Cohomological decomposition of tensor sheaves?

Question

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?

Answer 1

No, it is false, consider e.g. $E=F=\mathcal{O}_X$ on a curve $X$ of genus one.

(Here is something that is true though: let $p$ and $q$ be the two projection maps $X \times X \to X$. Then $H^\bullet(X\times X,p^\ast E \otimes q^\ast F) \cong H^\bullet(X,E) \otimes H^\bullet(X,F)$. There are lots of basic variants of this theorem and all of them are called the Kunneth formula.)

Answer 2

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