multiplicity of schemes in positive characteristic

let $K$ a field and $X$ a scheme of finite type over $K$. let $X_1,\dots,X_r$ the irreducible components of $X$ with generic points $\eta_i$. Let $\bar{X}=X\times Spec(\bar{K})$ where $\bar{K}$ is an algebraic closure, and $\bar{\eta_i}$ the points over $\eta_i$. My question are: 1) find an example in positive characteristic where $l(\mathcal{O}_{\bar{X}_i,\bar{\eta}_i})$ is $>1$ 2) why $l(\mathcal{O}_{\bar{X}_i,\bar{\eta}_i})l(\mathcal{O}_{\bar{X},\bar{\eta}_i})=l(\mathcal{O}_{X,\eta_i})$?

• What about the multiplication formula for the multiplicities? My example shows that as it stands it cannot be true. What is an example for $\ell (O_{\bar{X}_i ,\eta_i})<\ell (O_{\bar{X} ,\eta_i})$ ? – Hagen Knaf Apr 6 '11 at 14:07

Let $L$ be a purely inseparable extension field of $K$ of degree $p$; then $L\cong K[X]/(X^p-a)$ for some $a\in K$. The scheme $X:=\mathrm{Spec} (L)$ is integral and thus has multiplicity $1$, but $L\times_K\overline{K}\cong \overline{K}[X]/((X-b)^p)$ for $b\in\overline{K}$ with $b^p=a$ is irreducible and non-reduced. A composition series for the local ring $O_{\overline{X},\overline{\eta}}=\overline{K}[X]/(X-b)^p\overline{K}[X]$ is given by
$0\subset (X-b)^{p-1}\overline{K}[X]/(X-b)^p\overline{K}[X]\subset (X-b)^{p-2}\overline{K}[X]/(X-b)^p\overline{K}[X]\subset\ldots$
$\ldots\subset (X-b)\overline{K}[X]/(X-b)^p\overline{K}[X]\subset \overline{K}[X]/(X-b)^p\overline{K}[X].$
Hence the multiplicity of $\overline{X}$ equals $p$.
• Hagen, the first sentence is incorrect for all perfect fields: think for example of $K=\mathbb F_2$ and $L=\mathbb F_4$. – Georges Elencwajg Apr 6 '11 at 17:32