Exercise 3, § 3

(a) Let $\mathbf{A} = \bigoplus_{n \geq 0} \mathbf{A}_n$ be a graded algebra with positive degrees over a field $k$; suppose that $\mathbf{A}_0 = k$ and that $\mathbf{A}$ is a Krull domain. Show that for $\mathbf{A}$ to be factorial, it is necessary and sufficient that every graded prime ideal $p$ of $\mathbf{A}$ of height 1 be of the form $p = \mathbf{A}a$, where $a$ is a homogeneous element (use Exercise 16 of § 1). Show that every homogeneous element $\neq 0$ of $\mathbf{A}$ is a product of homogeneous extremal elements.

(b) Let $A$ be a graded $k$-algebra satisfying the conditions of (a). Let $k'$ be an extension of $k$ and suppose that $A \otimes_k k'$ is a factorial domain; show that $A$ is factorial. (If a graded ideal $a$ of $A$ is such that $a \otimes_k k'$ is principal in $A \otimes_k k'$, show that $a$ is principal; then use (a).)