Journal of the
Korean Mathematical Society JKMS

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$. We give some descriptions of the $\mathfrak{a}$-depth of $\mathfrak{a}$-relative Cohen-Macaulay modules by cohomological dimensions, and study how relative Cohen-Macaul-\\ayness behaves under flat extensions. As applications, the perseverance of relative Cohen-Macaulayness in a polynomial ring, formal power series ring and completion are given.

Keywords: Relative Cohen-Macaulay module, $\mathfrak{a}$-depth, cohomological dimension

MSC numbers: Primary 13C15, 13H10

Supported by: The authors thank the referee for important comments and suggestions on improving this paper. This research was partially supported by National Natural Science Foundation of China (11901463), graduate Research Fund project of Northwest Normal University (2021KYZZ01031) and Gansu Province outstanding graduate student ``Innovation Star'' project (2022CXZX-238).