The Stacks project

Lemma 39.23.4. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $U = \mathop{\mathrm{Spec}}(A)$ and $R = \mathop{\mathrm{Spec}}(B)$ are affine and $s, t : R \to U$ finite locally free. Let $C \subset A$ be as in ( Then $A$ is integral over $C$.

Proof. First, by Lemma 39.23.3 we know that $(U, R, s, t, c)$ is a disjoint union of groupoid schemes $(U_ r, R_ r, s, t, c)$ such that each $s, t : R_ r \to U_ r$ has constant rank $r$. As $U$ is quasi-compact, we have $U_ r = \emptyset $ for almost all $r$. It suffices to prove the lemma for each $(U_ r, R_ r, s, t, c)$ and hence we may assume that $s, t$ are finite locally free of rank $r$.

Assume that $s, t$ are finite locally free of rank $r$. Let $f \in A$. Consider the element $x - f \in A[x]$, where we think of $x$ as the coordinate on $\mathbf{A}^1$. Since

\[ (U \times \mathbf{A}^1, R \times \mathbf{A}^1, s \times \text{id}_{\mathbf{A}^1}, t \times \text{id}_{\mathbf{A}^1}, c \times \text{id}_{\mathbf{A}^1}) \]

is also a groupoid scheme with finite source and target, we may apply Lemma 39.23.2 to it and we see that $P(x) = \text{Norm}_{s^\sharp }(t^\sharp (x - f))$ is an element of $C[x]$. Because $s^\sharp : A \to B$ is finite locally free of rank $r$ we see that $P$ is monic of degree $r$. Moreover $P(f) = 0$ by Cayley-Hamilton (Algebra, Lemma 10.16.1). $\square$

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