Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold
This paper justifies an assertion in [Eld09] that Galois scaffolds make the questions of Galois module structure tractable. Let be a perfect field of characteristic and let . For the class of characteristic elementary abelian -extensions with Galois scaffolds described in [Eld09], we give a necessary and sufficient condition for the valuation ring to be free over its associated order in . Interestingly, this condition agrees with the condition found by Y. Miyata, concerning a class of cyclic Kummer extensions in characteristic zero.
Key words and phrases:Galois module structure, Associated Order
1991 Mathematics Subject Classification:11S15, 11R33
Let be a perfect field of characteristic , and let be a local function field over of dimension 1. For any finite extension of , we write for the valuation ring of and for the normalized valuation on . If is a Galois extension with Galois group , we write
for the associated order of in the group algebra . Then is an -order in containing , and is a module over . It is natural then to ask whether is a free over .
This question was investigated by Aiba [Aib03] and by de Smit and Thomas [dST07] when is an extension of degree (for the analogous results in characteristic zero, see [BF72, BBF72]). Ramified cyclic extensions of degree in characteristic are special in that they possess a particular property, a Galois scaffold. In [Eld09], a class of arbitrarily large fully ramified elementary abelian -extensions , the near elementary abelian extensions, was introduced. These extensions are similarly special. They too possess a Galois scaffold.
Let as above. An elementary abelian extension of of degree is a one-dimensional elementary abelian extension of if for elements and such that with , and , with the further condition that whenever for , the projections of into are linearly independent over the field with elements.
More generally, is a near one-dimensional elementary abelian extension of if
where , are as above, and the “error terms” satisfy
The purpose of this paper to use the Galois scaffold for near one-dimensional elementary abelian extensions (restated here as Theorem 2.1) to determine a necessary and sufficient condition for to be free over . So that we can state out main result (Theorem 1.1), we introduce additional notation.
As observed in [Eld09], any near one-dimensional elementary abelian extension is totally ramified, and its lower ramification numbers are the distinct elements in the sequence
where . This means that the first ramification number of is , and that all the (lower) ramification numbers are congruent modulo to , the least non-negative residue of .
Given any integer , let denote the base- digits of :
with and for large enough. Thus . Following [Byo08], we define a set .
Given with , let be the unique solution of , . Then consists of all integers with and satisfying the following property: For all , with there exists with
The main result of this paper is the following
Let be any near one-dimensional elementary abelian extension of degree . Then is free over its associated order if and only if .
The definition for is however difficult to digest. A simpler condition that focuses on the congruence class , containing all the (lower) ramification numbers of can be used to replace the condition involving , but at the expense of a weaker statement:
Let with . Then is free over if and only if divides .
Let with . Then is free over if divides for some .
If contains the field of elements with , , and where
then is a totally ramified elementary abelian extension of degree , with unique ramification break . And if denotes the least non-negative residue of modulo , then is free over its associated order if and only if . Thus
If then is free over if and only if divides .
If then is free over if divides for some .
If has characteristic 2 and and is any totally ramified biquadratic extension of (i.e. , then is free over .
1.1. Miyata’s result in characteristic zero
Let be a finite extension of the -adic field that contains a primitive root of unity. Again, for any finite Galois extension with Galois group , we may consider the valuation ring as a module over its associated order . A nice, natural class of extensions consists of those totally ramified cyclic Kummer extensions of degree with
These extensions have been studied in a series of papers by Miyata [Miy95, Miy98, Miy04]. In particular, Miyata gave a necessary and sufficient condition in terms of for to be free over . This condition can be expressed in terms of . See [Byo08].
Theorem 1.4 (Miyata).
Let be as above, satisfying (3), then is free over its associated order if and only if .
This suggests that we should regard near one-dimensional elementary abelian extensions in characteristic as somehow analogous to Miyata’s cyclic characteristic extensions. In particular, it seems natural to regard the families of extensions in Corollary 1.2 and Theorem 1.4 (both defined by a single equation) to be analogous. If this analogy has merit, then Theorem 1.1 suggests that there should be a larger family of Kummer extensions, “deformations” of Miyata’s family, for which, in some appropriate sense, Miyata’s criterion holds.
2. Proof of Main Theorem and its Corollaries
Recall that is an near one-dimensional elementary abelian extension of characteristic local fields.
2.1. Galois scaffold
The definition of Galois scaffold in [Eld09] is clarified in [BE]. There are two ingredients: A valuation criterion for a normal basis generator and a generating set for a particularly nice -basis of the group algebra .
In our setting, where is a near one-dimensional elementary abelian extension of degree , the valuation criterion is , which means that if then .
The second ingredient is a generating set of elements from the augmentation ideal of that satisfy a regularity condition, namely for , and for all that satisfy the valuation criterion, . And moreover, if we define for , then is a complete set of residues modulo .
Let be a near one-dimensional elementary abelian elementary abelian extension of degree , let and let be the largest lower ramification number of . Then for there exist elements in the augmentation ideal of such that and, for any with and any , we have
In [Eld09, Theorem 1.1], take . ∎
In the next two sections, we describe the associated order in terms of these , and show that is free over if and only if . To do so, we require nothing more than the existence of the described in Theorem 2.1.
2.2. Associated order
For the fixed prime power , there is a partial order on the integers defined as follows. Recall the -adic expansion of an integer: with . Define
Write for . Note that does not respect addition: if then is equivalent to (both say that no carries occur in the base- addition of and ) but these are not equivalent to (which always holds).
Recall that for , we have defined . Since for all we have
Now set for . This means that with . Let be any element with valuation . Recall , so . Set , so , and set
is an -basis for . Moreover is a -basis for the group algebra , and generates a normal basis for the extension .
The first assertion follows from the fact that since , takes all values in as does. From the definition of the , we then deduce that the elements span over . Comparing dimensions, it follows that generates a normal basis, and that the form a -basis for . ∎
2.3. Freeness over associated order
Then and (taking ), we have for all .
Let be any near one-dimensional elementary abelian extension of degree , with largest ramification number , and let be any element with . The associated order of has -basis . Moreover is a free module over if and only for all , and in this case is a free generator of over .
Since is a -basis of , any element of may be written with . Using (5) we have
Hence the elements form an -basis of .
Now suppose that for all . As , the definition of , preceding (5), yields , so the basis elements take to the basis elements of of . Hence is a free -module on the generator .
Conversely, suppose that is free over , say where with . Then is an -basis for , and using (5) we have , which is an -linear combination of the with . In other words, there is an upper triangular matrix with such that . This matrix is invertible, since is also an -basis for . Thus for , which means that , and thus as required. ∎
With the above notation, for all if and only if satisfies
for all integers , , with satisfying and .
The condition for all can be restated as
As this is symmetric in and , we may assume .
Let , and . So . The first step is to prove that and if and only if and . Observe that holds if and only if there are no carries in the base- addition of and (see for example [Rib89, p. 24]).
Observe that means that for all . Using the definition of , this means that and for all . So and means that and for all . So there are no carries occur in the base- addition of and .
On the other hand, assume that , for all , and for a contradiction that there is an such that . We may assume that is the smallest such subscript. Thus for all and where . This means that for all , and . So for all and .
It therefore remains to show that the inequality corresponds to , where . For we have , so that . Hence
Now since , and . Thus the last inequality is equivalent to where , as required. ∎
Proof of Theorem 1.1.
Note that, by (1), all the (lower) ramification numbers are congruent modulo . Therefore and for all integers , . As a result of Lemma 2.4, we conclude that is free over its associated order if and only if
for all integers , , with satisfying and . This is Miyata’s necessary and sufficient condition for to be free over (recall that is a cyclic extension in characteristic ). So our conclusion agrees with Miyata’s result, as recorded in [Byo08, Theorem 1.3] where , except that in this paper we have (instead of ).
Note that the purpose of [Byo08] was to translate Miyata’s necessary and sufficient condition, namely (7), into the condition given in [Byo08, Theorem 1.8]. Thus, it is the content of the proof for [Byo08, Theorem 1.8] that now allows us to conclude, in our situation, that is free over its associated order if and only if . ∎
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