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If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is the identity component of the special fibre also connected reductive and split?

Oh dear I'm getting very confused about this question. Is there a closed subgroup scheme of $GL(2)$ over $A$ whose generic fibre is trivial but whose special fibre is a Borel subgroup of $GL(2)$? Is life really as bad as that?

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  • $\begingroup$ So you're not assuming $G$ is even $A$-flat? What is the actual motivation? $\endgroup$
    – grghxy
    Jul 29, 2015 at 12:20
  • $\begingroup$ I'm trying to understand a question that an algebraist who knows nothing of algebro-geometric notions such as flatness is asking me. Currently they have G a closed subgroup of GL(n) over Q_p and are defining a scheme over Z_p by simply intersecting the ideal defining GL_n/Q_p with the ring of functions on GL_n/Z_p. I am completely failing to be able to say anything at all about what the corresponding scheme is. I guess this is just the closure of the generic fibre in GL_n? So maybe I know the generic fibre is dense in G? They want to know facts about the special fibre. $\endgroup$
    – slider
    Jul 29, 2015 at 12:22
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    $\begingroup$ See SGA3 XIX section 5 for a smooth affine closed subgroup $G$ of some GL$_N$ over $k[t]$ ($k$ any field of char. 0) that is PGL$_2$ over $k[t,1/t]$ and such that $G_0$ is disconnected with solvable identity component. The example works the same way over any dvr not of residue characteristic 2. But the entirety of Bruhat-Tits theory rests on phenomena of smooth affine groups with connected reductive generic fiber and non-reductive identity component of the special fiber; this makes life interesting, not bad. Can you say more about the motivating context? $\endgroup$
    – grghxy
    Jul 29, 2015 at 12:29
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    $\begingroup$ In your given setup such $G$ is exactly the schematic closure in GL$_n$ of the given subgroup of the generic fiber of GL$_n$, so it is flat over the dvr. But without more information one cannot say anything about the special fiber (other than that its dimension is the same as the generic fiber): it could be badly non-reduced, and if smooth then quite "bad" too. Indeed, every flat affine group scheme of finite type over a dvr is a closed subgroup scheme of some GL$_n$ over the dvr, so working inside GL$_n$ provides essentially no useful information to get interesting conclusions. $\endgroup$
    – grghxy
    Jul 29, 2015 at 12:34
  • $\begingroup$ Oh but this example is exactly what I needed to know. This is great -- many thanks. $\endgroup$
    – slider
    Jul 29, 2015 at 13:20

2 Answers 2

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This may also be SGA3's example, since I am too lazy to go check, but my standard example of a reductive group degenerating to something solvable is to define a multiplication $\ast_t$ on $2 \times 2$ matrices by $$\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \ast_t \begin{pmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{pmatrix} = \begin{pmatrix} x_{11} y_{11} + t x_{12} y_{21} & x_{11} y_{12} + x_{12} y_{22} \\ x_{21} y_{11} + x_{22} y_{21} & t x_{21} y_{12} + x_{22} y_{22} \end{pmatrix}.$$ For $t \neq 0$, this is simply the ordinary $2 \times 2$ matrices under the change of coordinates $\left( \begin{smallmatrix} z_{11} & z_{12} \\ t z_{21} & z_{22} \end{smallmatrix} \right)$, so it is associative, and it gives a group when restricted to the open set $z_{11} z_{22} - t z_{21} z_{12} \neq 0$. I hope this shows you that there is nothing mysterious in the given formulas.

In the limit $t=0$, the above formulas still define an associative multiplication, which forms a group $G_0$ on the open set $z_{11} z_{22} \neq 0$. Explicitly, $$\begin{pmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{pmatrix} \ast_0 \begin{pmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{pmatrix} = \begin{pmatrix} x_{11} y_{11} & x_{11} y_{12} + x_{12} y_{22} \\ x_{21} y_{11} + x_{22} y_{21} & x_{22} y_{22} \end{pmatrix}.$$ Clearly, there is a map $G_0 \to \mathbb{G}_m^2$ projecting a matrix onto its diagonal entries. The kernel, matrices of the form $\left( \begin{smallmatrix} 1 & x_{12} \\ x_{21} & 1 \end{smallmatrix} \right)$, is clearly a copy of $\mathbb{G}_a^2$. So we have a short exact sequence $1 \to \mathbb{G}_a^2 \to G_0 \to \mathbb{G}_m^2 \to 1$ and $G_0$ is solvable.

As grghxy points out, every flat affine group scheme of finite type over a dvr embeds in some $GL_N$, so you are not adding anything with that condition.

This example can be modified to suit many tastes: Replace $z_{11} z_{22} - t z_{12} z_{21} \neq 0$ by $z_{11} z_{22} - t z_{12} z_{21} = 1$ if you like simple Lie groups better than reductive ones; quotient by $\pm \mathrm{Id}$ if you like your groups in adjoint form; work with $2 \times 2$ matrices and $\ast_t$ commutator if you like Lie algebras. Replace $t$ by $p$ in the above formulas to work over $\mathbb{Z}_p$.

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SGA3 XIX section 5 has a terrifying example of $PGL(2)$ degenerating into something solvable. Thanks to grghxy.

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    $\begingroup$ That example is "beautiful" and "awesome", but not "terrifying" as you say (e.g., it is smooth!); this is entirely typical behavior that occurs in Bruhat-Tits theory all over the place. $\endgroup$
    – grghxy
    Jul 29, 2015 at 20:14

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