introduce and investigate the notions of $n$-exact sequences, $n$-projective (resp., $n$-injective) modules and $n$-exact complexes. In particular, we characterize the left little finitistic dimensions in terms of all above notions. Finally, we build a connection of the classical derived categories and $n$-derived categories.

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Xiaolei Zhang, Tiwei Zhao, Dingguo Wang. 2023-06-15. Little finitistic dimensions and generalized derived categories. https://arxiv.org/abs/2306.09140

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