Let $X, Y$ be smooth manifolds. The, for $p : X \times Y \to X$ the projection, is $p^*$ on complexes of sheaves (of abelian groups) exact?
In algebraic geometry, I know this is true for coherent sheaves (as long as the product is over a field), because then these projections are flat.
For context - I am trying to demonstrate that there is a soft resolution of the constant sheaf on $Z = X \times Y$ by the box product of the pullbacks of the De Rham complexes, in order to prove the Kunneth formula.
In my context I am using the inverse image functor for sheaves vector spaces on a topological space, not the pullback of sheaves of modules of ringed spaces. Hence it is always exact, as can easily seen by checking stalks.