Let $X$ be an affine variety defined by $xy=z^2$ in $\mathbb{A}^3$ and $D,E$ be Weil divisors of $X$ defined by $x,z=0$ and $y,z=0$ respectively.
I want to prove that the sheaf $\mathcal{O}_X(D)$ is not locally free of rank 1.
Let $U=X-E$. If $\mathcal{O}_X(D)$ is locally free of rank 1, then there is a generator $h$ of $\Gamma(U, \mathcal{O}_X(D))$. Since $\frac{1}{x}, \frac{1}{z} \in \Gamma(U, \mathcal{O}_X(D))$, I tried to show that $h$ cannot generate both $\frac{1}{x}, \frac{1}{z}$ but I've got stuck here. Can anyone help me how to proceed?
Consider $D = \{x = z = 0\}$. It is not hard to find a locally free resolution $$ \mathcal{O}_X^{\oplus 2} \stackrel{\left(\begin{smallmatrix}z & -y\\-x & z\end{smallmatrix}\right)}\to \mathcal{O}_X^{\oplus 2} \stackrel{(x,z)}\to \mathcal{O}_X(-D) \to 0. $$ Tensoring it by $\mathcal{O}_P$, where $P$ is the vertex of $X$, one obtains $$ \mathcal{O}_P^{\oplus 2} \stackrel{\left(\begin{smallmatrix}0 & 0\\0 & 0\end{smallmatrix}\right)}\to \mathcal{O}_P^{\oplus 2} \to \mathcal{O}_X(-D) \otimes \mathcal{O}_P \to 0, $$ which implies that $$ \mathcal{O}_X(-D) \otimes \mathcal{O}_P \cong \mathcal{O}_P^{\oplus 2}, $$ hence $\mathcal{O}_X(-D)$ cannot be locally free of rank 1. On the other hand, $2D$ is a Cartier divisor, so if $\mathcal{O}_X(D)$ would be locally free, the same would be true for $\mathcal{O}_X(-D)$.