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=pr such that xq∈K for all x∈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∈A, xq is in K and is integral over R so, by integral closure, xq∈R.
So, let 𝔞 be a nonzero ideal in A, and let 𝔟 be the ideal of R generated by terms of the form aq for a∈𝔞,
𝔟=(aq:a∈𝔞)R. |
Then, as R is a Dedekind domain, there is a fractional ideal 𝔟-1 of R such that 𝔟𝔟-1=R, and write 𝔟-1A for the fractional ideal of A generated by 𝔟-1. Then,
1∈R=𝔟𝔟-1⊆𝔞q𝔟-1A. | (1) |
On the other hand, if a1,…,aq∈𝔞 and b∈𝔟-1 then
(a1⋯aqb)q=(aq1b)⋯(aqqb)∈R, |
so a1⋯aqb is integral over R and is in A. Therefore, 𝔞q𝔟-1A⊆A. Combining with (1) gives 𝔞q𝔟-1A=A, so 𝔞 is invertible with inverse 𝔞q-1𝔟-1A.
Title | proof of finite inseparable extensions of Dedekind domains are Dedekind |
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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 |