TITLE: Cohomological decomposition of tensor sheaves?
QUESTION [6 upvotes]: My question is similar to this, but not identical.  I believe the following to be true, but I'd like a reference.
Given (quasicoherent?) sheaves of $\mathcal O_X$ modules $E$ and $F$ on a projective variety $X$, 
$$
H^n(X, E\otimes F) \cong \bigoplus_{p+q=n} H^p(X, E) \otimes H^q(X, F).
$$

Is this true, and what is a good reference or counterexample? If true, is quasicoherent necessary?

REPLY [3 votes]: It is false.  Take $E = \mathcal O_{\mathbb P^n}(2)$ and $F= \mathcal O_{\mathbb P^n}(-2)$.  Then,
$$
H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}) = 
H^0(\mathbb P^n, E\otimes F) \ne 
H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(2)) \otimes
H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(-2)).
$$