2
$\begingroup$

Let $\mathcal{O}$ be a discrete valuation ring, and $\mathcal{G}$ is a flat group scheme over $\mathcal{O}$, we may assume the generic fiber is reductive. Then can we define its Lie algebra, and whether it is flat?

PS: given a reductive group $G$ and a parabolic subgroup $P$, we can construct a groups scheme over $\mathcal{O}$ with generic fiber $G$ and special fiber $P$, it seems not to be flat group scheme

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ What do you mean by Lie algebra in this setting? If $\mathcal O = \mathbf Z_p$ and $\mathcal G = \mu_p$, then $\Omega_{\mathcal G/\mathcal O}$ is not flat over $\mathcal O$. But if you take linear dual it always becomes free by the structure theory of modules over a PID... $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 23, 2019 at 3:04
  • $\begingroup$ But in your example $\mathcal{G}$ is not flat over $\mathbb{Z}_{p}$. Can we define Lie algebra using derivations along the unit section? If $\mathcal{G}$ is smooth, I guess we can define the Lie algebra as dual of relative differential, this should coincide with derivation. $\endgroup$
    – Bin Wang
    Commented Oct 26, 2019 at 2:36
  • $\begingroup$ No, it is flat. The ring $\mathcal O[x]/(x^p-1)$ is free of rank $p$ over $\mathcal O$. But my question remains: what do you mean by dual? Because the dual $\operatorname{Hom}_{\mathcal O}(M,\mathcal O)$ of any finitely generated $\mathcal O$-module $M$ is flat (even free) by the structure theory of modules over a PID. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 26, 2019 at 4:17
  • $\begingroup$ You are absolutely right. If I define the Lie algebra as dual module, then it is automatically free module over $\mathcal{O}$ $\endgroup$
    – Bin Wang
    Commented Oct 26, 2019 at 8:30
  • $\begingroup$ Thank you very much, and I find I actually do not know how to define the Lie algebra properly. Can you provide a definition, or what's your opinion? $\endgroup$
    – Bin Wang
    Commented Oct 26, 2019 at 8:37

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .