Exercise 1, § 2

Let $A$ be a local noetherian ring, and let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of Macaulay A-modules. Now prove that one has $\dim(M') = \dim(M)$ and that $\dim(M'')$ is equal to $\dim(M)$ or to $\dim(M) - 1$.