Assume we have two "nice" schemes $X$ and $Y$ over $k=\mathbb{C}$, a finite flat map $f:X\rightarrow Y$ and a k-algbera $A$. Then we get an induced finite flat map $f\_A:X\times\_k A \rightarrow Y\times\_k A$.


Now given an $O\_{X\times\_k A}$-module $M$, flat over $A$ and an $A$-module $N$.


Is $(f\_A)\_{\*}(M\otimes\_A N)=((f\_A)\_{\*}M)\otimes\_A N$?


My idea was to use the projection formula: let $\pi:Y\times\_k A \rightarrow A$ and $\phi:X\times\_k A \rightarrow A$ be the structure morphisms, i.e. $\phi=\pi\circ f\_A$. Then: $M\otimes\_A N=M\otimes\_{O\_{X\times\_k A}} \phi^{\*} \tilde{N}$, where $\tilde{N}$ is the sheaf associated to $N$ on Spec(A). But $\phi^{\\*}=(f\_A)^{\\*}\circ\pi^{\\*}$, so we get $(f\_A)\_{\*}(M\otimes\_A N)=(f\_A)\_{\*}(M\otimes\_{O\_{X\times\_k A}} (f\_A)^{\\*}(\pi^{\\*}\tilde{N}))$. 


Now if the projection formula was an isomorphism in this case the last module would be $(f\_A)\_{\*}(M)\otimes\_{O\_{Y\times\_k A}} \pi^{\\*}\tilde{N}$, which is nothing else but $((f\_A)\_{\*}M)\otimes\_{A} N$.


So is the projection formula an isomorphism in this case?



Direct image sheaf and tensor product (is the projection formula an isomorphism?)
Most of this becomes obvious when translated using affine charts. For the first question, use a presentation of $N$ and the fact that $(f\_A)\_\*$ is exact to reduce to the case $N=A$. The projection formula is stated in EGA I (new edition), Corollaire 9.3.9.


