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. LinCommented Nov 17, 2010 at 20:32
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10$\begingroup$ I suggest you work it out when the base scheme is Spec of a field. $\endgroup$– Laurent Moret-BaillyCommented Nov 17, 2010 at 20:32
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12$\begingroup$ Linear functions on V are elements of V^*, so polynomial functions on V are elements of Sym(V^*). $\endgroup$– Dustin ClausenCommented Nov 17, 2010 at 21:21
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$\begingroup$ Functoriality . $\endgroup$– Martin BrandenburgCommented Nov 17, 2010 at 23:08
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$\begingroup$ Or maybe when you say "corresponding to" you're thinking of a different correspondence from the rest of us. $\endgroup$– Tom GoodwillieCommented Nov 18, 2010 at 11:38
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1 Answer
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Let $L$ be a locally free ${\cal O}_X$-module of finite rank. Define $V=Spec(Sym(L^\vee))$. Then $$Mor_X(X, V)={\cal O}_X-Alg(Sym(L^\vee), {\cal O}_X)=Hom(L^\vee, {\cal O}_X)=L(X).$$ The universal mapping property of the (global) Spec is in EGA II.1.
answered Nov 18, 2010 at 9:51