Connected Components
For an algebraic group \(G\) over a field \(k\), the underlying scheme decomposes into connected components, and these inherit a strong group structure: the component containing the identity is a closed normal subgroup.
The identity component \(G^\circ\)
The identity component of \(G\), denoted \(G^\circ\) or \(G^0\), is the connected component of the identity \(e \in G\). It is:
- a closed subscheme of \(G\);
- a closed normal subgroup of \(G\) (since conjugation is continuous and fixes the identity, it permutes connected components and fixes \(G^\circ\));
- a connected algebraic group in its own right;
- of finite index in \(G\) when \(G\) is of finite type (\(G/G^\circ\) is finite étale).
The component group \(\pi_0(G)\)
The quotient
\[ \pi_0(G) \;=\; G \,/\, G^\circ \]
is a finite étale group scheme over \(k\), called the group of connected components of \(G\). It fits in the short exact sequence
\[ 1 \to G^\circ \to G \to \pi_0(G) \to 1. \]
For \(k\) algebraically closed, \(\pi_0(G)\) is an étale group scheme, i.e. just an abstract finite group. Over a non-algebraically-closed field, \(\pi_0(G)\) carries a Galois action recording the descent data.
Examples
- \(\mathbb{G}_a, \mathbb{G}_m, \operatorname{GL}_n, \operatorname{SL}_n\): connected (\(\pi_0 = 1\)).
- \(\mu_n\) for \(n\) coprime to \(\operatorname{char}(k)\): étale, \(\pi_0(\mu_n) = \mu_n\), \(G^\circ = 1\).
- \(\mu_p\) over \(\mathbb{F}_p\): connected (it is the kernel of Frobenius on \(\mathbb{G}_m\)) but not reduced — a non-trivial infinitesimal group.
- Orthogonal group \(\mathrm{O}_n\): has two components, \(\mathrm{O}_n^\circ = \mathrm{SO}_n\) and the determinant-\(-1\) coset; \(\pi_0(\mathrm{O}_n) = \mathbb{Z}/2\).
- Normaliser of a torus \(N_G(T)\): \(N_G(T)^\circ = T\), and \(\pi_0(N_G(T)) = W(G, T)\), the Weyl group.
Connected vs. geometrically connected
\(G\) is connected if its underlying scheme is connected (as a topological space). \(G\) is geometrically connected if \(G_{\bar{k}}\) is connected for \(\bar{k}\) an algebraic closure. These can differ: for a non-perfect field, an irreducible scheme can split into components after base change to \(\bar{k}\) (Galois descent of components).
The Langlands roadmap uses geometric connectedness throughout (connected geometric fibres over the base).
Reduced vs. smooth components
In characteristic \(p\), an algebraic group can have non-reduced identity component (e.g. \(\mu_p\), \(\alpha_p\)). In characteristic zero (or for smooth algebraic groups in general), the identity component is automatically reduced and smooth.
The reduction \(G^\circ_{\mathrm{red}}\) is a subgroup of \(G^\circ\), and equals \(G^\circ\) iff \(G^\circ\) is reduced. Even in positive characteristic, \(G^\circ_{\mathrm{red}}\) is a closed subgroup (over a perfect base field; in general one must base change first).