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Connected smooth quasiprojective model with proper special fibre is proper

Statement

Assume AC and DC as inherited from the stated suppliers. Let $R$ be a discrete valuation ring and let $G$ be a smooth separated finite-type quasi-projective $R$ -scheme whose generic fibre is an abelian variety. If the special fibre $G_{k}$ is proper and geometrically connected, then $G$ is proper over $R$ . Applied to the connected identity model of a smooth group scheme with abelian generic fibre, $G$ becomes an abelian scheme.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with residue field $k$ , uniformizer $t$ , and a smooth separated finite-type quasi-projective $R$ -scheme $G$ with abelian generic fibre and proper geometrically connected special fibre.

[F1]

The quasi-projective identity model with abelian generic fibre is Divisor ampleness and quasi-projectivity of group models; schematic closures are contracted from the generic fibre, and a smooth scheme is flat hence schematically dense (Schematic closure and agreement on a dense open, Over a principal ideal domain flatness is equivalent to torsion-freeness, Every DVR is a PID).

[F2]

The perfect-complex machinery for a proper flat finitely presented morphism: the cohomology of $O$ is computed by a bounded finite projective complex concentrated in degrees $\geq 0$ , with naturality in base algebras (Universal finite projective cohomology complex over any base); Noetherian completion is flat and faithfully flat, and properness descends along faithfully flat maps (The completion of a Noetherian ring is flat, Completion of a finite module is extension of scalars, A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra, Properness descends through fpqc base change); a morphism from a proper source to a separated target is proper (Morphisms from a proper scheme to a separated one are proper), and proper integral fibres have constant functions (Global functions on proper integral schemes form a finite extension of the base field, Projective coherent finiteness and large twist vanishing).

Proof

technique · direct: prove properness over the completion by a connectedness/idempotent argument on the projective closure, then descend to $R$

1.1F1F2givenalgebra

Properness descends along the faithfully flat completion $R \to \hat{R}$ by [F2], so it suffices to treat a complete DVR. In that case view $G$ as an open subscheme of its projective schematic closure $X$ in $P_{R}^{m}$ by [F1]; the closure is $t$ -torsion-free on coordinate charts because its ideal is contracted from the generic fibre, so $X$ is flat over $R$ , and $X_{K} = G_{K}$ because the generic fibre $G_{K}$ is proper, hence closed in projective space. The finite $R$ -module $H^{0} (X, O_{X})$ injects into $H^{0} (X_{K}, O) = K$ by flatness, and every element is integral over the integrally closed ring $R$ , so $H^{0} (X, O_{X}) = R$ .

2.1F2step 1.1algebra

If $X_{k}$ were disconnected, a nontrivial clopen partition would define compatible idempotents $e_{n} \in H^{0} (X_{n}, O)$ on the nilpotent thickenings $X_{n} = X \times_{R} R / t^{n + 1}$ , which share the same underlying space. Applying the finite projective complex of [F2] to the proper flat $X$ and $O_{X}$ gives $H^{0} (X_{n}, O) = \ker (C_{0} / t^{n + 1} \to C_{1} / t^{n + 1})$ ; finite projective modules over complete $R$ are complete and inverse limits preserve kernels, so $\underset{\leftarrow}{l i m} H^{0} (X_{n}, O) = \ker (C_{0} \to C_{1}) = H^{0} (X, O) = R$ . The compatible idempotents would therefore produce a nontrivial idempotent in $R$ , impossible; hence $X_{k}$ is connected.

3.1F2step 2.1algebra

The open immersion $G_{k} \to X_{k}$ has proper source and separated target, hence is proper by [F2], so its image is closed and open and nonempty in the connected $X_{k}$ ; it is therefore all of $X_{k}$ , and $G_{k} \to X_{k}$ is an isomorphism. Since the generic fibres already coincide, the closed complement $X ∖ G$ has empty fibres over both points of $Spec R$ , hence is empty and $G = X$ is proper over $R$ .

4.1F1F2step 3.1algebra∎

Applied to the identity component $G = H^{0}$ of a smooth separated finite-type $R$ -group scheme $H$ with abelian generic fibre: $H^{0}$ is quasi-projective (Divisor ampleness and quasi-projectivity of group models), smooth separated finite type with geometrically connected fibres, so if its special fibre is proper then $G$ is proper by steps 1.1 and 2.1 and is an abelian scheme over $R$ . This lemma does not claim that arbitrary smooth connected group models are quasi-projective, and the argument retains the AC and DC assumptions of the perfect-complex supplier.

Depends on

Used by

The Neron-Ogg-Shafarevich criterion in residue characteristic prime to l Theorem

Dependency tree · two levels

157 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bosch, Lutkebohmert, Raynaud, Neron Models (1990), 6.5 and EGA IV3 15.7.10 (properness from the special fibre); local proof in owner-arithmetic-models/neron-source/closure-supplement.md (standard reference, not scraped)