depth and dimension – Thinking Bear

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].