Let $\mathcal F$ be a locally free $\mathcal O_X$-module. Then $\mathcal R := \bigoplus_n \mathcal F^{\otimes n}$Sym_{\mathcal O_X}(\mathcal F)$, the tensor product products being over$\mathcal O_X$, is a sheaf of rings, and we can take its$\bf Spec$to get a space over$X$. That space is the corresponding vector bundle.$\mathcal R$'s grading is what gives the dilation action on the fibers. The map$\mathcal F \to (\mathcal F \otimes \mathcal O_X) \oplus (\mathcal O_X \otimes \mathcal F)$,$f \mapsto (f\otimes 1) + (1\otimes f)$induces a cocommutative comultiplication$\mathcal R \to \mathcal R \otimes \mathcal R$, which gives the vector addition on${\bf Spec}\ \mathcal R$, I think. 1 Let$\mathcal F$be a locally free$\mathcal O_X$-module. Then$\mathcal R := \bigoplus_n \mathcal F^{\otimes n}$, the tensor product being over$\mathcal O_X$, is a sheaf of rings, and we can take its$\bf Spec$to get a space over$X$. That space is the corresponding vector bundle.$\mathcal R$'s grading is what gives the dilation action on the fibers. The map$\mathcal F \to (\mathcal F \otimes \mathcal O_X) \oplus (\mathcal O_X \otimes \mathcal F)$,$f \mapsto (f\otimes 1) + (1\otimes f)$induces a cocommutative comultiplication$\mathcal R \to \mathcal R \otimes \mathcal R$, which gives the vector addition on${\bf Spec}\ \mathcal R\$, I think.