The definition of a (geometric) vector bundle over a scheme $X$ can be rewritten as follows in terms of 'not-so-geometrical algebra' if $X=Spec R$ is affine and if I am not missing something.

A *vector bundle* of rank $n$ over $R$ is an $R$-algebra $A$ such that

for every $p\in Spec R$ there is a isomorphism (belonging to the data) $$ \phi_p:k(p)[X_1,...,X_n]\xrightarrow{\cong} A\otimes_R k(p) $$ where $k(p)$ is the residue field $R_p/m_p$ and

there are elements $\{a_i\}_{i\in I}$ of $R$ such that the $D(a_i)=\{ I\in Spec R\mid a\notin I \}$ cover $Spec R$ and for every $i\in I$ there is an $R_a$-algebra isomorphism $$ A\otimes_R R_{a_i}\xrightarrow{\cong}R[X_1,\ldots,X_n]\otimes_RR_{a_i} $$ which induces for every $p\in Spec R$ with $a_i\notin p$ a $k(p)$-

**linear**$k(p)$-algebra isomorphism $$ A\otimes_R k(p)~\xleftarrow{\phi_p}k(p)[X_1,...,X_n]\to k(p)[X_1,...,X_n]\cong R[X_1,...,X_n]\otimes_R k(p). $$

The (isomorphism classes) of such vector bundles over $R$ should correspond to (isomorphism classes) of finitely gererated projective modules over the ring $R$.

How can this correspondence be seen?

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