I am not an expert in algebraic geometry, so my question maybe be trivial for those who are.
All schemes are over $\mathbb C$. Let X be a scheme and $\mathcal E\rightarrow X$ be a vector bundle of rank $n$. I knew from a paper that there is a Frame Bundle $P(\mathcal E)\rightarrow X$ which has a natural $GL_n-$action associated to this vector bundle. And there is an equivalence of categories between the category of vector bundles over $X$ and the category of $GL_n-$principal bundles over $X$.
I searched on Bing and the terms are all about Manifolds, which I think is different from our context.
My question is: is there a good reference for the above construction? Or can anyone show me the detailed constructions? I know little about vector bundles, so any reference for it would also be appreciated.
I think now I figured it out. For a vector bundle $\mathcal E\rightarrow X$ of rank $n$, its frame bundle $P(\mathcal E)\rightarrow X$ is defined as follows:
Let $U_i=\mathrm{Spec}\ A_i$ be an open affine covering of $X$ such that $\mathcal E|_{U_i}$ is trivializable, namely we have $\rho_i: \mathcal E|_{U_i}\xrightarrow{\sim} \mathbb A^n_{U_i}$. Denote by $\mathcal E$ the associated locally free $\mathcal O_X$ module. Consider the sheaf Hom $\mathcal G=\mathcal{Hom}(\mathcal O_X^n,\mathcal E)$. Then $\mathcal G(U_i)=M_n(A_i)$. So the associated vector bundle $ \mathcal G$ is defined such that $\mathcal G|_{U_i}=\mathrm{Spec}\ \mathrm{Sym}(M_n(A_i))$. Consider its open sub scheme $D(det(x_{ij}))$ where $x_{ij}$ is a basis of $M_n(A_i)$ as $A_i-$module. This defines an open subscheme of the vector bundle $\mathcal G$ and this is the frame bundle that we want.