Let $G$ be an affine group scheme. Let $\pi : P\rightarrow X$ be a principal $G$-bundle over a scheme $X$ ( i.e. $\pi$ is surjective flat affine morphism, $\phi : P\times G\rightarrow P$ action of $G$ on $P$ such that $\pi \circ \phi = \pi \circ p_1$; where $p_1 : P\times G\rightarrow P$, and $\psi : P\times G\rightarrow P\times_X P$ defined by $p_1\times \phi$ is an isomorphism).

Let $V$ be a finite dimensional representation. Then How to get vector bundle (locally free sheaf of finite rank) on $X$ associated to $P$ and $V$. In the setting of manifold I know this, but in scheme setting, we can not take point wise action. I don't know any good reference for this.

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    $\begingroup$ I may be wrong, but maybe considering the sheaf of $G$-equivariant sections of $\pi_*\mathcal{O}_P\otimes_{\mathbb{K}}V$ could work. $\endgroup$ Dec 11 '10 at 8:24

Thinking in terms of stacks, the answer just jumps out at you. What you have is a map $X \to BG$ and a map $G \to GL_n$ and you want to cook up a map $X \to BGL_n$. And in general a map $G \to H$ induces a map $BG \to BH$ via the ''contracted product'' construction, mapping a $G$-bundle $P$ to $P \times^G H$.

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    $\begingroup$ The stackically challenged should focus on the final formula: Given a $G$-torsor $P\to X$ and a scheme $Y$ on which $G$ acts we can construct the quotient $P\times_GY=(P\times Y)/G$. When $Y$ is affine space on which $G$ acts this becomes a vector bundle in the geometric sense, one then gets a locally free sheaf by looking at its sheaf of sections. This coincides with domenico's answer. $\endgroup$ Dec 11 '10 at 11:08
  • $\begingroup$ I don't understand why the quotient $(P\times Y)/G$ exists in the category of scheme? is some condition on $G$ required? $\endgroup$
    – Yashica
    Dec 11 '10 at 12:13
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    $\begingroup$ Those worried about quotients may want to just get dirty and work out the description of $P$ in terms of $G$-valued cocycles in whatever topology they are using. The representation then gives you cocyles for the vector bundle. Going this way makes it slightly harder to see the functoriality. For that you need to start calling the cocycles 'descent data' or something. $\endgroup$ Dec 11 '10 at 12:35
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    $\begingroup$ can anyone give me good reference for this!! $\endgroup$
    – john
    Dec 11 '10 at 13:54
  • $\begingroup$ Jantzen, Representations of Algebraic Groups, section 1.5. $\endgroup$ Dec 11 '10 at 14:13

Do the construction usual construction (as in manifolds) to get a morphism of schemes $\mathbb{V}\to X$. Now use the exercise in Hartshorne, Chapter 2 section 5, 5.18 to get a locally free sheaf of finite rank.

  • $\begingroup$ Didn't notice Ekadahl's comment above, my answer is the same. $\endgroup$
    – Rex
    Dec 11 '10 at 11:51

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