Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f\_A:X\_A \to Y\_A$ be the trivial deformation of $f$, where $X\_A:=X \times \mbox{Spec}(A)$, $Y\_A:=Y \times \mbox{Spec}(A)$ and $f\_A$ is simply $f \times \mbox{id}$. Let $\mathcal{F}\_A$ be a coherent sheaf on $X\_A$, which is $A$-flat. Is $(f\_A)\_\*\mathcal{F}\_A$ going to be $A$-flat?



Push-forward of flat module under a finite, flat morphism
Yes, it follows from the projection formula <https://stacks.math.columbia.edu/tag/08EU> For any $A$-module $M$ we have $$Rf\_{A\*}\mathcal{F}\_A\otimes^{L}\_{\mathcal{O}\_Y}p^\*M\simeq Rf\_{A\*}(\mathcal{F}\_A\otimes^L\_{\mathcal{O}\_X}Lf\_A^\*(p^\*M))$$ where $p:Y\_A\to Spec\, A$ is the structure morphism. Since $f\_A$ is flat, $Lf\_A^\*(p^\*M)=q^\*M$ where $q:X\_A\to Spec\, A$ is the structure morphism of $X\_A$ and the RHS of the projection formula is the derived pushforward of a quasi-coherent sheaf $\mathcal{F}\_A\otimes\_{\mathcal{O}\_X}f\_A^\*p^\*M$(we can replace the derived tensor product by the usual tensor product since $\mathcal{F}\_A$ is $A$-flat) along the affine morphism $f\_A$, so it is concentrated in degree 0. 


In other words, this shows that the derived tensor product of $f\_{A\*}\mathcal{F}\_A=Rf\_{A\*}\mathcal{F}\_A$ with any $A$-module is concentrated in degree $0$. It means that $f\_{A\*}\mathcal{F}\_A$ is $A$-flat.


