Suppose that $A$ is an affine algebraic variety and $P$, and $Q$ are subvarieties.  It is easy
to see that the coordinate rings $C[P]$ and $C[Q]$ of $P$ and $Q$ are modules over $C[A]$ and
$C[P]\otimes_{C[A]} C[Q]$ is frequently the coordinate ring of $P\cap Q$.  For instance,
if the sum of the ideals of $P$ and $Q$ is its own radical.



If $X$ is a topological space, and $P$ and $Q$ are subspaces then their singular cohomology
groups  $ H^{\*}(P) $ and $ H^{\*}(Q) $ are modules over $H^{|*}(X)$.  

To make it simple, assume that
$X$ is a smooth manifold and $P$ and $Q$ are smooth submanifolds.  Are there simple conditions
that imply that 
$H^{\*}(P\cap Q)=H^{\*}(P)\otimes_{H^{\*}(X)} H^{\*}(Q)?$