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Prime to characteristic multiplication is etale

Statement

Assume AC and DC as inherited from the stated suppliers. Let $S$ be a locally Noetherian scheme, let $G \to S$ be a smooth, separated, commutative group scheme of finite type over $S$ (Group schemes over a base scheme, Smooth morphism of schemes), and let $n \geq 1$ be invertible on $S$ (that is, a unit of $O_{S}$ locally). Then the multiplication-by- $n$ endomorphism $[n] : G \to G$ and the kernel $G [n] \to S$ are etale. If moreover $S = Spec R$ for a discrete valuation ring with strict henselization $R^{s h}$ and separably closed residue field $k^{s}$ and $k^{s}$ has characteristic not dividing $n$ , then reduction gives a bijection $G [n] (R^{s h}) \to G [n] (k^{s})$ .

Facts & Assumptions

Given: AC and DC, a locally Noetherian base $S$ , a smooth separated commutative finite-type $S$ -group scheme $G$ , an integer $n$ invertible on $S$ , and, for the last clause, a DVR $R$ with strict henselization $R^{s h}$ and separably closed residue field $k^{s}$ .

[F1]

On a smooth group scheme the tangent space at every point is identified with the translation of the tangent space at the identity, and the differential of a group homomorphism is translation-equivariant; the differential of $[n]$ at the identity is $n$ times the identity because $[n]$ is the sum of $n$ copies of the identity morphism in the group law (Group schemes over a base scheme, Smooth morphism of schemes).

[F2]

On smooth schemes of equal relative dimension the relative Jacobian criterion makes a morphism etale exactly where its differential determinant is invertible; standard smooth presentations and base change give the same over each affine open of the base (Relative Jacobian criterion with its presentation hypothesis, Base change and composition of standard smooth presentations, Differentials of a smooth morphism, Étale equals flat and unramified in finite presentation).

[F3]

Over a strictly henselian local ring with separably closed residue field, reduction is a bijection on points of a separated etale finite-type scheme (Strict henselian etale sections, Unramified residue extensions are finite separable).

Proof

technique · direct: compute the differential of multiplication by $n$, apply the Jacobian criterion, and specialise

1.1F1givenalgebra

At the identity the differential $d [n]_{e}$ is multiplication by $n$ on the tangent space, because $[n]$ is the composite of the $n$ -fold group law and the differential of the group law at the identity is addition; translation identifies the tangent space at every other point with the tangent space at the identity and conjugates $d [n]$ at that point with the corresponding tangent map, so the differential of $[n]$ is everywhere multiplication by the unit $n$ on the locally free tangent sheaf.

2.1F2step 1.1algebra

Since $G$ is smooth over $S$ of constant relative dimension on each connected component and $[n]$ is a morphism between smooth schemes of the same relative dimension, the relative Jacobian criterion in [F2] applies: $[n]$ is etale exactly where the determinant of its differential is a unit, which by step 1.1 holds everywhere since $n$ is invertible on $S$ . Hence $[n]$ is etale. The scheme $G [n]$ is the pullback of $[n]$ along the identity section, so it is etale over $S$ as a base change of an etale morphism; it is separated and of finite type because $G$ is.

3.1F2step 2.1algebra

Negative $n$ is handled by composing with the inversion, which is an isomorphism of $G$ over $S$ . The graph-Jacobian computation of step 2.1 works over each affine open of $S$ , so it does not need $S$ to be a DVR or Noetherian beyond the local Noetherian hypothesis.

4.1F3step 3.1algebra∎

In the DVR case, $G [n] \to Spec R$ is separated etale of finite type, and after the base change to the strictly henselian ring $R^{s h}$ with separably closed residue field $k^{s}$ the general section result [F3] gives that reduction $G [n] (R^{s h}) \to G [n] (k^{s})$ is bijective. This specialization statement is asserted only in this DVR/strictly henselian setting.

Depends on

Used by

Dependency tree · two levels

78 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), 7.3/2(b) (multiplication by n prime to the residue characteristic) (standard reference, not scraped)