Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients

Use the right-coset relation $R=G\times H$, with $s(g,h)=g$ and $t(g,h)=gh$. Projection is flat of finite presentation because every $k$-scheme is $k$-flat, and the change of variables $(g,h)\mapsto (gh,h^{-1})$ gives the same property for $t$. The map $R\to G\times G$ is a closed immersion: under the isomorphism $(g,g^{\mathrm{\prime }})\mapsto (g,g^{-1}{g}^{\mathrm{\prime }})$ it becomes $G\times H\hookrightarrow G\times G$. Thus it is an equivalence relation and [F1] supplies a nonempty dense saturated quotientable open of $G$.

For every finite extension $L\mathrm{/}k$ in an algebraic closure, let $O[L]$ be the union of the saturated opens of $G_L$ having scheme fppf quotients. These quotients glue: on an intersection, its image is open under the faithfully flat quotient map and its inverse image is the intersection by saturation; that open represents the restricted quotient sheaf. The two open quotients are consequently uniquely isomorphic, with the cocycle following from uniqueness. The union therefore has a quotient. Base change preserves the covering projection and its kernel pair, so $O[L{]}_{L^{\mathrm{\prime }}}\subset O[L^{\mathrm{\prime }}]$. Left translation preserves right cosets and hence preserves $O[L]$. Choose a closed point in the initial generic open and enlarge $k$ finitely to make it rational. Thereafter $O[L]$ contains every point of $G(L)$, since translation moves that rational point to any other.

There is a finite extension $K\mathrm{/}k$ for which $O[K]=G_K$. To prove this, if the complement is nonempty, choose a closed point in each of its finitely many irreducible components. Enlarge the field finitely to split all their finite residue extensions, including their inseparable parts. Every point above a selected point is now rational and belongs to the enlarged $O$. Every irreducible component of the old complement has lost a point on every component above it: finite field extension is flat and finite, so each such component maps onto an old component. Thus the dimension of the complement strictly decreases. Repetition terminates. Gluing in step 2.1 gives a finite-type quotient $q_K:G_K\to Y$: finite type follows from a finite subcover of $G_K$ by quotientable opens. It is faithfully flat of finite presentation with kernel pair $R_K$.

The scheme $Y$ is separated. Pulling its diagonal back along the fppf cover $G_K\times G_K\to Y\times Y$ gives the closed immersion $R_K\hookrightarrow G_K\times G_K$. Closed immersions descend in this situation: the diagonal is a separated finite-type morphism (every diagonal is separated), its base change is affine, so affine descent in [F2] makes it affine; on affine target charts its coordinate map becomes surjective after faithful flat extension, and the cokernel vanishes faithfully flatly. Thus the diagonal is closed. The same argument applies to each field base change.

Every finite subset of $Y$ lies in an affine open. First replace its points by closed specializations and lift those closed points to closed points $g_i$ of $G_K$. Choose a dense affine open $V\subset Y$: in each of the finitely many irreducible components choose a nonempty affine open avoiding the other components, and take their disjoint union. Since $q_K$ is open, $A=q_K^{-1}(V)$ is dense in $G_K$. Enlarge $K$ to a finite $L$ splitting all residue fields of the $g_i$; list all rational lifts $g_j^{\mathrm{\prime }}$. The intersection ${\bigcap }_j{g}_j^{\mathrm{\prime }}{A}_L^{-1}$ is dense open, since translation and inversion preserve density and finite intersections of dense opens are dense. Choose a closed point in it and enlarge $L$ again to make that point $a$ rational. Then $g_j^{\mathrm{\prime }}\in a{A}_L$ for every $j$, so the affine translate $a{V}_L$ in $Y_L$ contains all points above the selected $Y$-points. The left action on $Y_L$ exists by morphism descent in [F2]. These points are a union of orbits of the canonical finite relation $Y_L\rightrightarrows Y$. The saturated affine-neighbourhood construction in [F2], applied simultaneously to that finite union (the same prime-avoidance and norm proof applies), gives a saturated affine open inside $a{V}_L$ containing them. Its affine finite quotient is its open image in $Y$, as follows from its fppf lifting and kernel-pair property. That image is the required affine neighbourhood.

The quotient sheaf and its represented kernel pair commute with scalar extension, since their local-lifting description is preserved by base change. Thus the two base changes of $Y$ to $K{\otimes }_{k}K$ represent the same coset sheaf of the base-changed $G,H$; their unique isomorphism is a descent datum satisfying the cocycle over the triple tensor algebra. The resulting finite locally free descent relation on the underlying $k$-scheme $Y$ has finite orbits, and step 5.1 puts them in affine opens. The finite-field descent lemma [F2] yields a separated finite-type $k$-scheme $Q$ with $Q_K\cong Y$. The map $q_K$ descends to $q:G\to Q$ by morphism descent, and flatness descends. Surjectivity and finite presentation follow from faithful field base change and the finite-type Noetherian setting. The isomorphism $R\cong G{\times }_{Q}G$ descends from that over $K$ by uniqueness of compatible morphisms and their inverses. The covering and kernel-pair description proves that $Q$ represents the original fppf coset sheaf.

Normality defines multiplication on the quotient sheaf: for local representatives, $(gH)(g^{\mathrm{\prime }}H)=g{g}^{\mathrm{\prime }}H$ is independent of representatives because $g^{\mathrm{\prime }-1}H{g}^{\mathrm{\prime }}=H$ as subgroup schemes, after every base change. Identity and inverse are also well-defined. Since $Q$ represents the sheaf, these operations are morphisms. Associativity, identity and inverse identities follow after the fppf covering by representatives, giving the unique group structure for which $q$ is a homomorphism. The kernel-pair identity identifies its fibre at the identity with $H$ and gives the displayed torsor isomorphism. Base changing by $G\to Q$ trivializes the torsor, so it is fppf locally trivial. Any homomorphism killing $H$ is constant on the relation and descends uniquely by [F2]; it remains a homomorphism since that identity can be checked after the product cover. The same local description proves compatibility with every field extension. The continuous surjection $G\to Q$ preserves connectedness.

If $G$ is smooth, then after any field extension $Q$ is reduced: faithful flatness injects its local affine function rings into rings on an affine fppf cover from the reduced scheme $G$, so nilpotents vanish. In particular $Q$ is geometrically reduced. Over an algebraic closure a reduced finite-type group scheme has a smooth point by [F3]; translation moves it to the identity and then to every closed rational point. The nonsmooth locus is closed and, if nonempty, has a closed rational point, a contradiction. Thus $Q$ is smooth after algebraic closure and hence over $k$ by [F3]. If $H$ is smooth, the fppf local trivialization makes $q$ smooth: equivalently each geometric fibre is an $H$-torsor and becomes smooth after extension to an algebraically closed residue field where it has a point; finite presentation and flatness then give smoothness by the geometrically regular fibre criterion. This argument does not assert that $q$ is smooth when $H$ is nonsmooth. AC enters through the closure, closed-point and local quotient suppliers.