Let $S$ be any quasi-compact scheme and $\mathcal{E}$ be a locally free sheaf of rank $r$ on $S$ and $X = \mathbb{P}(\mathcal{E})$ be its associated projective bundle with structure map $f : X \rightarrow S$. Let $\mathcal{F}$ be a quasi coherent sheaf on $X$. I am trying to show that $ f_{\ast} \mathcal{F} \otimes_{\mathcal{O}_{S}} \mathcal{S}^{n}(\mathcal{E}) \cong f_{\ast} \mathcal{F}(n-1) \otimes_{\mathcal{O}_{S}} \mathcal{E}$ for $n \geq 1$, where $\mathcal{S}^{n}(\mathcal{E})$ denote the $n$-th symmetric module. Firstly, I am expecting it to be true. It is clear for $n=1$. But I don't know for higher $n$.
My try:
$f_{\ast} \mathcal{F}(n-1) \otimes_{\mathcal{O}_{S}} \mathcal{E} \cong f_{\ast} \mathcal{F}(n-1) \otimes_{\mathcal{O}_{S}} f_{\ast}\mathcal{O}_{X}(1) \cong f_{\ast}( \mathcal{F}(n-1) \otimes_{\mathcal{O}_{S}} \mathcal{O}_{X}(1) \cong f_{\ast}( \mathcal{F} \otimes_{\mathcal{O}_{S}} \mathcal{O}_{X}(n)) \cong f_{\ast}( \mathcal{F}) \otimes_{\mathcal{O}_{S}} f_{\ast}(\mathcal{O}_{X}(n)) \cong f_{\ast} \mathcal{F} \otimes_{\mathcal{O}_{S}} \mathcal{S}^{n}(\mathcal{E}) $. The only problem I see in my proof is that pushforward functor need preserve tensor product of two sheaves. Can some one tell whether it is correct or not?
Any help would be great.