Effective ample-pair and group descent from a strict henselization

For an ample pair $(X^{\mathrm{\prime }},{\mathcal{L}}^{\mathrm{\prime }})$ with its compatible pair descent datum, form the graded section algebra $B={⨁}_{n\ge 0}\mathrm{\Gamma }(X^{\mathrm{\prime }},{\mathcal{L}}^{\mathrm{\prime }\otimes n})$. Flat base change of sections on a finite affine cover and its intersections follows by tensoring their equalizer; hence the pair datum induces compatible descent data on each graded piece and on multiplication. Ampleness gives a finite cover of $X^{\mathrm{\prime }}$ by nonvanishing opens of positive-degree sections, each quasi-affine. These opens, rather than an asserted identity with $\mathrm{Spec}B$, will construct the descended scheme.

By effective affine algebra descent [F3] applied to the graded pieces, the datum descends $B$ to a graded $R$-algebra $B_0$ with $R^{\mathrm{s}\mathrm{h}}{\otimes }_R{B}_0\cong B$ compatibly with the datum. Every section of ${\mathcal{L}}^{\mathrm{\prime }\otimes n}$ over $R^{\mathrm{s}\mathrm{h}}$ is a finite sum of $R^{\mathrm{s}\mathrm{h}}$-multiples of descended sections of the same degree, because the tensor identification is an isomorphism of graded modules; hence if a section generates ${\mathcal{L}}^{\mathrm{\prime }}$ at a point, at least one descended section does too.

Here is the required morphism descent for the faithfully flat quasi-compact base extension $R\to R^{\mathrm{s}\mathrm{h}}$, without a local finite presentation assumption on that extension. Given a compatible morphism $v^{\mathrm{\prime }}:P_{R^{\mathrm{s}\mathrm{h}}}\to Q_{R^{\mathrm{s}\mathrm{h}}}$ between descended schemes, cover $Q$ by affine opens $W$. The opens $(v^{\mathrm{\prime }}{)}^{-1}(W_{R^{\mathrm{s}\mathrm{h}}})$ are saturated under the cover's kernel pair because the two pullbacks of $v^{\mathrm{\prime }}$ agree. Their images are open in $P$ by fpqc submersivity, and saturation makes their pullbacks exactly the original opens. On an affine open $V$ in such an image, $v^{\mathrm{\prime }}$ corresponds to a ring map $\mathrm{\Gamma }(W,\mathcal{O})\to \mathrm{\Gamma }(V,\mathcal{O}){\otimes }_R{R}^{\mathrm{s}\mathrm{h}}$. Compatibility puts its image in the faithfully flat equalizer $\mathrm{\Gamma }(V,\mathcal{O})$, by [F3]; it therefore descends a unique morphism $V\to W$. The descended maps agree on overlaps because their pullbacks agree, and glue to $v:P\to Q$. This also descends isomorphisms by descending their inverses. A compatible open immersion is a stable open upstairs, whose open image descends by the same saturated-open argument, and the induced isomorphism descends as just proved. No fppf assertion is applied to $R\to R^{\mathrm{s}\mathrm{h}}$.

For each descended homogeneous section $s$ of positive degree, its nonvanishing locus $D(s)\subseteq X^{\mathrm{\prime }}$ is a quasi-affine open subscheme stable under the descent datum, and it descends: embed $D(s)$ into $\mathrm{Spec}$ of its global-section algebra, descend that algebra by [F3], and take the image of this stable open in the descended spectrum under the faithfully flat spectrum map; stability makes the inverse image of the image exactly $D(s)$, and submersivity [F4] makes the image open. These descended opens glue compatibly because compatible morphisms and open immersions descend by step 3.1, producing a finite-type $R$-scheme $X$; the invertible sheaf descends on each affine chart by module descent and glues by uniqueness of descent, giving $\mathcal{L}$ on $X$ with $(X,\mathcal{L}{)}_{R^{\mathrm{s}\mathrm{h}}}\cong (X^{\mathrm{\prime }},{\mathcal{L}}^{\mathrm{\prime }})$. To verify ampleness downstairs, cover each quasi-affine $X_s$ by distinguished affine opens of its global-section spectrum contained in $X_s$. Their defining functions extend after multiplication by powers of $s$ by [F2]; multiplying once more by $s$ makes their global nonvanishing loci lie inside $X_s$, where they are exactly those affine opens. These affine section loci cover $X$, so $\mathcal{L}$ is ample by its definition. This proves (a).

In (b), choose a fibre-dense affine open $V\subseteq U$ downstairs using the codimension-one neighbourhood lemma in [F2]. Its pullback $V^{\mathrm{\prime }}\subseteq H$ is stable under the descent datum. On the regular smooth $R^{\mathrm{s}\mathrm{h}}$-scheme $H$, its reduced complement is an effective Cartier divisor $D$ with no vertical components by [F2]; equivalently it is the closure of its generic boundary. Stability of $V^{\mathrm{\prime }}$ makes $D$ and $\mathcal{O}(D)$ stable with their canonical pair cocycle. Since $H$ has abelian generic fibre and $V^{\mathrm{\prime }}$ is affine and fibre-dense, the ampleness theorem in [F2] makes $\mathcal{O}(D)$ ample. Apply (a) to this ample pair to obtain the scheme $G$ with its given descent datum. Multiplication, inverse and unit now descend as compatible morphisms by step 3.1, and their group identities hold downstairs since they hold after the faithfully flat extension. The original stable open $U_{R^{\mathrm{s}\mathrm{h}}}$ descends to the specified $U\subseteq G$ by morphism and open-immersion descent.

The descended $G$ is finite type and separated by fpqc descent of these properties [F4], and flat over $R$ because $H$ is flat and flatness descends [F4]; since $R$ is Noetherian and $G$ is finite type, $G$ is locally of finite presentation. For each residue field, geometric regularity of the fibres descends along the faithfully flat map $R\to R^{\mathrm{s}\mathrm{h}}$ by [F4]; the smoothness criterion now gives that $G$ is smooth over $R$. The descended open $U\subseteq G$ is open by descent of open immersions and is fibre-dense because fibre-density is checked after the faithfully flat base change and stabilized by the datum.

Finally consider the case (c): $H$ is the group completion of the strict law on $U_{R^{\mathrm{s}\mathrm{h}}}$, and $U$ is the stable fibre-dense open model of the given abelian generic variety. Its generic completion is that abelian variety, so $H$ has the abelian generic fibre required in (b). Over each of the two pullbacks $R^{\mathrm{s}\mathrm{h}}{\otimes }_R{R}^{\mathrm{s}\mathrm{h}}$ and the triple tensor product, the pulled-back completions solve the same strict birational law; by the uniqueness part of [F1] they are canonically isomorphic, so the canonical isomorphism on $U$ extends uniquely to a group isomorphism of completions and the triple cocycle holds automatically since the triple-pullback completion is unique. This extends the datum on $U_{R^{\mathrm{s}\mathrm{h}}}$ uniquely to a group descent datum on $H$, which is effective by (b); the descended open remains fibre-dense by step 6.1.