Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=\mathcal{I}_1\cdots\mathcal{I}_k$ the product. My question is whether $\mathcal{I}$ is the ideal sheaf of the union $Z_1\cup\ldots\cup Z_k$. In other words, I need to know if $\mathcal{I}$ is locally principal. You may assume that $Z_1\cap\ldots\cap Z_k=\emptyset$ or, equivalently, $\mathcal{I}_1+\ldots+\mathcal{I}_k=\mathcal{O}_X$. You may also assume that the $Z_i$ intersect transversally, if that helps.

Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=\mathcal{I}_1\cdots\mathcal{I}_k$ the product. My question is whether $\mathcal{I}$ is the ideal sheaf of the union $Z_1\cup\ldots\cup Z_k$. You may assume that $Z_1\cap\ldots\cap Z_k=\emptyset$ or, equivalently, $\mathcal{I}_1+\ldots+\mathcal{I}_k=\mathcal{O}_X$. You may also assume that the $Z_i$ intersect transversally, if that helps.