Exercise 3, § 3

(a) Let $A = \bigoplus_{n \geq 0} A_n$ be a graded algebra with positive degrees over a field $k$; suppose that $A_0 = k$, and that $A$ is a Krull ring. Prove that, in order that $A$ be factorial, it is necessary and sufficient that every graded prime ideal $p$ of height 1 in $A$ be of the form $p = Aa$, where $a$ is a homogeneous element (use exerc. 16 of § 1). Prove that every homogeneous element $\neq 0$ of $A$ is a product of homogeneous extreme elements.

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