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I believe your second approach is Proposition 2 (p.6) of Heinloth's Uniformization of $G$-bundles available from Heinloth's website: http://staff.science.uva.nl/~heinloth/Uniformization_17-8-09.pdf

Does every algebraic space have a Nisnevich cover by a scheme? (Assume that the algebraic space is quasi-separated, quasi-compact and over a scheme $S$.) Background: Every algebraic space has an et …

Let $k$ be a field, and let $T$ an $n$-dimensional split torus over $k$. Let $X$ be a $k$-scheme with algebraic $T$-action. Solve for X: $$X / T \cong \mathbf{P}^1_k$$ (The quotient should be a cate …

Let $k$ be a field, let $G$ be an algebraic group scheme over $k$ and let $T = \textrm{Spec } k[x, x^{-1}]$ be a one-dimensional torus. Does there exist a scheme $X$ over $k$, an algebraic $T$-actio …

If I have a Deligne-Mumford stack $\Pi : X \to (\mathrm{Sch}/k)$ for some field $k$, can it be reconstructed from $\Pi^{-1}(C) \subset X$ for some "small" subcategory $C \subset (\mathrm{Sch}/k)$? For …

Ben gave much of the answer, but I'll try to make it precise. Toen says there is no Riemann-Roch for the naive (rational) chow ring (Remark 4.3 in Theoremes de R-R) (EDIT: unless you also take the nai …

Let $F$ be a Deligne-Mumford stack that is of finite type, smooth and proper over $\mathrm{Spec~}k$ for a perfect field $k$. Consider $K_m$, the presheaf of $m$-th $K$-groups on $F_{et}$, the etale si …