When is $G(R)\rightarrow G^{\textrm{ad}}(R)$ surjective?

Question

Consider a reductive group $G$ over a field $k$. The adjoint group $G^{\textrm{ad}}$ is defined by the exact sequence $$1\rightarrow Z(G)\rightarrow G\rightarrow G^{\textrm{ad}}\rightarrow 1$$The sequence $$1\rightarrow Z(G)(R)\rightarrow G(R)\rightarrow G^{\textrm{ad}}(R)\rightarrow 1$$ fails to be exact for a general $k$-algebra $R$, but is exact for $R=K$, the algebraic closure of $k$. Is there a criterion to determine for which $R$ the above sequence is exact? I'm mainly interested in the case that $k$ is a non-Archimedean local field, $G$ is a split $k$-group (namely $G=\textrm{SO}_{2n}$, $G^{\textrm{ad}}=\textrm{PSO}_{2n}$), and $R=k$.

Answer 1

$\require{AMScd} $In this answer, $k$ is a nonarchimedean local field.

Lemma 1. Consider a short exact sequence of linear algebraic $k$-groups $$ 1\to C\overset i\longrightarrow G\overset j\longrightarrow G'\to1$$ where $G$ and $G'$ are reductive, and $C\subset G$ is central and smooth. Then we have an exact sequence of groups \begin{equation}\label{exact} C(k)\overset i\longrightarrow G(k)\overset j\longrightarrow G'(k)\overset\delta\longrightarrow H^1(k,C)\overset {i_*}\longrightarrow H^1(k,G)\overset {j_*}\longrightarrow H^1(k,G') \end{equation} in which the group structures on $H^1(k,G)$ and $H^1(k,G')$ are defined as in Section 6 of [BK].

Proof. The exact sequence of groups and sets is constructed in [S], Section I.5. It remains to show that $i_*$ and $j_*$ are homomorphisms.

Let $T\subseteq G$ and $T'\subseteq G'$ be compatible maximal tori. Let $G^{\rm sc}$ denote the universal cover of the commutator subgroup $[G,G]$ of $G$. Consider the homomorphism $$ \rho\colon G^{\rm sc}\twoheadrightarrow [G,G]\hookrightarrow G.$$ Write $T^{\rm sc}=\rho^{-1}(T)\subset G^{\rm sc}$. We identify $(G')^{\rm sc} =G^{\rm sc}$ and $(T')^{\rm sc} =T^{\rm sc}$. Consider the commutative diagrams with exact rows $$ \begin{CD} G'(k) @>\delta >>H^1(k,C) @>{i_*}>> H^1(k,G) @>{j_*}>> H^1(k,G')\\ @. @| @V\sim V{\rm ab}V @V\sim V{\rm ab}V\\ @. H^1(k,C) @>{i_*}>>{\Bbb H}^1(k,T^{\rm sc}\to T) @>{j_*}>>{\Bbb H}^1(k,T^{\rm sc}\to T') \end{CD} $$ in which the vertical arrows are bijections defining the abelian group structure on $H^1(k,G)$ and $H^1(k,G')$; see [B] and Section 6 of [BK]. Since the horizontal arrows in the bottom row are group homomorphisms, the horizontal arrows in the top row are homomorphisms as well, as required.

Now ${\rm char}(k)\neq 2$, $G={\rm SO}_{2n}$, $C=Z_G$, $G'=G^{\rm ad}:= G/Z_G$.

Lemma 2. $\#H^1(k,C)\ge 4$.

Proof. Since $C\cong\mu_2$, we have $H^1(k,C)={\rm Hom}({\rm Gal}(k^s/k),\mu_2)$. We know from Section II.5 of [N], that $k^\times/(k^\times)^2$ has cardinality at least 4. The lemma follows.

Lemma 3. $\#H^1(k,G)=2$.

Proof. The Galois cohomology set $H^1(k,G)=2$ classifies
the isomorphism classes of quadratic forms of dimension $2n$ with given determinant. There is exactly two isomorphism classes: with Hasse invariant $1$ and $-1$, see Chapter VI of [O'M], and the lemma follows.

Alternatively, consider the algebraic fundamental group $M=\pi_1^{\rm alg}(G)=X_*(T)/\rho_*X_*(T^{\rm sc})$. Then $M\cong{\Bbb Z}/2{\Bbb Z}$, and by [B] or [BK] we have $$H^1(k,G)\cong (M_{\Gamma_k})_{\rm Tors},$$ the torsion subgroup of the group of coinvariants of the Galois group $\Gamma_k={\rm Gal}(k^s/k)$ in $M$. Clearly, we have $$ (M_{\Gamma_k})_{\rm Tors}= M_{\rm Tors}=M={\Bbb Z}/2{\Bbb Z},$$ and the lemma follows.

Calculation. Let $G={\rm SO}_{2n}$ for $2n\ge 4$ over a non-archimedean local field $k$ of characteristic not 2. Then by Lemmas 2 and 3, the homomorphism $i_*$ in the exact sequence (1) is not injective. It follows that the homomorphism $j\colon G(k)\to G^{\rm ad}(k)$ is not surjective.

References

[B] Mikhail Borovoi, Abelian Galois cohomology of reductive groups, Mem. Amer. Math. Soc. 132 (1998), no. 626.

[BK] Mikhail Borovoi and Tasho Kaletha, Galois cohomology of reductive groups over global fields, arXiv:2303.04120.

[N] Jürgen Neukirch, Algebraic number theory, Springer-Verlag, Berlin, 1999.

[O'M] O. Timothy O'Meara, Introduction to quadratic forms, Springer-Verlag, Berlin, 2000.

[S] Jean-Pierre Serre, Galois Cohomology, Springer-Verlag, Berlin, 1997.