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Do equivalences induced by finite flat pushforward preserve flatness Direct image sheaf and commute with ideal multiplcationtensor product (is the projection formula an isomorphism?)

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 $f_A:X\times_k A \rightarrow Y\times_k A$.

Since $f_A$ is finite flat, it inducesNow given an equivalence of categories between quasi-coherent $O_X\otimes_k A$-modules and quasi-coherent $(f_A)_{*}(O_X\otimes_k A)$-modules. Let us call the inverse equivalence $i$.

Let $E$ be a $(f_{*}O_X)\otimes_k A$$O_{X\times_k A}$-module $M$, flat over $A$. By flat base change we get $(f_{*}O_X)\otimes_k A\cong (f_A)_{*}(O_X\otimes_k A)$. So $E$ corresponds to and an $O_X\otimes_k A$$A$-module $i(E)$ via $i$$N$.

Is $i(E)$ also flat over $A$$(f_A)_{*}(M\otimes_A N)=((f_A)_{*}M)\otimes_A N$?

IsMy idea was to use the projection formula: let $i(I\otimes_A E)=I\otimes_A i(E)$$\pi:Y\times_k A \rightarrow A$ and $\phi:X\times_k A \rightarrow A$ be the structure morphisms, i.e. $\phi=\pi\circ f_A$. Then: $M\otimes_A N=M\otimes_{O_{X\times_k A}} \phi^{*} \tilde{N}$, where $I$$\tilde{N}$ is an ideal in $A$?

Now given anthe sheaf associated to $O_X\otimes_k A$-module$N$ on Spec(A). But $M$$\phi^{\*}=(f_A)^{\*}\circ\pi^{\*}$, flat over $A$ and an ideal $I$ inso we get $A$$(f_A)_{*}(M\otimes_A N)=(f_A)_{*}(M\otimes_{O_{X\times_k A}} (f_A)^{\*}(\pi^{\*}\tilde{N}))$.

Is $(f_A)_{*}M$ aNow if the projection formula was an isomorphism in this case the last module would be $(f_A)_{*}(O_X\otimes_k A)=(f_{*}O_X)\otimes_k A$-module$(f_A)_{*}(M)\otimes_{O_{Y\times_k A}} \pi^{\*}\tilde{N}$, which is flat over $A$?

Isnothing else but $(f_A)_{*}(I\otimes_A M)=I\otimes_A (f_A)_{*}M$?$((f_A)_{*}M)\otimes_{A} N$.

Or do we need extra assumptions on $E$ and $M$ forSo is the projection formula an isomorphism in this to be truecase?

Do equivalences induced by finite flat pushforward preserve flatness and commute with ideal multiplcation?

Since $f_A$ is finite flat, it induces an equivalence of categories between quasi-coherent $O_X\otimes_k A$-modules and quasi-coherent $(f_A)_{*}(O_X\otimes_k A)$-modules. Let us call the inverse equivalence $i$.

Let $E$ be a $(f_{*}O_X)\otimes_k A$-module, flat over $A$. By flat base change we get $(f_{*}O_X)\otimes_k A\cong (f_A)_{*}(O_X\otimes_k A)$. So $E$ corresponds to an $O_X\otimes_k A$-module $i(E)$ via $i$.

Is $i(E)$ also flat over $A$?

Is $i(I\otimes_A E)=I\otimes_A i(E)$, where $I$ is an ideal in $A$?

Now given an $O_X\otimes_k A$-module $M$, flat over $A$ and an ideal $I$ in $A$.

Is $(f_A)_{*}M$ a $(f_A)_{*}(O_X\otimes_k A)=(f_{*}O_X)\otimes_k A$-module which is flat over $A$?

Is $(f_A)_{*}(I\otimes_A M)=I\otimes_A (f_A)_{*}M$?

Or do we need extra assumptions on $E$ and $M$ for this to be true?

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

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?