# Push-forward of flat module under a finite, flat morphism

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?

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.