There is a well-known classification of finite abelian groups into products of cyclic groups.

What about finite abelian group schemes, where we may put in the qualifiers "affine", "etale", or "connected" if it helps?

There are some easy examples showing that the theory is richer than that of groups:

  • The constant group scheme $\mathbb{Z}/n\mathbb{Z}$,
  • The roots of unity $\mu_n$,
  • The group $\alpha_p$ over $k$ of characteristic $p$.

One might hope to obtain more finite abelian group schemes from abelian varieties, but over fields of characteristic $p$, the $p$-torsion of an abelian variety is a product of the finite group schemes already mentioned. We also know that any affine etale finite abelian group scheme becomes constant after base change.

Are there other (fundamentally different) examples of finite abelian group schemes, and is there some sort of classification of them all?

  • $\begingroup$ Just to make sure I know what you mean: By finite you mean having a finite dimensional algebra of regular functions, right? $\endgroup$ – Tobias Kildetoft Jan 3 '13 at 19:19
  • $\begingroup$ Yes. I mean "finite" in the same sense as "finite scheme over the base," not "finite group." $\endgroup$ – Tony Jan 3 '13 at 19:22
  • $\begingroup$ Over a field or over an arbitrary ring ? $\endgroup$ – user18119 Jan 4 '13 at 21:11
  • $\begingroup$ I'm happy with just fields. $\endgroup$ – Tony Jan 5 '13 at 16:41
  • 2
    $\begingroup$ The $p$-torsion of a supersingular elliptic curve is none of the three above, but a group scheme of rank $p^2$ with a subgroup and quotient of $\alpha_p$, but not split. $\endgroup$ – Lubin Jan 25 '14 at 17:25

I don't know of any classification for general finite group schemes over an arbitrary base, but there is one if we restrict ourselves to the category of finite abelian group schemes over a perfect field $k$. A group scheme $G$ of this form splits canonically into $$ G = G_{(e,e)} \times G_{(e,c)} \times G_{(c,e)} \times G_{(c,c)}, $$ where $G_{(e,e)}$ is etale with etale Cartier dual, $G_{(e,c)}$ is etale with connected Cartier dual, and so on. Furthermore, this is a categorical decomposition into four subcategories corresponding to each type of scheme above, as there are no maps between different types.

These four categories have nice descriptions, except for that of finite connected-connected groups over $k$. The etale-etale part is groups with order coprime to $p$, etale-connected is (roughly) products $\prod_i \mathbb{Z}/p^{n_i}\mathbb{Z}$, and by duality the connected-etale category is products $\prod_i \mu_{p^{n_i}}$. The group scheme $\alpha_p$ is a connected-connected example.

If $\mathrm{char}(k) = 0$ then all finite group schemes over $k$ are etale, so this is only useful when we're in characteristic $p$. A reference for some of the above is Waterhouse's Introduction to Affine Group Schemes section 6.8.


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