Timeline for On the structure of commutative group schemes
Current License: CC BY-SA 3.0
13 events
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| Sep 5, 2012 at 18:14 | comment | added | grp | @S. Carnahan: True, but Witt groups are nonetheless "split" (in the sense that they have a composition series over $k$ with successive quotients $k$-isomorphic to $\mathbf{G}_a$), so a more striking kind of example answering Will's question in the negative is one which doesn't even contain $\mathbf{G}_a$ as a $k$-subgroup. Of course there are no such examples when $k$ is perfect, but for imperfect $k$ Rosenlicht gave the classic example $y^p = x - a x^p$ for $a \in k - k^p$ with any $p = {\rm{char}}(k) > 0$. Tits studied this phenomenon in detail (in any dimension) in his Yale lecture notes. | |
| Sep 5, 2012 at 9:23 | comment | added | S. Carnahan♦ | @Will: Witt vectors of length $n>1$ form counterexamples in positive characteristic. | |
| Sep 4, 2012 at 22:12 | comment | added | grp | @Pooya: Consider two affine finite type $X$-groups $G$ and $H$ (e.g., $H$ could be $\mathbf{G}_a$). By general spreading-out principles, any $\eta$-scheme map $H_{\eta} \rightarrow G_{\eta}$ spreads out to a $U$-scheme morphism $H_U \rightarrow G_U$ for some dense open $U$ in $X$. If the given $\eta$-map is a homomorphism (resp. closed immersion) then we can arrange the same for the $U$-map by shrinking $U$ a bit. Here we use that "homomorphism" amounts to a commutative diagram (i.e., an equality of some maps) and "closed immersion" is one of those properties which can be "spread out". | |
| Sep 4, 2012 at 18:11 | comment | added | Pooya | Yes sorry grp, I rather meant on an open subset too. I guess my question is becoming naive then? I understand that one can spread out certain "properties" of morphisms but why should a morphism $\mathbb G_{a, U} \to G_U$ for an open $U$ exist at the first place (pulling back to the one on the generic fiber)? | |
| Sep 4, 2012 at 11:22 | comment | added | grp | @Pooya: No, not over the entirety of $X$. For example the underlying additive group of a line bundle with no nonzero global section is a counterexample to the global statement. But in your question you only asked for the property over some (dense) open subset of the base, and that much follows from standard "spreading out" arguments. | |
| Sep 4, 2012 at 7:39 | comment | added | Pooya | Dear grp: so is it the case that if $G$ is a group scheme over base scheme $X$ with generic point $\eta$, and $G_\eta$ contains a subgroup isomorphic to $\mathbb G_{a, \eta}$ then $G$ has a subgroup scheme isomorphic to $\mathbb G_{a,X}$ pulling back to the former one? why? | |
| Sep 4, 2012 at 6:19 | comment | added | grp | The quotient $G/T$ exists as a smooth affine group, and by design it is unipotent on geometric fibers over the integral base in char. 0. The generic fiber is unipotent over the char. 0 function field of the base. Your definition of unipotence is wrong in char. $> 0$ (where there are many smooth connected commutative unipotent groups containing no $\mathbf{G}_a$), but in char. 0 it is equivalent to the "right" definition (the filtration condition on geometric fibers). Such a composition series on the actual (not just geometric) generic fiber spreads out over some dense open of the base. | |
| Sep 4, 2012 at 3:54 | history | edited | Pooya | CC BY-SA 3.0 |
For my specific final question $X$ is a $\mathbb C$-scheme.
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| Sep 4, 2012 at 3:48 | comment | added | Pooya | Dear Will: I think they are vector spaces (i.e. Zariski locally trivial) by my definition [Kambayashi-Miyanishi, on flat fibrations by the affine line]. | |
| Sep 4, 2012 at 3:38 | comment | added | Pooya | Dear Emerton: yes! A maximal torus is a subgroup scheme $T$ of $G$ which is a torus (i.e. fpqc locally isomorphic to $\mathbb G_m^r$) and point-wise maximal: for any point $x \in X$, $T_{\bar x}$ is the maximal torus of $G_{\bar x}$ where $\bar x$ is the spectrum of the algebraic closure of $\kappa (x)$ [SGA 3-IIX, Definition 1.3]. Thanks, | |
| Sep 4, 2012 at 3:18 | comment | added | Will Sawin | Are there smooth commutative unipotent groups that are not vector spaces? | |
| Sep 4, 2012 at 3:08 | comment | added | Emerton | Dear Pooya, It might help to clarify exactly what you mean by "maximal torus" in the relative context. Regards, | |
| Sep 4, 2012 at 2:50 | history | asked | Pooya | CC BY-SA 3.0 |