# Schematic closure of maximal torus over a discrete valuation ring: smoothness of the special fibre

Let $R$ be a complete DVR (or just Henselian) with field of fractions $F$ and algebraically closed residue field $k$ (of characteristic $p>0$). Let $G$ be a reductive group scheme over $R$ and let $T$ be a maximal torus of the generic fibre $G_F=G\times_{R} F$.

Let $\overline{T}$ be the schematic closure of $T$ in $G$. Explicitly, if $T=\mathrm{Spec}\,F[x_1,\dots,x_n]/I$, then $$\overline{T}=\mathrm{Spec}\,R[x_1,\dots,x_n]/(IF[x_1,\dots,x_n]\cap R[x_1,\dots,x_n]).$$ Then $\overline{T}$ is a flat group scheme over $R$ with generic fibre $T$.

Assume that $k$ has very good characteristic for $G$. Is $\overline{T}$ smooth over $R$? In other words, does $\overline{T}$ have smooth/reduced special fibre?

For $G=\mathrm{GL}_{n}$ this is true because $T$ is then the multiplicative group of an algebra. I think I can also prove this for $G=\mathrm{SL}_{n}$, but the argument does not generalise.

A weaker property than smoothness which is still enough for my purposes, is the following:

Is the canonical map $\rho:\overline{T}(R)\rightarrow\overline{T}(k)$ surjective?

• Does the Lie algebra of $\overline{T}$ coincide with the Zariski closure in the relative Lie algebra $\mathfrak{g} = \text{Hom}_R(\Omega_{G/R,e},R)$ of the Lie algebra $\mathfrak{t}$ of $T$, i.e., does it coincide with the intersection in $\mathfrak{g}\otimes_R F$ of $\mathfrak{g}$ and $\mathfrak{t}$? If so, then maybe you can extend to the general case your "algebra argument" in the case of $\text{GL}_n$. – Jason Starr Feb 28 '17 at 13:19
• @Jason Starr: Do you mean whether the Lie algebra of $\overline{T}$ is the schematic closure in $\mathfrak{g}$ of $\mathfrak{t}$? This would indeed solve the problem, but I don't know whether it is true. I think this is more or less equivalent to my question. – A Stasinski Feb 28 '17 at 21:49
• As observed in Bruhat-Tits second paper, section 4.4, a necessary condition for a counterexample is that $T$ should not be induced (i.e. a product of various Weil restriction of $\mathbf{G}_m$). For a quasi-split group $G$ and $T$ a centraliser of a maximal split torus, $T$ is always induced when $G$ is simply connected or adjoint. So maybe you could try to see what happens for the standard torus of a ramified quasi-split $\textrm{SU}_4/\mu_2$ in residue characteristic $2$ (but this might be a silly suggestion, I don't have time to think more about it). – thierry stulemeijer Mar 2 '17 at 12:53
• @thierry stulemeijer: This part of Bruhat-Tits is relevant, but even for $\mathrm{GL}_n$, there are $T$ which are not "induced" but for which the answer is still positive. Regarding $\mathrm{SU}_4$ over residue char $2$, note that I assume that the residue char is very good (it is not hard to find a counterexample for $\mathrm{SL}_2$ over residue char $2$). – A Stasinski Mar 3 '17 at 8:54
• @A Stasinski Ah, sorry, I had in mind that the very good characteristic assumption was on $F$. – thierry stulemeijer Mar 3 '17 at 9:17