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ag.algebraic geometry – Functorial raise of inescapable vector bundles to the ambient projective area

Given an very copious line bundle $L$ on a round we embed it right into a projective area such that pullback of $mathcal{O}(1)$ is $L$. Then we are able to establish the 2. So within the class of vector bundles of the figure $bigoplus mathcal{O}(n_i)$ the place $n_igeq 0$, clearly we are able to raise every objective from the round to the ambient projective area. The identical is undoubted for the morphisms, because the morphisms crack right down to direct sum of $textual content{Hom}(mathcal{O}(n),mathcal{O}(m))=Gamma(mathcal{O}(m-n))$. My query: is there a functorial route to raise this class from the round to the projective area? If so is it workable to make this functor require on the neighborhood of the round?

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