When is a functor the associated bundle construction

algebraic-geometrydifferential-geometryprincipal-bundlesvector-bundles

Let $G$ be a group and $X$ be a space. I am principally interested in these objects in the category of schemes over some base $S$, but an answer in another geometric category would be welcome.

Suppose I have a functor $F$ from $Rep(G)$ to bundles on $X$. What are necessary and sufficient conditions for $F$ to be the associated bundle construction for some principal $G$-bundle on $X$? Necessary conditions include that $F$ is an exact tensor functor and $\mathrm{rank}\, F(V) = \dim V$.

Best Answer

An answer is given in [1, ยง2.2] when $S = spec(k)$ and $G/S$ is an affine algebraic group. Let $Rep(G)_{fd}$ denote the category of finite-dimensional representations. $F: Rep(G)_{fd}\to Qcoh(X)$ must satisfy the following properties:

  1. $F$ is $k$-additive and exact;
  2. $F$ is a strict tensor functor;
  3. $F(\text{rank one trivial representation}) = \mathcal{O}_X$;
  4. $F(V)$ is locally free of rank $\dim V$ for all $V$.

One then extends $F$ to $Rep(G)$; then $\mathbf{Spec}(F(\mathcal{O}_G))$ is then a principal $G$-bundle whose associated bundle functor is $F$.

[1] Nori, Madhav V. "The fundamental group-scheme." Proc. Indian Acad. Sci. Math. Sci 91, no. 2 (1982): 73-122.

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