Affine quotients by products of general linear groups are algebraic stacks

Let $T\to [X\mathrm{/}G]$ correspond to a left torsor $P\to T$ and an equivariant map $\varphi :P\to X$. An object of $X{\times }_{[X\mathrm{/}G]}T$ is a point $x\in X$ and an equivariant isomorphism from the trivial left torsor to $P$, compatible with $\varphi$. Such an isomorphism is determined by the image $p\in P$ of the identity, and sends $g$ to $gp$; the compatibility condition forces $x=\varphi (p)$. Thus $X{\times }_{[X\mathrm{/}G]}T\cong P$. Since $P\to T$ is a torsor, it is a smooth surjective scheme morphism by [F3] and the local triviality in [F1]. Hence $X\to [X\mathrm{/}G]$ is a representable smooth surjective atlas.

For two objects $(P,\varphi )$ and $(P^{\mathrm{\prime }},{\varphi }^{\mathrm{\prime }})$ over a scheme $T$, their isomorphism functor is the subfunctor of ${\mathrm{Isom}}_G(P,P^{\mathrm{\prime }})$ on which ${\varphi }^{\mathrm{\prime }}\circ f=\varphi$. After a common fppf cover trivializing both torsors, and after restricting to affine opens of the cover, ${\mathrm{Isom}}_G(P,P^{\mathrm{\prime }})\cong G\times T$. With the chosen trivializing sections $p,p^{\mathrm{\prime }}$, the condition is the inverse image of the diagonal $X\hookrightarrow X\times X$ under the two evaluation maps $f\mapsto ({\varphi }^{\mathrm{\prime }}(f(p)),\varphi (p))$. The diagonal of affine $X$ is closed, so this is a closed subscheme of $G\times T$, affine over $T$. The isomorphism functor therefore has affine local representatives with their canonical descent data; [F4] makes this affine descent datum effective. Thus the isomorphism functor is represented by an affine scheme over $T$, which proves the diagonal is affine, hence representable.

The torsor-with-map construction is a category fibred in groupoids by pullback. To verify the stack condition, take an fppf cover of $T$ and torsors with equivariant maps and compatible isomorphisms on its overlaps. The torsors are affine over their bases by [F4]. On affine opens of $T$, refine the cover by faithfully flat affine covers and apply the effective algebra descent of [F4]; its full faithfulness glues these descended affine schemes on overlaps. The action, torsor isomorphism $G\times P\simeq P{\times }_{T}P$, and maps to $X$ descend as scheme morphisms by [F1]. Their equivariance and compatibility identities can be checked after the cover, and the resulting $P\to T$ is fppf-locally the given torsor, hence is a torsor. Compatible isomorphisms descend uniquely in the same way, with inverses descending too. Thus morphisms form sheaves and every object descent datum is effective, exactly the stack condition of [F1].

By steps 1.1–1.3, the quotient stack has a smooth surjective representable atlas and representable diagonal. Step 1.3 establishes the stack condition, so the criterion in [F2] applies and $[X\mathrm{/}G]$ is an algebraic stack. The inherited AC appears only in [A1] through those defining suppliers.