Can anybody explain why the vector bundle corresponding to a locally free sheaf F is global spec of sym of the dual of F and not just F? How does a section get identified with a polynomial in the dual?
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$\begingroup$ See en.wikipedia.org/wiki/Symmetric_algebra $\endgroup$ – Kevin H. Lin Nov 17 '10 at 20:32

10$\begingroup$ I suggest you work it out when the base scheme is Spec of a field. $\endgroup$ – Laurent MoretBailly Nov 17 '10 at 20:32

11$\begingroup$ Linear functions on V are elements of V^*, so polynomial functions on V are elements of Sym(V^*). $\endgroup$ – Dustin Clausen Nov 17 '10 at 21:21

$\begingroup$ Functoriality . $\endgroup$ – Martin Brandenburg Nov 17 '10 at 23:08

$\begingroup$ Or maybe when you say "corresponding to" you're thinking of a different correspondence from the rest of us. $\endgroup$ – Tom Goodwillie Nov 18 '10 at 11:38
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Let $L$ be a locally free ${\cal O}_X$module of finite rank. Define $V=Spec(Sym(L^\vee))$. Then $$Mor_X(X, V)={\cal O}_XAlg(Sym(L^\vee), {\cal O}_X)=Hom(L^\vee, {\cal O}_X)=L(X).$$ The universal mapping property of the (global) Spec is in EGA II.1.