Lemma 29.30.13. Let $R$ be a ring. Let $R \to A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ be a relative global complete intersection. Set $S = \mathop{\mathrm{Spec}}(R)$ and $X = \mathop{\mathrm{Spec}}(A)$. Consider the morphism $f : X \to S$ associated to the ring map $R \to A$. The function $x \mapsto \dim _ x(X_{f(x)})$ is constant with value $n - c$.

**Proof.**
By Algebra, Definition 10.136.5 $R \to A$ being a relative global complete intersection means all nonzero fibre rings have dimension $n - c$. Thus for a prime $\mathfrak p$ of $R$ the fibre ring $\kappa (\mathfrak p)[x_1, \ldots , x_ n]/(\overline{f}_1, \ldots , \overline{f}_ c)$ is either zero or a global complete intersection ring of dimension $n - c$. By the discussion following Algebra, Definition 10.135.1 this implies it is equidimensional of dimension $n - c$. Whence the lemma.
$\square$

## 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Your email address will not be published.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)