Depth and Dimension of the Fibre of a Local Ring Homomorphism

An important information of a homomorphism of local rings (R,\mathfrak{m}) \to (S,\mathfrak{n}) is its fibre S/\mathfrak{n}S. For example it relates the dimensions and depth of R and S. 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].