Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties

If $X=\mathrm{Spec}B$ is affine, [F1] and [F2] identify $B$ with $k$ as a $k$-algebra. Taking spectra gives $X\cong \mathrm{Spec}k$.

For an arbitrary affine target $T=\mathrm{Spec}B$, a $k$-morphism $X\to T$ corresponds by [F2] to a $k$-algebra map $B\to \mathrm{\Gamma }(X,{\mathcal{O}}_X)=k$. That map defines a $k$-rational point of $T$, and naturality in [F2] gives the desired factorization through $X\to \mathrm{Spec}k$. By [F3], an affine abelian variety would be a point by step 1.1, so a positive-dimensional abelian variety cannot be affine. AC is used precisely through [F1].

Fix a complex point $x$ of the product, and choose algebraic local parameters $t,s$ of the two curves at its coordinates using [F1]. They are holomorphic manifold coordinates at $x$ and a cotangent basis of the two-dimensional regular local algebraic ring $R={\mathcal{O}}_{C_1\times C_2,x}$. By [F2], the map $\mathbf{C}\left[\text{\hspace{0.17em}⁣}[T,S]\text{\hspace{0.17em}⁣}\right]\to \widehat{R}$ sending $T,S$ to $t,s$ is an isomorphism: its maps on successive homogeneous quotients are the graded isomorphism in [F2], so lifting successively in the complete filtrations proves surjectivity and the first nonzero homogeneous term proves injectivity. Formal Taylor expansion of analytic germs, justified by [F3], agrees with this identification on every element of $R$. Indeed it sends the parameters to $T,S$, respects ring operations, and gives the same finite jets on the polynomial representatives of each $R\mathrm{/}{\mathfrak{m}}^n$ supplied by [F2].

Choose an affine open of $Y$ containing $h(x)$ and finitely many coordinate-ring generators. By holomorphy and continuity their pullbacks under $h$ are holomorphic on a manifold neighbourhood of $x$. On the nonempty rational domain they are rational functions, so write one as $a\mathrm{/}b\in \mathrm{Frac}R$. The identity $a=b{h}^{\ast }(z)$ holds on that domain in the neighbourhood and hence as analytic germs by continuity; the complement of the rational domain has no manifold interior, since a nonzero algebraic defining function gives a nonzero holomorphic germ and cannot vanish on a manifold open. Taking formal Taylor series via step 1.1 gives $a\in b\widehat{R}$. Faithful flatness in [F2] forces $a\in bR$: the nonzero class of $a$ in $R\mathrm{/}(b)$ could not become zero after a faithful flat extension. Thus every target coordinate pulls back to an element of $R$.

The finitely many pulled-back coordinates are regular on an algebraic neighbourhood of $x$, satisfy the target's algebraic relations by generic agreement, and therefore define an algebraic extension there. They agree analytically with $h$ near $x$ by step 2.1. The same argument applies at every complex point. The complement of the union of these algebraic neighbourhoods is closed and has no complex closed point, hence is empty by the weak Nullstellensatz. Separatedness glues the extensions and makes them unique, since they agree on the dense original rational domain. This constructs the asserted algebraic morphism. AC is inherited from the regularity/completion suppliers.

First consider an affine cover with a section. For the equivalent scheme map $g:T\to S$ with section $s$, set $M=s^{\ast }N$. Pull transport back along $T\to T{\times }_{S}T$, $t\mapsto (s(g(t)),t)$; it gives an isomorphism $g^{\ast }M\to N$. Compatibility with the original transport is the cocycle identity pulled back along $(s(g(t_1)),t_1,t_2)$. Diagonal transport is an invertible idempotent, hence the identity, and reverse transport is its inverse by the same cocycle. Pulling back a compatible map along $s$ recovers its unique map on $M$. Therefore descent is effective and fully faithful for this split cover. The argument applies equally to algebra transports.

The displayed $D$ is the kernel of the difference of two $A$-linear maps from $N$ into $B{\otimes }_{A}N$. Flat scalar extension preserves this kernel. After extending $A$ to $B$, the cover becomes $\mathrm{Spec}(B{\otimes }_{A}B)\to \mathrm{Spec}B$, with diagonal section supplied by multiplication $B{\otimes }_{A}B\to B$. The datum becomes split, so step 1.1 says its invariant module recovers its downstairs module, and the base extension of the natural map $B{\otimes }_{A}D\to N$ is an isomorphism. Faithful flatness in [F1] reflects that isomorphism, proving the displayed descent map is an isomorphism before extension.

For the canonical datum on $B{\otimes }_{A}Q$, the invariant equalizer is $Q$. Indeed after tensoring by $B$, the sequence $0\to Q\to B{\otimes }_{A}Q\rightrightarrows B{\otimes }_{A}B{\otimes }_{A}Q$ becomes split exact, using multiplication and the split-cover argument of step 1.1. Exactness and detection in [F1] give the original assertion. A compatible map $N\to N^{\mathrm{\prime }}$ preserves invariant equalizers; step 2.1 identifies it uniquely with the base extension of its restriction $D\to D^{\mathrm{\prime }}$. This proves full faithfulness as well as effectiveness for modules.

For algebra transport, the invariant subset is closed under unit, sums, scalar multiplication, and products, because transport preserves these operations. Thus $D$ is an $A$-algebra. The module isomorphism in step 2.1 preserves multiplication and unit and is an algebra isomorphism. Compatible algebra maps restrict to algebra maps on invariants by step 3.1, giving the asserted algebra equivalence and effective affine scheme descent through [F2]. AC is inherited from the module and affine-scheme suppliers.

For any faithfully flat ring map $A\to B$, $A\to B\rightrightarrows B{\otimes }_{A}B$ is an equalizer. Check exactness after the faithful flat tensor extension by $B$. The extended sequence is $0\to B\to B{\otimes }_{A}B\to B{\otimes }_{A}B{\otimes }_{A}B$, with the first arrow $b\mapsto b\otimes 1$. That arrow is split by multiplication. If $x=\sum b_i\otimes c_i$ has equal images in the triple tensor product, applying multiplication to its first two factors gives $x=(\sum b_i{c}_i)\otimes 1$, proving exactness. Flatness preserves kernels and cokernels, and their vanishing descends by [F1]; thus the original sequence is exact.

For an affine open $V\subset Z$, the open $W=(f^{\mathrm{\prime }}{)}^{-1}(V)$ is stable under the two relation projections. Any two points of $X^{\mathrm{\prime }}$ over the same point of $X$ lift to a common point of the fibre product: their residue-field tensor product over the base residue field is nonzero. Hence each fibre lies entirely in $W$ or entirely outside it. The image $U=p(W)$ is open by [F1], and $W=p^{-1}(U)$. Such $U$ cover $X$ as $V$ ranges over an affine target cover.

On an affine open $T=\mathrm{Spec}A\subset U$, choose a finite affine open cover of $p^{-1}(T)$; quasi-compactness gives finiteness. Its disjoint union is affine, say $\mathrm{Spec}B$, and maps faithfully flat to $T$, since restriction of $p$ to each open remains flat and the union is onto. The restriction of $f^{\mathrm{\prime }}$ to this cover, with affine target $V$, gives a ring map $\mathcal{O}(V)\to B$ whose two composites into $B{\otimes }_{A}B$ agree. By step 1.1 it takes values in $A$, yielding a unique morphism $T\to V$. The same equalizer shows that its pullback agrees with $f^{\mathrm{\prime }}$ on the whole $p^{-1}(T)$, by checking on these affine opens.

The descended morphisms agree on overlaps: their pullbacks agree, and the equalizer argument on affine source covers detects equality. Thus they glue uniquely on $X$. Conversely every pulled-back morphism satisfies compatibility. For a family of fppf covers, work over an affine open of $X$, take affine source opens whose open images cover it, and select finitely many by quasi-compactness. Their disjoint union is a faithfully flat affine refinement of the family to which the same argument applies. This proves the sheaf condition and uniqueness for represented functors. AC is inherited from [F1] and the scheme-affine-cover suppliers.

Work over an affine open $U=\mathrm{Spec}A\subset S$. Choose finitely many affine opens covering $S_U^{\mathrm{\prime }}$; their disjoint union is affine, $V=\mathrm{Spec}B$, and is a faithfully flat affine cover of $U$. The pullback $X_V$ is affine over $V$, say $\mathrm{Spec}C$. Its two pullbacks to $V{\times }_{U}V$ have the canonical isomorphism coming from the scheme $X_U$, and satisfy the cocycle. By [F1], $C$ descends to an $A$-algebra $D$, with $D{\otimes }_{A}B\cong C$ as algebras with datum.

This algebra isomorphism gives a compatible isomorphism $X_V\cong (\mathrm{Spec}D{)}_V$. The map from $X_V$ to $\mathrm{Spec}D$ descends along $X_V\to X_U$ by the morphism part of [F1], and the inverse map descends along $(\mathrm{Spec}D{)}_V\to \mathrm{Spec}D$. The resulting two maps are inverse because their composites become identities after the faithful cover and uniqueness in [F1] detects equality. Thus $X_U\cong \mathrm{Spec}D$, proving affineness of $f$ locally on its target and hence globally.

If $f_{S^{\mathrm{\prime }}}$ is finite, $C$ is a finite $B$-module. By [F2], $D$ is a finite $A$-module: equivalently express finitely many generators of $D{\otimes }_{A}B$ using finitely many tensor coefficients in $D$, let $D_0$ be their $A$-span, and use faithfulness to deduce $D\mathrm{/}{D}_0=0$. Hence $X_U\to U$ is finite. Finiteness is affine local on the target by this module description, so $f$ is finite. AC is inherited from [F1]–[F2]; quasi-compactness supplies the finite affine subcover used in step 1.1.

Fix $v\in M$ and write $\rho (v)={\sum }_{i=1}^r{v}_i\otimes a_i$ with the $a_i$ linearly independent. Let $C\subset A$ be the finite-dimensional space containing the $a_i$ and every second-factor coefficient of the finitely many $\mathrm{\Delta }(a_i)$. Choose linear functionals ${\lambda }_j:C\to k$ with ${\lambda }_j(a_i)={\delta }_{ij}$ by extending this finite independent list to a basis of $C$. Apply $1\otimes 1\otimes {\lambda }_j$ to coassociativity. It gives $\rho (v_j)={\sum }_i{v}_i\otimes (1\otimes {\lambda }_j)\mathrm{\Delta }(a_i)$. Thus $W={\mathrm{span}}_k\{v_1,\dots ,v_r\}$ is a finite-dimensional subcomodule. The counit gives $v={\sum }_iϵ(a_i)v_i\in W$. Finite sums of subcomodules are subcomodules, proving the filtered-union assertion. Both sides lie in $M\otimes A\otimes C$, so all these contractions are defined on finite coefficient spaces and require only finite choices.

Apply step 1.1 to the comodule $(A,\mathrm{\Delta })$ and to a finite algebra-generating list for $A$. Taking the sum gives a finite-dimensional subcomodule $V$ containing that list. In a basis $e_1,\dots ,e_n$ of $V$, write $\mathrm{\Delta }(e_j)={\sum }_i{e}_i\otimes a_{ij}$. Coassociativity and counit give $\mathrm{\Delta }(a_{ij})={\sum }_l{a}_{il}\otimes a_{lj}$ and $ϵ(a_{ij})={\delta }_{ij}$. The antipode gives an inverse for the matrix $(a_{ij})$. Hence these entries define a homomorphism $G\to \mathrm{GL}(V)$: for any $k$-algebra $R$ and point $g:A\to R$, its matrix is $(g(a_{ij}))$. This is the right regular action $(gf)(x)=f(xg)$, formulated on all algebras.

The image of $k[\mathrm{GL}(V)]\to A$ contains every $a_{ij}$. Applying $ϵ\otimes 1$ to $\mathrm{\Delta }(e_j)$ gives $e_j={\sum }_iϵ(e_i)a_{ij}$, so that image contains $V$, hence the algebra generators of $A$. The ring map is onto and therefore the group morphism is a closed immersion. In particular it is injective on $R$-points for every $R$, including nonreduced algebras.

First suppose $I^2=0$. The sheaf $I$ is a quasi-coherent module on $Y$, because $I$ annihilates itself. The closed immersion does not change the underlying topological space. Thus [F1] gives $H^1(X,I)=H^1(Y,I)=0$, and the exact sequence $0\to I\to {\mathcal{O}}_X\to {\mathcal{O}}_Y\to 0$ gives a surjection $A=\mathrm{\Gamma }(X,{\mathcal{O}}_X)\to B=\mathrm{\Gamma }(Y,{\mathcal{O}}_Y)$ with square-zero kernel.

Around each point choose an affine open $V\subset X$, and then a principal open $D(b)\subset Y=\mathrm{Spec}B$ contained in $V\cap Y$. Lift $b$ to $a\in A$ by step 1.1. Its nonvanishing open $X_a$ has underlying space $D(b)$, lies in $V$, and is the principal open of the restriction of $a$ to $V$, hence affine. By [F2], $\mathrm{\Gamma }(X_a,\mathcal{O})=A_a$: surjectivity follows by clearing powers of $a$, and a section of $A$ zero on $X_a$ is annihilated by a power of $a$, by the same extension/localization argument on the finite affine cover of $X$. Choose finitely many of these opens covering $X$. Their corresponding $D(a)\subset \mathrm{Spec}A$ cover that spectrum, since $\mathrm{Spec}A$ and $\mathrm{Spec}B$ have the same underlying space under the square-zero quotient. The canonical map $X\to \mathrm{Spec}A$ therefore is an isomorphism on this affine-open cover, and hence globally.

For general $I^m=0$, start with $X_1=Y$, and successively thicken to the closed schemes $X_j$ defined by $I^j$, for $j=2,\dots ,m$. The ideal of $X_{j-1}$ in $X_j$ is $I^{j-1}\mathrm{/}{I}^j$, whose square is zero because $2(j-1)\ge j$. Steps 1.1 and 2.1 show inductively that each $X_j$ is affine; $X_m=X$. AC is inherited from the cohomology and section-extension suppliers.

Flatness identifies ${\mathfrak{m}}^{j}B\mathrm{/}{\mathfrak{m}}^{j+1}B$ with $({\mathfrak{m}}^j\mathrm{/}{\mathfrak{m}}^{j+1}){\otimes }_{A\mathrm{/}\mathfrak{m}}(B\mathrm{/}\mathfrak{m}B)$, by tensoring the inclusions of the ideals of $A$. Multiplication by $f$ is injective on every such graded piece, since it is injective on the second factor and the first is a vector space. If $fb=0$, injectivity first gives $b\in \mathfrak{m}B$, then successively $b\in {\mathfrak{m}}^{j}B$ for every $j$. Krull intersection in the local ring $B$ gives $b=0$. The exact sequence $0\to B\stackrel{f}{\to }B\to B\mathrm{/}fB\to 0$ and $A$-flatness of $B$ identify ${\mathrm{Tor}}_1^A(A\mathrm{/}\mathfrak{m},B\mathrm{/}fB)$ with the kernel of $f$ on $B\mathrm{/}\mathfrak{m}B$, which is zero. The finite-over-target criterion [F1] proves that $B\mathrm{/}fB$ is $A$-flat. No finiteness over $A$ is used.

If $u(Y_z)$ is already finite, take $F=X$. Otherwise its image is constructible by applying [F2] on a finite affine cover of $Y_z$, and contains infinitely many closed points: a nonempty locally closed positive-dimensional piece has infinitely many closed points, while a zero-dimensional finite-type scheme has only finitely many points. Here $\kappa (z)\mathrm{/}k$ is finite, so $Y_z$ is finite type over $k$. Choose a closed image point $x$ different from the images of the finitely many associated points of $Y_z$. Write $X=\mathrm{Spec}C$ and ${\mathfrak{p}}_i$ for their image primes. Since the maximal ideal ${\mathfrak{m}}_x$ is not contained in any ${\mathfrak{p}}_i$, prime avoidance supplies $h\in {\mathfrak{m}}_x\setminus {\bigcup }_i{\mathfrak{p}}_i$. Thus $V(h)$ meets $u(Y_z)$ at $x$ and avoids all associated points after pullback. At every point of the cut over $z$, its defining element is regular in the fibre local ring, and step 1.1 proves flatness of the cut over $Z$.

Repeat inside the affine closed subscheme $V(h)$ if its fibre image is still infinite, using the new fibre's associated points. Each cut has nonempty fibre and is proper: its defining element avoids the fibre's associated points, so cannot vanish identically on that fibre. Thus the defining ideals in the original Noetherian affine ring strictly increase at every repetition. The ascending chain condition forces termination. The last image is finite and nonempty, and step 1.1 preserves the required flatness at each stage. AC is inherited from the associated-prime, Krull-intersection and local-flatness suppliers.

Choose a closed point $z\in X$ and an affine neighbourhood $E$ of the source of an arrow targeting $z$ (the identity arrow suffices). Apply [F1] to $s:t^{-1}(z)\cap s^{-1}(E)\to E$ and the flat map $t:s^{-1}(E)\to X$. It gives a closed $F\subset E$ with nonempty fibre and finite source image over $z$, while $a=t:s^{-1}(F)\to X$ is flat along that fibre. This fibre has finitely many points: over a fixed source point $f$ and target $z$, the possible product points lie in $\mathrm{Spec}(\kappa (f){\otimes }_k\kappa (z))$, which is finite over $\kappa (f)$ because $\kappa (z)\mathrm{/}k$ is finite. A monomorphism $R\to X\times X$ has at most one point over each such product point. The fibre of $a$ is therefore a finite-type $\kappa (z)$-scheme with finitely many points, hence zero-dimensional with finite residue fields. Thus $a$ is also quasi-finite at those fibre points.

Let $P\subset s^{-1}(F)$ be the locus where $a$ is flat and quasi-finite. Composition gives the following invariance: for arrows $f\to x$ and $x\to y$, composing identifies the space of arrows $f\to x$ with that of arrows $f\to y$ after base change to the arrow scheme parametrizing $x\to y$. The two target base changes use $s,t:R\to X$, both faithfully flat and of finite presentation. Flatness descends by [F2], and quasi-finiteness is detected by geometric fibre dimension, which is unchanged by residue-field extension. Consequently the two inverse images of $P$ on $s^{-1}(F){\times }_F{s}^{-1}(F)$ coincide. The map $s:s^{-1}(F)\to F$ is open and onto, so $P=s^{-1}(F^{\mathrm{\prime }})$ for an open $F^{\mathrm{\prime }}\subset F$. It contains the entire fibre over $z$. Replace $F$ by $F^{\mathrm{\prime }}$; now $a$ is flat, quasi-finite and separated everywhere. Its open image $D$ contains $z$ and is saturated by composition.

Inside $D$ take the union $W_z$ of all opens over which $a$ is finite. This open contains the generic points of every irreducible component of $X$ through $z$. Indeed those points lie in the open image $D$; over their Artinian local rings a quasi-finite finite-type separated scheme is finite. To see this, its reduced closed fibre is a finite discrete scheme, so its finitely many affine point neighbourhoods are disjoint and cover the scheme, and lifting finite module generators through the nilpotent maximal ideal proves module finiteness. A finite compactification from [F2], replaced by the schematic closure of its source, then has no boundary over that local scheme; the finite image of the closed boundary can be removed from a neighbourhood of the generic point, making $a$ finite there.

The finite locus just defined is invariant along $R$: its two inverse images are the finite loci of the two isomorphic base changes of $a$ given by composition. Here finiteness descends under our faithfully flat open covers. An explicit verification is as follows. If a separated quasi-finite finite-type map becomes finite after such a cover, it becomes universally closed. For every further base change and closed source subset, its image pulls back to a closed set on the covering target; an open surjective map detects closed sets, so that image is closed downstairs. The original map is therefore proper, and [F2] makes it finite. This also proves equality of the maximal finite loci, by descending each covering open's saturated image. Hence $W_z$ is saturated. Set $U_z=F^{\mathrm{\prime }}\cap W_z$. Saturation identifies $s^{-1}(U_z)$ with $a^{-1}(W_z)$, so its target map is finite, flat and onto. Its base change by $U_z\subset W_z$ is one projection of $R_{U_z}$, and inversion gives the other.

If $W_z$ is not dense, repeat in the interior of $X\setminus W_z$. This interior is saturated: openness of the relation projections implies that the closure of a saturated subset is saturated, since the inverse image of its closure equals the closure of its inverse image for an open map. Each repetition meets a previously missed irreducible component at its generic point, and there are only finitely many components. We obtain finitely many disjoint $W_i$ with dense union. Finally $U_i$ is open in the affine $F$: for any finite subset, the ideal defining the complement of $U_i$ avoids its point primes; prime avoidance gives a principal open in $F$ containing the subset and contained in $U_i$. This proves the affine-neighbourhood assertion. AC is inherited from [F1]–[F2].

The rank of the outgoing-relation fibre is locally constant on $U$ and constant along every relation arrow: composing with that arrow and its inverse gives mutually inverse maps between the outgoing fibres, preserving their target point. Thus the finitely many constant-rank clopen pieces of $U$ are saturated. Their defining idempotents satisfy $s(e)=t(e)$ and belong to $C$. Split by them; it suffices to prove the claim when the two ranks have one positive constant value $r$. Positivity follows from the identity arrow. For $a\in A$, take the characteristic polynomial of multiplication by $t(a)$ on the locally free $A$-module $B$ via $s$. Its coefficients lie in $C$: after either pullback to $R$, the composition/inverse isomorphism between the two outgoing-relation fibres preserves target evaluation $t(a)$, so the corresponding multiplication operators are conjugate and have the same characteristic polynomial. Cayley–Hamilton holds here over the coordinate ring: locally write $\mathrm{adj}(ZI-T)={\sum }_{j=0}^{r-1}{B}_j{Z}^j$ for the multiplication matrix $T$. The adjugate identity in [F1] yields $-T{B}_0=c_{0}I$, $B_{j-1}-T{B}_j=c_{j}I$, and $B_{r-1}=I$; multiplying by successive powers of $T$ and adding telescopes to ${\sum }_{j=0}^r{c}_j{T}^j=0$. This identity glues for the locally free module. Follow it by the identity-arrow map $B\to A$ to obtain a monic equation for $a$ over $C$.

Hence $A$ is integral over $C$. Since $A$ is generated by finitely many elements over $k\subset C$, it is a finite $C$-module. Also $C$ is a finite-type $k$-algebra: choose finite $C$-module generators ${\alpha }_j$ of $A$, express their products, the unit, and finite $k$-algebra generators of $A$ in this generating family. Let $C_0\subset C$ be generated over $k$ by those finitely many coefficients. The $C_0$-span of the ${\alpha }_j$ contains the unit and algebra generators and is closed under multiplication, so equals $A$. The Noetherian ring $C_0$ makes its submodule $C\subset A$ finite over $C_0$. Thus $C$ is finite type and Noetherian. Injective integrality makes $\mathrm{Spec}A\to \mathrm{Spec}C$ onto by lying-over.

The map $j:R\to U\times U$ is finite: its graph into $R\times U$ is closed, and the map of that product to $U\times U$ is the base change of one finite projection. It is a monomorphism by the equivalence-relation assumption. A finite monomorphism is a closed immersion here: after any residue-field base change its affine coordinate algebra $D$ has diagonal multiplication $D\otimes D\to D$ an isomorphism; finite dimension gives $(\dim D{)}^2=\dim D$, so $D$ is zero or the residue field. Nakayama applied to the finite cokernel of the original ring map then gives surjectivity. Consequently $A{\otimes }_{C}A\to B$, $a\otimes b\mapsto s(a)t(b)$, is onto.

Localize at any prime of $C$ and faithfully flat extend this local ring, if necessary, to $C[T{]}_{\mathfrak{m}C[T]}$, whose residue field is the infinite field $\kappa (\mathfrak{m})(T)$. Formation of $C$ as the kernel of $s-t$ commutes with this flat extension. The extended $A$ is finite over the local extended $C$, so is semilocal, and $B$ is finite projective of rank $r$ over $A$ via $s$. Such a module is free: choose residue-field bases at the finitely many maximal ideals, lift them simultaneously by the Chinese remainder theorem, use Nakayama to obtain a surjection $A^r\to B$, split it by projectivity, and apply Nakayama to its finite kernel.

In this semilocal situation, the $C$-submodule $t(A)$ generates $B$ as an $s(A)$-module by step 3.1. It contains an $A$-basis of $B$. To prove this, reduce modulo the Jacobson radical of $A$ and consider the finitely many residue-field vector spaces of dimension $r$. Choose finitely many elements of $t(A)$ which generate $B$ over $A$. Because the residue field of $C$ is infinite and maps into every residue field of $A$, a linear combination with coefficients in that common field can be chosen nonzero in every factor: each forbidden condition is a proper linear subspace, and finitely many such subspaces cannot cover a vector space over an infinite field. Lift the coefficients to $C$. The resulting element generates a free direct summand of rank one, by Nakayama and the splitting argument of step 3.2. Apply the same argument to the quotient, and induct on $r$. Lifting the quotient basis elements from $t(A)$ gives a basis $t(x_1),\dots ,t(x_r)$ of $B$ over $s(A)$.

Write $t(a)={\sum }_{i}s(c_i)t(x_i)$, with $c_i\in A$. Let $c:B\to B{\otimes }_{s,A,t}B$ be the composition comorphism. The groupoid identities give $c(t(a))=t(a)\otimes 1$ and $c(s(a))=1\otimes s(a)$. Comparing composition of the displayed expansion with its first-factor pullback, and using $s(c_i)\otimes 1=1\otimes t(c_i)$, gives ${\sum }_{i}t(x_i)\otimes (s(c_i)-t(c_i))=0$. Since the $t(x_i)$ are a basis in the first tensor factor, all $c_i$ lie in $C$. Applying the identity-arrow map shows $a={\sum }_i{c}_i{x}_i$. Independence follows by applying $t$ and the basis independence in $B$. Therefore $A={⨁}_{i}C{x}_i$, and the surjection in step 3.1 is an isomorphism, since it takes the corresponding basis in the second tensor factor to $t(x_i)$.

The conclusions in step 5.1 descend to each local ring of the original $C$ by faithful flatness in [F1]. Thus $A{\otimes }_{C}A\to B$ is an isomorphism globally, and $A$ is flat over $C$. It is finite and finitely presented over the Noetherian $C$, so is finite locally free by [F1]. It is faithful because its spectrum is onto by step 2.1. The kernel pair is now exactly $R$. Every morphism into $M$ lifts after base change by the finite locally free covering $U\to M$; any two local lifts differ by the kernel pair. Since $M$ is an fppf sheaf by [F2], these statements identify it with the fppf quotient sheaf. Recombine the saturated rank pieces from step 1.1 to finish. AC is inherited from [F1] and the prime/basis selections.

The saturation $s(t^{-1}(X\setminus V))$ is closed, since $s$ is finite, and is a union of entire orbits by composition. Let $V^{\mathrm{\prime }}$ be its complement. It is the largest saturated open contained in $V$ and contains $E$. Write $V=\mathrm{Spec}A$. The closed subset $V\setminus V^{\mathrm{\prime }}$ is defined by an ideal $I$. No prime of any point of the finite orbit $E$ contains $I$, so [F2] gives $f\in I$ nonzero at every point of $E$. Hence $V_f\subset V^{\mathrm{\prime }}$ and contains $E$.

On the saturated $V^{\mathrm{\prime }}$, the restricted relation is still finite locally free. Its norm $N={\mathrm{Norm}}_s(t^{\ast }f)$ is a regular function on $V^{\mathrm{\prime }}$. Its nonvanishing locus consists exactly of the points all of whose relation targets lie in $V_f$: on a residue-field fibre the determinant is nonzero exactly when multiplication by $t^{\ast }f$ is invertible in its finite algebra, equivalently when that element vanishes at none of the fibre's points. Thus this locus is saturated and contains $E$. It is contained in $V_f$, because the identity arrow is one of those targets. Therefore it equals the principal nonvanishing locus of the restriction $N{\mathrm{\mid }}_{V_f}$ in the affine scheme $V_f$, and is affine. It is the required $W$. The norm invariance in [F1] also verifies saturation scheme theoretically. AC is inherited from [F1]–[F2].

By [F1], cover $X$ by saturated affine opens $W_i$ and let $Y_i$ be their affine finite locally free quotients. For each intersection $W_i\cap W_j$, its image under $W_i\to Y_i$ is open, since finite locally free maps are open, and its inverse image is exactly the intersection by saturation. That open represents the quotient of the restricted relation, by [F2] and the local lifting/kernel-pair description. The corresponding open in $Y_j$ represents the same sheaf, so is uniquely isomorphic. These isomorphisms satisfy the cocycle identity by uniqueness; glue the $Y_i$ along them to a scheme $Y$.

The local quotient maps glue. Finite local freeness is local on the target, so $q:X\to Y$ is finite locally free and onto, and the local kernel-pair isomorphisms give $R=X{\times }_{Y}X$. Since every target point lifts after the covering $q$, and two lifts agree in the quotient precisely when related by that kernel pair, [F2] identifies $Y$ with the fppf quotient. A finite affine subcover of $X$ by the $W_i$ gives a finite affine cover of $Y$ by finite-type $k$-algebras, so $Y$ is finite type.

The image of the closed diagonal of $X$ under the finite closed map $q\times q$ is the diagonal of $Y$ as a subset, since $q$ is onto. Thus that diagonal has closed image. A finite-type $k$-scheme is locally separated: around every point of its diagonal, choose an affine open $T$ containing the point, and the restriction of the diagonal to $T\times T$ is closed. Hence its diagonal is an immersion; closed image makes this immersion closed, by checking the affine quotient ideals on those neighbourhoods and the open complement of the image. Therefore $Y$ is separated. AC is inherited from [F1]–[F2].

The source and target projections of $R_U$ are finite locally free: its target projection is the base change of $V\to X$ by $U\subset X$, and inversion exchanges the two. By [F1] it has a scheme quotient $Y$, with $U\to Y$ finite locally free and onto. The morphism $V\stackrel{s}{\to }U\to Y$ has equal pullbacks along $V{\times }_{X}V$: two arrows with equal target determine, by composition and inverse, a relation arrow between their source points in $U$. Therefore [F2] descends it to a morphism $q:X\to Y$.

As fppf sheaves, the quotient of $X$ by $R$ equals that of $U$ by $R_U$. Indeed every point of $X$ lifts locally along $V\to X$ and is then related to its source point in $U$, proving local surjectivity of the latter quotient into the former. Two points of $U$ are identified exactly when related by $R_U$, by restriction of the original equivalence relation. Thus [F2] gives $Y\cong X\mathrm{/}R$. Since $R$ is an equivalence relation, its arrows are unique when source and target are specified, so this equality of sheaves identifies $R$ with the representable kernel pair $X{\times }_{Y}X$. In particular $X{\times }_{Y}U\cong V$, by the arrow/source description.

Base change of $q$ by the finite locally free cover $U\to Y$ is $V\to U$, which is flat as a base change of the original relation's source map. Thus [F2] makes $q$ flat. It is onto since the composite $U\to X\to Y$ is onto, and is of finite presentation: both $X$ and $Y$ are finite-type $k$-schemes, so any $k$-morphism between them is finite type, and over their Noetherian affine charts finite type implies finite presentation. Therefore $q$ is faithfully flat of finite presentation with the asserted kernel pair and quotient. AC is inherited from [F1]–[F2].

Apply [F1] and write its dense open as the finite disjoint union of saturated opens $W_i$, with quasi-sections $U_i$. The induced relation has finite locally free projections, so every orbit is finite; [F1] puts it in an affine open of $U_i$. Thus every hypothesis of [F2] holds on each $W_i$, and it supplies a finite-type scheme $Y_i$ representing $W_i\mathrm{/}{R}_{W_i}$, with the asserted projection and kernel pair.

Set $Y={\coprod }_i{Y}_i$. Since the $W_i$ are disjoint and saturated there are no relation arrows between different pieces, so this disjoint union represents the fppf quotient of $W={\coprod }_i{W}_i$. Flatness, finite presentation, surjectivity and the kernel-pair identity hold piecewise and hence globally. AC is inherited from [F1]–[F2].

The datum makes $D=Y{\times }_{\mathrm{Spec}k}\mathrm{Spec}K$ into a relation on the underlying $k$-scheme $Y$: its first map is projection, and its second map uses the given isomorphism between the two $K{\otimes }_{k}K$ base changes. Both projections are finite locally free of rank $[K:k]$. The cocycle gives composition, and the diagonal and exchange of the two scalar factors give identity and inverse. The map $D\to Y{\times }_{k}Y$ is a monomorphism: the source point together with the scalar structure of the target uniquely determines the scalar point in the second factor, and the datum then uniquely determines the target. Thus [F1] gives $p:Y\to Y_0=Y\mathrm{/}D$, finite locally free, onto, with kernel pair $D$.

The map $(p,\text{structure}):Y\to Y_0{\times }_{k}K$ becomes an isomorphism after the faithfully flat cover $Y\to Y_0$: its pullback is $Y{\times }_{Y_0}Y=D$, which is exactly $Y{\times }_{k}K$, with the isomorphism supplied by the datum. Its inverse descends by [F2], so the map itself is an isomorphism. The same morphism descent gives uniqueness and descent of compatible morphisms. This uses the entire cocycle over the tensor algebras; automorphism invariance alone is insufficient for inseparable $K\mathrm{/}k$. AC is inherited from [F1]–[F2].