Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations — Examples

$h_{{\mathbb{A}}_k^1}$ is an algebraic space over $k$ by [F1] applied to $T={\mathbb{A}}_k^1$, with the identity as its etale scheme cover.

The presentation $R\rightrightarrows U\to {\mathbb{A}}_k^1$ with $U={\mathbb{A}}_k^1$, $R=\mathrm{\Delta }({\mathbb{A}}_k^1)$ and the two projections is exactly the kernel pair of the identity: the projections are isomorphisms, the comparison map $R=U{\times }_{{\mathbb{A}}_k^1}U$ is the diagonal, and the coequalizer of the two projections is ${\mathbb{A}}_k^1$ itself; by [F2] the diagonal relation is an equivalence relation on $U$ over $k$, and both projections are etale because they are isomorphisms. This exhibits the asserted presentation, and the final claim that every $k$-scheme is an algebraic space is [F1].

Every torsor is finite etale. The given kernel-pair isomorphism trivializes $P\to T$ over the cover $P\to T$. Over an affine open $T_0\subseteq T$, choose finitely many affine opens in $P_{T_0}$ whose open images cover $T_0$, using [F2]. Their disjoint union $V\to T_0$ is an affine faithfully flat quasi-compact fppf refinement. Over $V$ the torsor is $G_V$, hence finite etale. Its canonical finite-etale descent datum is effective by [F1], giving a finite etale $T_0$-scheme $Q$. The local isomorphism $Q_V\cong P_V$ and its inverse descend as morphisms by [F1], and their composites are identities by uniqueness. Thus $P_{T_0}\cong Q$, and these identifications glue over target opens. Every torsor is therefore finite etale and surjective over any $T$, with no Noetherian hypothesis. Base change preserves its torsor kernel-pair isomorphism, so the stated fibred category exists.

$BG$ is a stack in groupoids. For a compatible family of torsors on any fppf covering, work over an affine target open and choose a finite affine refinement of that covering by the open-image argument of [F2]. The underlying finite-etale covers descend effectively by [F1] and step 1.1. Their action maps and the inverse of the torsor kernel-pair isomorphism descend by morphism descent in [F1]; their identities hold because they do after pullback. Surjectivity is detected on the cover, so the descended scheme is a torsor. Equivariant morphisms descend uniquely for the same reason, and uniqueness glues these constructions over target affine opens. This proves both effective object descent and the sheaf condition for morphisms, hence $BG$ is a stack.

Diagonal and presentation. For two torsors $P,Q$ over $T$, the finite etale surjective cover $P{\times }_{T}Q\to T$ trivializes both by step 1.1. The sheaf of equivariant isomorphisms becomes $G$ on this cover, with transitions induced by the two trivializations. Its cocycle is canonical, so [F1] represents this Isom sheaf by a finite etale $T$-scheme; this is precisely the base change of the diagonal of $BG$. The trivial torsor supplies $\mathrm{Spec}k\to BG$. Its base change along a torsor $P\mathrm{/}T$ is $P$ itself: an equivariant map $G_{T^{\mathrm{\prime }}}\to P_{T^{\mathrm{\prime }}}$ is uniquely determined by the image of the identity. Thus the base change is finite etale and surjective, hence smooth and surjective. These representable base changes establish a scheme presentation and representable diagonal, so $BG$ is algebraic by [F3]. This checks arbitrary schemes $T$, including schemes with nontrivial torsors, rather than asserting that all global trivializations exist.

Automorphisms and inertia. For the trivial torsor $G_T$ over $T$, every automorphism of right torsors is left multiplication by an element of $G$: if $\phi \text{\hspace{0.17em}⁣}:G_T\to G_T$ satisfies $\phi (pg)=\phi (p)g$, then $\phi (p)=\phi (1)p$ for a locally constant $\phi (1)\in G(T)$, and conversely every such left multiplication is an automorphism. Hence the automorphism sheaf of the trivial torsor is $G_T$ by left multiplication. If $G$ is abelian, then also every automorphism of an arbitrary torsor $P$ is right translation by a section of $G_T$: an automorphism is $\phi (p)=p\cdot g(p)$ with $g(ph)=h^{-1}g(p)h=g(p)$ by commutativity, so $g$ descends along the fppf map $P\to T$ to a section of $G_T$ by [F1]. Consequently, for abelian $G$ the inertia stack satisfies ${\mathcal{I}}_{BG}\cong G_k{\times }_{k}BG$, and over the trivial torsor in $BG(\mathrm{Spec}k)$ the inertia objects are exactly the pairs $(\text{trivial torsor},g)$ with $g\in G$; for nonabelian $G$ the automorphism sheaf of an arbitrary torsor is only a form of $G$, and the stronger product description is asserted only when $G$ is abelian, as in the trivial-torsor case.

$BG$ has nontrivial inertia. By [F1] the trivial torsor $G_k$ over $\mathrm{Spec}k$ has automorphism group $G(\mathrm{Spec}k)=G$ acting by left multiplication, and $G$ is nontrivial by hypothesis. Hence the fibre category of $BG$ over $\mathrm{Spec}k$ is not a setoid.

$BG$ is not the stack of a scheme. Suppose $BG$ were equivalent to ${\mathcal{S}}_X$ for some $k$-scheme $X$. Then by [F2] every fibre category of $BG$ would be a setoid, contradicting step 1.1. Hence no $k$-scheme $X$ has $BG\simeq {\mathcal{S}}_X$, even though $BG$ is an algebraic stack by [F1].

Separation of the sheaf and stack levels. For the trivial action of $G$ on $\mathrm{Spec}k$, the fppf quotient sheaf $(\mathrm{Spec}k)\mathrm{/}G$ is the representable sheaf $h_{\mathrm{Spec}k}$ by [F3], so at the level of quotient sheaves the quotient is a scheme, namely $\mathrm{Spec}k$; at the level of quotient stacks the same data present the classifying stack $BG$, which is not a scheme by step 2.1. This witnesses that passing from quotient sheaves to quotient stacks genuinely enlarges the category, and the failed conclusion is exactly the identification $BG\simeq {\mathcal{S}}_X$ for some scheme $X$. The supplier definition Quotient sheaves and representable quotients for pre-relations and group actions is now authored, and this field-level use is reconciled directly by [F3]: the quotient presheaf is terminal on the big fppf site and therefore already a sheaf.