Pullback of locally free sheaf over the associated vector bundle

Let $$X$$ be a scheme over $$k$$, $$\mathcal{F}$$ be a locally free sheaf on $$X$$ of rank $$r$$, $$F$$ be the vector bundle associated to $$\mathcal{F}$$ i.e. $$F=\operatorname{Spec}\operatorname{Sym}(\mathcal{F}^{*})$$, and $$\pi\colon F\rightarrow X$$ be a natural projection.
There is a canonical locally free sheaf $$\pi^{*}\mathcal{F}$$ on F, and I want an explicit description of $$\pi^{*}\mathcal{F}$$.

Take affine open set $$U=\operatorname{Spec}A$$ of $$X$$, and assume $$V=F\mid_{U}=U\otimes k[t_{1},\dots, t_{r}]$$.
Then $$\pi^{*}\mathcal{F}(V)=\pi^{-1}\mathcal{F}(V)\otimes_{\pi^{-1}\mathcal{O}_{X}(V)} \mathcal{O}_{F}(V)=\mathcal{F}(U)\otimes_{\mathcal{O}_{X}(U)} \operatorname{Sym}(\mathcal{F}^{*})(U)$$.

But what is $$\operatorname{Sym}(\mathcal{F}^{*})(U)$$ ? I'm stuck here. Thanks in advance.

• Over an affine, the $\mathrm{Sym}$ construction is just the usual module-theoretic one, right? So this is just $\mathrm{Sym}(V^*)$. Jan 7, 2021 at 22:41

Let $$U:=Spec(A)$$ and $$F:=A\{e_1,..,e_1\}$$ the free $$A$$-module on $$e_i$$ with $$F^*:=A\{x_1,..,x_n\}$$. Let $$\mathcal{F}$$ be the sheafification of $$F$$.

Question: "But what is $$Sym(F∗)(U)$$?"

Answer: $$Sym_A^*(F^*)(U)\cong A[x_1,..,x_n]$$ is the polynomial ring on $$x_i$$ over $$A$$.

There is a canonical map

V2. $$\pi: \mathbb{V}(F^*):=Spec(Sym_A^*(F^*)):=\mathbb{A}^n_X \rightarrow X$$

defined by the canonical map

V3. $$\phi: A \rightarrow A[x_1,..,x_n]:=B$$

and by definition $$\pi^*\mathcal{F}$$ is the sheafification of the $$B$$-module $$F\otimes_A B:=F\otimes_A A[x_1,..,x_n]$$.