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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.

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

1 Answer 1

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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 $F\otimes_A B:=F\otimes_A A[x_1,..,x_n]$.

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