I currently am working on trying to compute some Hochschild cohomology of some scheme. However, I should be able to do it as soon as I know/have a reference for the following natural statement:
Let $X$ be a scheme and $F,G\in \mathcal{D}^b\left(X\right)$ then $$\mathrm{supp}\left(F\otimes G\right)\subset \mathrm{supp} \left(F\right)\cap \mathrm{supp}\left(G\right)$$
where the tensor product is the left derived tensor product.
Tensor products (derived) commute with pullbacks (also derived), i.e., $$ \phi^*(F \otimes G) \cong \phi^*F \otimes \phi^*G. $$ Now, if $\phi$ is the embedding of the complement of the support of $F$, the RHS is zero, hence the LHS is zero. The same is true for the complement of the support of $G$, hence $$ \mathrm{supp}(F \otimes G) \subset \mathrm{supp}(F) \cap \mathrm{supp}(G). $$ On the other hand, if $\phi$ is the embedding of a point in $\mathrm{supp}(F) \cap \mathrm{supp}(G)$, the RHS is nonzero, hence the LHS is nonzero, hence $$ \mathrm{supp}(F \otimes G) \supset \mathrm{supp}(F) \cap \mathrm{supp}(G). $$