An important information of a homomorphism of local rings is its fibre . For example it relates the dimensions and depth of and . We introduce two main theorems that relate the depth and dimensions and some corollaries. Finally we apply them to show that the polynomial ring or formal power series ring over a Cohen-Macaulay ring is Cohen-Macaulay. See also Theorem A.11 in [Bruns, Herzog] and Theorem 23.2 in [Matsumura].