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arXiv · 2304.00527

G-dimensions for DG-modules over commutative DG-rings

Jiangsheng Hu·Xiaoyan Yang·Rongmin Zhu
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Abstract

We define and study a notion of G-dimension for DG-modules over a non-positively graded commutative noetherian DG-ring $A$. Some criteria for the finiteness of the G-dimension of a DG-module are given by applying a DG-version of projective resolution introduced by Minamoto [Israel J. Math. 245 (2021) 409-454]. Moreover, it is proved that the finiteness of G-dimension characterizes the local Gorenstein property of $A$. Applications go in three directions. The first is to establish the connection between G-dimensions and the little finitistic dimensions of &\mathcal{A}&. The second is to characterize Cohen-Macaulay and Gorenstein DG-rings by the relations between the class of maximal local-Cohen-Macaulay DG-modules and a special G-class of DG-modules. The third is to extend the classical Buchwtweiz-Happel Theorem and its inverse from commutative noetherian local rings to the setting of commutative noetherian local DG-rings.Our method is somewhat different from classical commutative ring.

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Jiangsheng Hu, Xiaoyan Yang, Rongmin Zhu. 2023-04-02. G-dimensions for DG-modules over commutative DG-rings. https://doi.org/10.1017/s0013091525100886

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