proof of finite inseparable extensions of Dedekind domains are Dedekind

Let $R$ be a Dedekind domain with field of fractions $K$ and $L/K$ be a field extension. We suppose that $K$ has characteristic (http://planetmath.org/characteristic) $p>0$ and that there is a $q=p^{r}$ such that $x^{q}\in K$ for all $x\in L$ . In particular, this is satisfied if it is a purely inseparable and finite extension.

We show that the integral closure $A$ of $R$ in $L$ is a Dedekind domain.

We cannot apply the same method of proof as for the proof of finite separable extensions of Dedekind domains are Dedekind, because here $A$ does not have to be finitely generated as an $R$ -module.

We use the characterization of Dedekind domains as integral domains in which all nonzero ideals are invertible (see proof that a domain is Dedekind if its ideals are invertible). Note that for any $x\in A$ , $x^{q}$ is in $K$ and is integral over $R$ so, by integral closure, $x^{q}\in R$ .

So, let $\mathfrak{a}$ be a nonzero ideal in $A$ , and let $\mathfrak{b}$ be the ideal of $R$ generated by terms of the form $a^{q}$ for $a\in\mathfrak{a}$ ,

\mathfrak{b}=\left(a^{q}:a\in\mathfrak{a}\right)_{R}.

Then, as $R$ is a Dedekind domain, there is a fractional ideal $\mathfrak{b}^{-1}$ of $R$ such that $\mathfrak{b}\mathfrak{b}^{-1}=R$ , and write $\mathfrak{b}^{-1}_{A}$ for the fractional ideal of $A$ generated by $\mathfrak{b}^{-1}$ . Then,

1\in R=\mathfrak{b}\mathfrak{b}^{-1}\subseteq\mathfrak{a}^{q}\mathfrak{b}^{-1}% _{A}.

(1)

On the other hand, if $a_{1},\ldots,a_{q}\in\mathfrak{a}$ and $b\in\mathfrak{b}^{-1}$ then

(a_{1}\cdots a_{q}b)^{q}=(a_{1}^{q}b)\cdots(a_{q}^{q}b)\in R,

so $a_{1}\cdots a_{q}b$ is integral over $R$ and is in $A$ . Therefore, $\mathfrak{a}^{q}\mathfrak{b}^{-1}_{A}\subseteq A$ . Combining with (1) gives $\mathfrak{a}^{q}\mathfrak{b}^{-1}_{A}=A$ , so $\mathfrak{a}$ is invertible with inverse $\mathfrak{a}^{q-1}\mathfrak{b}^{-1}_{A}$ .

Title	proof of finite inseparable extensions of Dedekind domains are Dedekind
Canonical name	ProofOfFiniteInseparableExtensionsOfDedekindDomainsAreDedekind
Date of creation	2013-03-22 18:35:42
Last modified on	2013-03-22 18:35:42
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	5
Author	gel (22282)
Entry type	Proof
Classification	msc 13A15
Classification	msc 13F05