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Base change and products of abelian schemes

Statement

Assume AC, inherited from the smoothness and properness stability suppliers. Let $A \to S$ and $B \to S$ be abelian schemes over $S$ of relative dimensions $g_{A}, g_{B}$ (Abelian schemes over a base), and let $S^{'} \to S$ be a morphism (Base change of objects, morphisms and properties). Then:

(a) the base change $A_{S^{'}} = A \times_{S} S^{'} \to S^{'}$ is an abelian scheme of relative dimension $g_{A}$ , with $S^{'}$ -group structure induced by that of $A$ , and for $s^{'} \in S^{'}$ with image $s \in S$ the fibre $(A_{S^{'}})_{s^{'}}$ is the base change $A_{s} \times_{κ (s)} κ (s^{'})$ of abelian varieties (Fibres after base change);

(b) the product $A \times_{S} B \to S$ is an abelian scheme of relative dimension $g_{A} + g_{B}$ with the product group law;

(c) kernels of homomorphisms of abelian schemes commute with arbitrary base change by their fibre-product definition, and the base change of a finite locally free subgroup scheme is again finite locally free of the same rank.

Scheme-theoretic images are not asserted to commute with arbitrary base change.

Facts & Assumptions

Given: AC and abelian schemes $A \to S$ , $B \to S$ of relative dimensions $g_{A}, g_{B}$ , and a morphism $S^{'} \to S$ .

[F1]

An abelian scheme is a smooth proper finitely presented $S$ -group scheme with connected geometric fibres of constant dimension (Abelian schemes over a base); assuming AC for the named stability suppliers, smoothness, properness, local finite presentation, flatness and relative dimension are stable under base change and preserved by products (Flatness is stable under arbitrary base change, Smoothness survives base change and composition, Properness survives arbitrary base change, Local finiteness conditions under base change, Relative dimension of a smooth morphism at a point).

[F2]

Fibres of a base change are computed by the fibre product of fibres (Fibres after base change); the group operations of an $S$ -group scheme base change to give the induced $S^{'}$ -group structure, and products inherit the componentwise group law.

Proof

technique · direct: check the defining properties of an abelian scheme after base change and for products

1.1F1F2givenalgebra

The base change $A_{S^{'}} \to S^{'}$ is smooth, proper and locally of finite presentation by the stability statements in [F1]; its geometric fibres are base changes of geometric fibres of $A \to S$ , hence nonempty and connected of dimension $g_{A}$ , and the relative dimension is $g_{A}$ . The base-changed group operations give an $S^{'}$ -group scheme structure. For a point $s^{'} \mapsto s$ the fibre identification is the base-change compatibility of fibres in [F2].

1.2F1givenalgebra

For the product, $A \times_{S} B \to S$ is smooth, proper and locally of finite presentation, its geometric fibres are products of nonempty connected smooth proper schemes over an algebraically closed field, and a product of nonempty connected schemes over an algebraically closed field is connected (indeed the product of geometrically connected schemes over a field with a rational point in the appropriate sense is geometrically connected); the relative dimension is $g_{A} + g_{B}$ . The componentwise group law makes it an $S$ -group scheme. This proves (b).

2.1F1F2step 1.1algebra∎

Kernels of $S$ -group scheme homomorphisms are defined by the fibre product with the unit section, so they commute with arbitrary base change by associativity of fibre products, as asserted in (c); a finite locally free subgroup scheme of rank $r$ pulls back to a finite locally free subgroup scheme of the same rank because finite locally free modules and isomorphisms pull back along the base change. Scheme-theoretic images are not claimed to commute with arbitrary base change, and nothing here asserts that.

Depends on

Used by

Dependency tree · two levels

43 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

B. Edixhoven, G. van der Geer, B. Moonen, Abelian Varieties (preliminary version 2012), Chapter 6 sections 1-3 (standard reference, not scraped)