Sections of box tensor product of sheaves on the diagonal Let $A,B$ be two (say coherent) sheaves on some scheme/variety $X$ and consider their product $A\boxtimes B$ on the diagonal $X\times X$. Is it always

$$H^0(A\boxtimes B)\simeq H^0(A)\otimes H^0(B) $$

(I guess this is a general fact but I am really interested in the case $A$,$B$ locally free and $X$ smooth)
If $X = \text{Spec}(R)$ then $X \times X = \text{Spec}(R \otimes R)$, $A,B$ are just $R$-modules and $A \boxtimes B $ is just $A \otimes_R B$. If $X$ is not affine, there is an affine cover of $X$ and one can apply the previous argument again.
If $X = \text{Spec}(R)$ then $X \times X = \text{Spec}(R \otimes R)$, $A,B$ are just $R$-modules and $A \boxtimes B $ is just $A \otimes_R B$. If $X$ is not affine, there is an affine cover of $X$ and one can apply the previous argument again.