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 (39.23.0.1). 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

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). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: