Algebraic Group Actions, Orbits, Stabilizers, and Controlled Quotients

For every $k$-scheme $T$ the map $U(T)\to M(T)$, $u\mapsto q(u)$, is constant on each generating pair $(s(x),t(x))$ of the relation on $U(T)$ by (1), hence constant on the equivalence relation it generates, so it descends to a map $P_{U\mathrm{/}R}(T)\to M(T)$; these maps are natural in $T$ and define a morphism of presheaves $P_{U\mathrm{/}R}\to h_M$. By [F2] the functor $h_M$ is an fppf sheaf, so the universal property in [F1] factors this morphism uniquely through a morphism $\phi :U\mathrm{/}R\to h_M$.

Every section of $U\mathrm{/}R$ is locally represented by a section of $U$. Let $V(T)\subseteq (U\mathrm{/}R)(T)$ be the set of sections $s$ for which there are an fppf covering $\{T_i\to T\}$ and elements $u_i\in U(T_i)$ whose images in $(U\mathrm{/}R)(T_i)$ equal the restrictions of $s$. Restrictions of such sections lie in $V$, so $V$ is a subpresheaf of $U\mathrm{/}R$; and $V$ is a sheaf, because an fppf covering of each member of an fppf covering of $T$ composes to an fppf covering of $T$. Since $U(T)\to P_{U\mathrm{/}R}(T)$ is surjective for every $T$, the canonical morphism $P_{U\mathrm{/}R}\to U\mathrm{/}R$ takes values in $V$, so there is a factorization $P_{U\mathrm{/}R}\to V$ through the inclusion $V\hookrightarrow U\mathrm{/}R$. Applying the initiality in [F1] to the target sheaf $V$ and to the target sheaf $U\mathrm{/}R$ shows that the composite $U\mathrm{/}R\to V\hookrightarrow U\mathrm{/}R$ is the identity: both maps make the triangle from $P_{U\mathrm{/}R}$ commute, and such a map is unique. Therefore $V=U\mathrm{/}R$, which is the claim.

The morphism $\phi$ is surjective as a map of fppf sheaves. Let $T$ be a $k$-scheme and $m\in M(T)=h_M(T)$. By (2) there are an fppf covering $\{T_i\to T\}$ and $u_i\in U(T_i)$ with $q(u_i)=m{\mathrm{\mid }}_{T_i}$. For every pair $i,j$, the restrictions $u_i{\mathrm{\mid }}_{T_{ij}}$ and $u_j{\mathrm{\mid }}_{T_{ij}}$ have the same image in $M(T_{ij})$, so by [F3] they define a section of $h_{U{\times }_{M}U}(T_{ij})$. Hypothesis (3) lifts this section, after an fppf refinement of $T_{ij}$, to $r$ with $(t(r),s(r))=(u_j,u_i)$. Thus $[u_i]$ and $[u_j]$ agree in the quotient presheaf locally on $T_{ij}$ and hence agree in its sheafification $U\mathrm{/}R$ on $T_{ij}$. They are therefore a matching family in the sheaf $U\mathrm{/}R$, so glue to $s\in (U\mathrm{/}R)(T)$. Step 1.1 gives $\phi ([u_i])=q(u_i)=m{\mathrm{\mid }}_{T_i}$; since $h_M$ is a sheaf, $\phi (s)=m$. Thus every section of $h_M$ lies in the image of $\phi$.

The morphism $\phi$ is injective as a map of presheaves. Let $\xi ,{\xi }^{\mathrm{\prime }}\in (U\mathrm{/}R)(T)$ satisfy $\phi (\xi )=\phi ({\xi }^{\mathrm{\prime }})=m$. By step 1.2 there are an fppf covering $\{T_i\to T\}$ of $T$ and elements $u_i,u_i^{\mathrm{\prime }}\in U(T_i)$ with $\xi {\mathrm{\mid }}_{T_i}=[u_i]$ and ${\xi }^{\mathrm{\prime }}{\mathrm{\mid }}_{T_i}=[u_i^{\mathrm{\prime }}]$. Then $q(u_i)=\phi ([u_i])=m{\mathrm{\mid }}_{T_i}=\phi ([u_i^{\mathrm{\prime }}])=q(u_i^{\mathrm{\prime }})$, so $(u_i^{\mathrm{\prime }},u_i)$ is a $T_i$-point of $U{\times }_{M}U$ by [F3]. Hypothesis (3) provides, after refining the covering, an element $r_i\in R(T_i)$ with $(t,s)(r_i)=(u_i^{\mathrm{\prime }},u_i)$, that is $t(r_i)=u_i^{\mathrm{\prime }}$ and $s(r_i)=u_i$. Then $(s(r_i),t(r_i))=(u_i,u_i^{\mathrm{\prime }})$ is a generating pair of the relation defining $P_{U\mathrm{/}R}(T_i)$, so $[u_i]=[u_i^{\mathrm{\prime }}]$ in $(U\mathrm{/}R)(T_i)$. Hence $\xi$ and ${\xi }^{\mathrm{\prime }}$ agree on the members of an fppf covering, and separatedness of the sheaf $U\mathrm{/}R$ forces $\xi ={\xi }^{\mathrm{\prime }}$.

A morphism of fppf sheaves that is surjective and injective on sections is an isomorphism, so steps 2.1 and 2.2 show that $\phi :U\mathrm{/}R\to h_M$ is an isomorphism; hence $h_M\cong U\mathrm{/}R$ naturally and $M$ represents the quotient sheaf $U\mathrm{/}R$. For the parenthetical instances, if $q$ is faithfully flat and locally of finite presentation then $h_U\to h_M$ is a surjection of fppf sheaves by [F2], and the same argument with $X=R$ and $Y=U{\times }_{M}U$ gives the second instance; this finishes the proof.

The equivalence-relation hypothesis includes that $j$ is a monomorphism. Reflexivity supplies the diagonal arrow $e:U\to R$ with $j\circ e={\mathrm{\Delta }}_{U\mathrm{/}k}$, symmetry supplies the unique arrow $i:R\to R$ with $j\circ i=\sigma \circ j$ for the factor swap $\sigma$, and transitivity supplies the unique composition arrow through $j$; these are exactly the groupoid-scheme arrows dual to the identities of the relation, so $(R\rightrightarrows U)$ is an equivalence-relation groupoid with finite locally free source and target.

Applying [F1] to the groupoid produced in step 1.1 gives that $C=\{a\in A:s^{\mathrm{♯}}(a)=t^{\mathrm{♯}}(a)\}$ is a finite-type $k$-algebra, $U\to M=\mathrm{Spec}C$ is finite locally free and surjective, and $R\to U{\times }_{M}U$ is an isomorphism; all hypotheses coincide because both statements use the comorphisms $s^{\mathrm{♯}},t^{\mathrm{♯}}$ of $s,t$ on coordinate rings.

The representing claim is a claim about the same object: by [F2] the phrase "$M$ represents the fppf quotient sheaf $U\mathrm{/}R$" means that $h_M$ is naturally isomorphic to the sheafification of the naive quotient presheaf of $s,t:R\to U$ in the fppf topology, which is exactly the conclusion recorded here; no further hypothesis is added and no step of the published proof is repeated.

Since $X$ is separated over $k$, the point $x$ is a closed immersion by [F1], and $H=G{\times }_{X,{\varrho }_x,x}\mathrm{Spec}k$ is the pullback of this closed immersion along ${\varrho }_x$, hence a closed subscheme of $G$ by [F4].

For every $k$-algebra $R$ the universal property in [F2] identifies the fiber of $(g,z)\mapsto (gz,z)$ over $(y,x)$ with $\{(g,z)\in G(R)\times X(R):gz=y_R,z=x_R\}=\{g\in G(R):g{x}_R=y_R\}=F_y(R)$; these identifications are natural in $R$ and therefore identify the two schemes. This is (c).

If $y=g_{0}x$ with $g_0\in G(k)$, left translation by $g_0$ is an automorphism of $G$ carrying $H$ onto the fiber $F_y$: for every $R$ it identifies $H(R)=\{h:h{x}_R=x_R\}$ with $\{g_{0}h:(g_{0}h)x_R=g_0{x}_R=y_R\}=F_y(R)$, and conversely $g{x}_R=y_R$ implies $(g_0^{-1}g)x_R=x_R$. Similarly, for $g\in G(k)$ and every $R$ one has $G_{gx}(R)=gH(R)g^{-1}$, because $g^{\mathrm{\prime }}(g{x}_R)=g{x}_R$ is equivalent to $(g^{-1}{g}^{\mathrm{\prime }}g)x_R=x_R$; the closed subschemes $G_{gx}$ and $gH{g}^{-1}$ have the same functor of points and hence coincide.

The morphism $\phi :G\times H\to G{\times }_{X}G$, $(g,h)\mapsto (g,gh)$, is well defined because $(gh)x=g(hx)=gx$; the morphism $\psi :G{\times }_{X}G\to G\times H$, $(g,g^{\mathrm{\prime }})\mapsto (g,g^{-1}{g}^{\mathrm{\prime }})$, is well defined because equal images $gx=g^{\mathrm{\prime }}x$ imply $(g^{-1}{g}^{\mathrm{\prime }})x=x$, that is $g^{-1}{g}^{\mathrm{\prime }}\in H$. On $R$-points for every $k$-algebra $R$ the two composites are the identity: $\psi \phi (g,h)=(g,g^{-1}gh)=(g,h)$ and $\phi \psi (g,g^{\mathrm{\prime }})=(g,g(g^{-1}{g}^{\mathrm{\prime }}))=(g,g^{\mathrm{\prime }})$. Hence $\phi$ and $\psi$ are inverse isomorphisms of $k$-schemes. This proves the first assertion of (d).

A fiber can be nonempty without having a $k$-point. Let $k=\mathbb{R}$, let $G={\mathbf{G}}_m=\mathrm{Spec}\mathbb{R}[t,t^{-1}]$ with the comultiplication $t\mapsto t\otimes t$ of [F6], let $X={\mathbf{G}}_m$, and let $G$ act on $X$ by $g\cdot z=g^{2}z$, an action because $(gh{)}^2=g^2{h}^2$ and $1^2=1$. Take $x=1$, $y=-1$. The fiber $F_y=G{\times }_X\mathrm{Spec}\mathbb{R}$ over $y$ has coordinate ring $\mathbb{R}[t,t^{-1}]{\otimes }_{\mathbb{R}[s,s^{-1}]}\mathbb{R}$, where the right factor is the residue field at $s=-1$; this tensor product is $\mathbb{R}[t,t^{-1}]\mathrm{/}(t^2+1)\cong \mathbb{R}[t]\mathrm{/}(t^2+1)$, since $t^2=-1$ makes $t$ invertible. That ring is nonzero because $t^2+1$ is nonconstant, hence not a unit of $\mathbb{R}[t]$ by [F7]. But $F_y(\mathbb{R})=\mathrm{\varnothing }$: an $\mathbb{R}$-point would give $z\in \mathbb{R}$ with $z^2=-1$, whereas $z^2\ge 0$ for every $z$ by the ordered-field fact of [F7] while $-1<0$, since $0-(-1)=1$ is positive. Thus a nonempty fiber need not have a $k$-point.

For every $k$-algebra $R$ the set $H(R)=\{g:g{x}_R=x_R\}$ is a subgroup of $G(R)$: it contains the identity; if $g{x}_R=x_R$ and $g^{\mathrm{\prime }}{x}_R=x_R$ then $(g{g}^{\mathrm{\prime }})x_R=g(g^{\mathrm{\prime }}{x}_R)=g{x}_R=x_R$; and if $g{x}_R=x_R$ then $g^{-1}{x}_R=g^{-1}(g{x}_R)=x_R$. The subsets are natural in $R$, so by the valued point criterion [F3] the closed subscheme $H$ of step 1.1 carries the unique structure of a closed subgroup scheme with this functor of points, which is (a).

For every $k$-algebra $R$ the map $F_y(R)\times H(R)\to F_y(R)\times F_y(R)$, $(g,h)\mapsto (g,gh)$, is a bijection: it is injective since $g=g^{\mathrm{\prime }}$ and $gh=g^{\mathrm{\prime }}{h}^{\mathrm{\prime }}$ give $h=h^{\mathrm{\prime }}$, and a pair $(g,g^{\mathrm{\prime }})\in F_y(R)\times F_y(R)$ has $g^{-1}{g}^{\mathrm{\prime }}\in H(R)$ because $(g^{-1}{g}^{\mathrm{\prime }})x_R=g^{-1}(g^{\mathrm{\prime }}{x}_R)=g^{-1}(g{x}_R)=x_R$, with $(g,g^{-1}{g}^{\mathrm{\prime }})\mapsto (g,g^{\mathrm{\prime }})$. By Yoneda this exhibits the action morphism $F_y{\times }_{k}H\to F_y{\times }_k{F}_y$ as an isomorphism. Consequently, if $F_y(L)$ is nonempty for some field extension $L\mathrm{/}k$, base change to $L$ identifies $(F_y{)}_L$ with $H_L$.

Suppose $F_y$ is nonempty. Then $F_y$ has a nonempty affine open $\mathrm{Spec}A$ with $A\mathrm{\ne }0$ a finitely generated $k$-algebra; choose a maximal ideal $\mathfrak{m}\subset A$ and put $L=A\mathrm{/}\mathfrak{m}$. By [F5] the field $L$ is a finite extension of $k$, and the resulting $L$-point of $F_y$ is an element of $F_y(L)$. Step 2.2 then identifies $(F_y{)}_L$ with $H_L$; since a finite field extension is a faithfully flat and finitely presented base change, $F_y$ is an fppf right $H$-torsor, trivialized by the fppf cover $\mathrm{Spec}L\to \mathrm{Spec}k$, and it has a $k$-point exactly when $F_y\cong H$. Step 1.5 shows that the latter can fail, since the fiber $F_{-1}$ constructed there is nonempty and the trivialization just produced applies to it.

The remaining clause of (d) follows because a locally closed immersion is a monomorphism by [F4]: if ${\varrho }_x$ factors through $O_x\hookrightarrow X$, then for every test scheme a pair of points of $G$ has equal images in $O_x$ if and only if it has equal images in $X$, so $G{\times }_{O_x}G=G{\times }_{X}G$ and step 1.4 identifies it with $G\times H$. Statement (c) is step 1.2; statement (a) is step 2.1; statement (b) consists of step 1.3, step 2.2 and step 3.1; the first assertion of (d) is step 1.4. This completes the proof.

The orbit map ${\varrho }_x$ is quasi-compact, so by [F1] its scheme-theoretic image $Y=V(\mathcal{I})\hookrightarrow X$ exists. Let $Z_k:={\varrho }_x(\mathrm{\mid }G\mathrm{\mid })$. Its closure is the underlying space of $Y$: on an affine target chart with finitely many source charts, if a basic open $D(a)$ misses the image then every source algebra localized at $a$ is zero. A power of $a$ vanishes in each of the finitely many algebras, so a common power belongs to the kernel defining $Y$, and $D(a)$ misses $Y$. The reverse inclusion follows because the image lies in $Y$. On an affine chart $\mathrm{Spec}A\subseteq X$ with ${\varrho }_x^{-1}(\mathrm{Spec}A)$ covered by finitely many affine charts $\mathrm{Spec}{B}_j$, the ideal is $I=\ker (A\to {\prod }_j{B}_j)$, and for every field extension $K\mathrm{/}k$ one has $I{\otimes }_{k}K=\ker (A_K\to {\prod }_j(B_j{\otimes }_{k}K))$ because $k\to K$ is flat and tensor products are right exact and commute with finite products; hence $Y_K:=Y{\times }_{k}K$ is the scheme-theoretic image of ${\varrho }_{x,K}$ and $Y_K$ is the closure of $Z_K:={\varrho }_{x,K}(\mathrm{\mid }{G}_K\mathrm{\mid })$ in $X_K$.

For $K=\bar{k}$ the set $Z_K$ is constructible in $X_K$: cover the quasi-compact $X_K$ by finitely many affine charts $\mathrm{Spec}A$, cover each preimage by finitely many affine charts $\mathrm{Spec}{B}_j$, and each $A$ is Noetherian by [F12], and $B_j$ is a finite-type $A$-algebra because it is generated by finitely many elements over $K$. Hence [F2] gives finite presentation, so apply [F2] to $A\to B_j$ and to $b=1$; the full scheme-point image is the finite union of these constructible images.

The projection $\pi :Y_K\to Y$ is surjective and closed. It is the base change of $\mathrm{Spec}K\to \mathrm{Spec}k$, which is surjective; and $K\mathrm{/}k$ is algebraic, hence integral, so $Y_K\to Y$ is integral and, by [F3], the image of any closed $V(J)\subseteq Y_K$ is the closed set $V(J\cap Y)$; a surjective closed map is a quotient map.

The scheme $Y$ is geometrically reduced: by step 1.1 it suffices to note that each $B_j{\otimes }_{k}K$ is reduced, since $G_K$ has regular local rings by [F4], and that $A_K\mathrm{/}I{\otimes }_{k}K$ embeds into the product of the reduced rings $B_j{\otimes }_{k}K$.

Since $Z_K$ is constructible by step 1.2 and dense in $Y_K$ by step 1.1, it contains a dense open $U$ of $Y_K$: writing $Z_K$ as a finite union of locally closed subsets and intersecting with the finitely many irreducible components of the Noetherian space $Y_K$, one piece is dense in each component, and a locally closed subset dense in an irreducible space contains an open dense subset of it; remove from $Y_K$ the finitely many closed complements of those relative opens and all intersections of distinct components. The remaining subset is open and dense in $Y_K$ and contained in $Z_K$. Moreover every nonempty constructible subset of $Y_K$ contains a point closed in $Y_K$: a nonempty locally closed piece has a nonempty open subset of an affine chart, and by [F6] that open contains a $K$-point of the chart, which is closed in $Y_K$ because its residue field is $K$.

The image is saturated for $\pi$: for a point $y\in Y_K$ with $q=\pi (y)$ one has $G_K{\times }_{X_K,y}\mathrm{Spec}\kappa (y)\cong G{\times }_X\mathrm{Spec}\kappa (y)$, because the morphism $\mathrm{Spec}\kappa (y)\to X$ factors through $\mathrm{Spec}\kappa (q)\to X$; this in turn is isomorphic to $G_q{\times }_{\mathrm{Spec}\kappa (q)}\mathrm{Spec}\kappa (y)$ with $G_q=G{\times }_{X,q}\mathrm{Spec}\kappa (q)$, and the latter is nonempty whenever $G_q$ is, by [F7]. Since $G_q$ is nonempty exactly when $q$ lies in $Z_k={\varrho }_x(\mathrm{\mid }G\mathrm{\mid })$, this shows $y\in Z_K$ if and only if $\pi (y)\in Z_k$, so ${\pi }^{-1}(\pi (Z_K))=Z_K$.

Each translation by $g\in G(K)$ preserves $Y_K$: translating ${\varrho }_{x,K}$ is precomposing it with left translation on $G_K$, so its scheme-theoretic image is unchanged by [F1]. Set $O_K:=Z_K$ with the open subscheme structure it has in $Y_K$; this is legitimate because $Z_K$ is open in $Y_K$: every closed point $c$ of $Z_K$ lifts to a $K$-point $h$ of $G_K$ by [F6] applied to the nonempty finite-type fibre of ${\varrho }_{x,K}$ over $c$, and with a $K$-point $v$ of the nonempty open ${\varrho }_{x,K}^{-1}(U)$ one has $c={\varrho }_{x,K}(h)=(h{v}^{-1})\cdot {\varrho }_{x,K}(v)\in (h{v}^{-1})U$, so that ${\bigcup }_{g\in G(K)}gU$ is an open subset of $Z_K$ containing every closed point of $Z_K$; its constructible complement in $Z_K$ would otherwise contain a closed point of $Y_K$ by step 2.2, so $Z_K={\bigcup }_{g\in G(K)}gU$ is open in $Y_K$. By step 2.1 the open subscheme $O_K$ is reduced, and it is finite type over $K$ because it is locally of finite type as a locally closed subscheme of the finite-type $X_K$ and quasi-compact as the continuous image of the quasi-compact space $G_K$.

The morphism ${\varrho }_{x,K}:G_K\to O_K$ is surjective by construction, and ${\varrho }_x(\mathrm{\mid }G\mathrm{\mid })=\pi (Z_K)$ is open in $Y$: by step 1.3 and step 2.3 the set ${\pi }^{-1}(\pi (Z_K))=Z_K$ is open, and $\pi$ is a quotient map, so $\pi (Z_K)$ is open. Define $O_x:=\pi (Z_K)$ with the induced open subscheme structure in $Y$; it is finite type over $k$, since it is locally of finite type as a locally closed subscheme of the finite-type $X$ and quasi-compact as the image of the quasi-compact space $G$ under ${\varrho }_x$, and $(O_x{)}_K=O_K$ as open subschemes of $Y_K$.