Lemma 79.9.11. In Situation 79.9.2 assume s, t are locally of finite type. Let G = \mathop{\mathrm{Spec}}(k) \times _{\Delta , \mathop{\mathrm{Spec}}(k) \times _ B \mathop{\mathrm{Spec}}(k), t \times s} R be the stabilizer group algebraic space. Then we have \dim (R) = \dim (G).
Proof. Since G and R are equidimensional (see Lemmas 79.9.9 and 79.9.10) it suffices to prove that \dim _ e(R) = \dim _ e(G). Let V be an affine scheme, v \in V, and let \varphi : V \to R be an étale morphism of schemes such that \varphi (v) = e. Note that V is a Noetherian scheme as s \circ \varphi is locally of finite type as a composition of morphisms locally of finite type and as V is quasi-compact (use Morphisms of Spaces, Lemmas 67.23.2, 67.39.8, and 67.28.5 and Morphisms, Lemma 29.15.6). Hence V is locally connected (see Properties, Lemma 28.5.5 and Topology, Lemma 5.9.6). Thus we may replace V by the connected component containing v (it is still affine as it is an open and closed subscheme of V). Set T = V_{red} equal to the reduction of V. Consider the two morphisms a, b : T \to \mathop{\mathrm{Spec}}(k) given by a = s \circ \varphi |_ T and b = t \circ \varphi |_ T. Note that a, b induce the same field map k \to \kappa (v) because \varphi (v) = e! Let k_ a \subset \Gamma (T, \mathcal{O}_ T) be the integral closure of a^\sharp (k) \subset \Gamma (T, \mathcal{O}_ T). Similarly, let k_ b \subset \Gamma (T, \mathcal{O}_ T) be the integral closure of b^\sharp (k) \subset \Gamma (T, \mathcal{O}_ T). By Varieties, Proposition 33.31.1 we see that k_ a = k_ b. Thus we obtain the following commutative diagram
As discussed above the long arrows are equal. Since k_ a = k_ b \to \kappa (v) is injective we conclude that the two morphisms a and b agree. Hence T \to R factors through G. It follows that R_{red} = G_{red} in an open neighbourhood of e which certainly implies that \dim _ e(R) = \dim _ e(G). \square
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