Let $S$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $S$. We give several bounds of $\mathfrak{a}$-depth of $S$-complexes and $S$-modules, investigate the behavior of $\mathfrak{a}$-depth and $\mathfrak{a}$-Cohen-Macaulay modules under tensor product with a faithfully flat $S$-module. Furthermore, we establish the Foxby equivalence of $\mathfrak{a}$-Cohen-Macaulay $S$-modules.