Direct image sheaf and tensor product (is the projection formula an isomorphism?)

Question

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?

Answer 1

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.