Let $f:X\to Y$ be a birational morphism between two integral and normal projective schemes over a field, such that $f_*\mathcal{O}_X=\mathcal{O}_Y$ and $R^if_*\mathcal{O}_X=0$ for $i>0$. Now let $A$ be a locally free sheaf on $X$.
My general question is whether there is a relation between $f_*(A)\otimes f_*(A)$ and $f_*(A\otimes A)$.
I know there is a map $$ f_*(A) \otimes f_*(A) \to f_*(A\otimes A) $$ but can one say more? For example, if $A=f^*B$ for a locally free sheaf $B$ on $X$, then the projection formula and the hypotheses on $f$ give that the above map is an isomorphism, but what happens when $A$ is not a pullback?
I am especially interested in computing the derived functors $R^if_*(A\otimes A)$: is there maybe a spectral sequence involving the terms $R^pf_*(A)$ which abuts to $R^if_*(A\otimes A)$?
It is easy to give an example when $Rf_*(A) = 0$, but $Rf_*(A \otimes A) \ne 0$ (for instance, let $A = O_E(-1)$, where $E$ is the exceptional line on a blowup of a plane in a point). So, you cannot hope to have a description of $Rf_*(A \otimes A)$ in terms of $Rf_*(A)$.
On the other hand, a reasonable relation can be obtained as follows. Consider the canonical map $Lf^*(Rf_*(A)) \to A$ and let $C$ be its cone: $$ Lf^*(Rf_*(A)) \to A \to C. $$ Tensoring (all tensor products are derived) this triangle by $A$ we get $$ A \otimes Lf^*(Rf_*(A)) \to A \otimes A \to A \otimes C. $$ Pushing forward and using the projection formula, we get $$ Rf_*(A) \otimes Rf_*(A) \to Rf_*(A \otimes A) \to Rf_*(A \otimes C). $$ This shows what controls the difference.