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Fulton's intersection theory book at Chapter 4 makes the following claim: If $\mathcal{E}$ is a locally free sheaf (on $X$), and $E:=Spec(Sym(\mathcal{E}))$ a total space of some cone/bundle on $X$, then the sheaf of sections of $E$ is $\mathcal{E}^{\vee}$. However, I have trouble understanding why the claim is true, as Fulton does not make any remarks about it. Thinking this locally does not seem to help either, so there must be some global way to think about this. Can you help me on this?

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    $\begingroup$ isn't this the universal property of the symmetric algebra? CAlg_A(Sym_A M, A) = Hom_A(M,A) = M^\vee $\endgroup$
    – pro
    Commented Dec 26, 2015 at 20:52
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    $\begingroup$ Oh now I am embarassed. I never thought of these as $A$-morphisms. Now everything is crystal clear. Thanks. $\endgroup$
    – chhan92
    Commented Dec 26, 2015 at 21:05
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    $\begingroup$ great! (15char) $\endgroup$
    – pro
    Commented Dec 26, 2015 at 21:19

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