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. 



Products of Ideal Sheaves and Union of irreducible Subvarieties
For each $i$ we have a short exact sequence 
$$
0 \to I\_i \to O\_X \to O\_{Z\_i} \to 0.
$$
Let us think about it as about a resolution of $I\_i$. Then it follows that
$$
I\_1 \otimes^L I\_2\otimes^L \dots \otimes^L I\_k = 
(O\_X \to O\_{Z\_1}) \otimes^L (O\_X \to O\_{Z\_2}) \otimes^L \dots \otimes^L (O\_X \to O\_{Z\_k}).
$$
The transversality condition ensures that
$$
O\_{Z\_{i\_1}} \otimes^L O\_{Z\_{i\_2}}\otimes^L \dots \otimes^L O\_{Z\_{i\_p}} \cong 
O\_{Z\_{i\_1} \cap Z\_{i\_2} \cap \dots \cap Z\_{i\_p}}.
$$
Therefore we conclude that $I\_1 \otimes^L I\_2\otimes^L \dots \otimes^L I\_k$ is quasiisomorphic to the complex
$$
O\_X \to \oplus O\_{Z\_i} \to \oplus O\_{Z\_i \cap Z\_j} \to \dots \to O\_{Z\_1 \cap Z\_2 \cap \dots Z\_k}.
$$
It follows that 
$$
I\_1\cdot I\_2\cdot \dots \cdot I\_k = Im(I\_1 \otimes^L I\_2\otimes^L \dots \otimes^L I\_k \to O\_X)
$$
is equal to the kernel of the map $O\_X \to \oplus O\_{Z\_i}$ that is to $I\_{Z\_1 \cup Z\_2 \cup \dots \cup Z\_k}$.


EDIT. The importance of the transversality condition can be seen at the following simple example. Let $X = A^3$ and $Z\_1,Z\_2$ be two lines intersecting at a point $P$ (the simplest nontransversal intersection). Then one has $O\_{Z\_1} \otimes O\_{Z\_2} = O\_P$, $Tor\_1(O\_{Z\_1},O\_{Z\_2}) = O\_P$. Therefore one has a spectral sequence with the first term being
$$
\begin{array}{ccccc}
O\_X & \to & O\_{Z\_1} \oplus O\_{Z\_2} & \to & O\_P \cr
&&&& O\_P
\end{array}
$$
which converges to $I\_{Z\_1} \otimes^L I\_{Z\_2}$. The second term then gives exact sequence $$
0 \to I\_{Z\_1}\otimes I\_{Z\_2} \to I\_{Z\_1 \cup Z\_2} \to O\_P \to 0,
$$ 
which shows that $I\_{Z\_1}\cdot I\_{Z\_2}$ is the ideal of the union of lines with an additional nilpotent at the point of intersection.


