The Stacks project

Definition 39.9.1. Let $k$ be a field. An abelian variety is a group scheme over $k$ which is also a proper, geometrically integral variety over $k$1.

[1] For equivalent definitions see Remark 39.9.12.

Comments (6)

Comment #78 by Keenan Kidwell on

The section title should be Picard groups of curves, right?

Comment #7838 by Zhenhua Wu on

There are equivalent definitions of abelian varieties. Let {projective, proper}, {geometrically irreducible, irreducible, geometrically connected, connected}, {smooth, geometrically reduced} be three sets of properties, pick one from each of them, and let be a group scheme with the chosen properties. Then it agrees with the definition of abelian variety. I suggest we add this somewhere just for the sake of new students who read too many books and get confused with the different definitions of abelian varieties. Actually I have the following proof in my thesis.

Proof: Clearly projectivity implies properness; and geometrically irreducibility implies the other three which in turn implies connectivity. By tag 056T we know that smoothness of K-schemes implies geometrically reducedness. Thus it suffices to show that "abelian varieties are projective, geometrically irreducible and smooth", and "proper connected and geometrically reduced group schemes over is an abelian variety".

By tag 0BF9, abelian varieties are projective and smooth. Geometrically irreducibility comes from the definition. Next we show that proper connected and geometrically reduced group schemes over are abelian varieties, i.e. they are proper and geometrically integral.

It suffices to show geometrically irreducibility. A group scheme over the field must contain a -rational point, the unit section. In this case, is connected implies it is geometrically connected by tag 04KV. Hence after base change to any field extension of , is connected. Plus, since every connected group scheme over a field is irreducible by tag 0B7Q, is irreducible. So is geometrically irreducible. The result follows.

Comment #8117 by Zhenhua Wu on

Cool! But have you added this remark on the site? I don't see it in this section.

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