Scaffolds and generalized integral Galois module structure
Let be a finite, totally ramified -extension of complete local fields with residue fields of characteristic , and let be a -algebra acting on . We define the concept of an -scaffold on , thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where was Galois and for . When a suitable -scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, for a given fractional ideal to be free over its associated order in . We also show how to determine the number of generators required when it is not free, along with the embedding dimension of the associated order. In the Galois case, the numerical parameters are the ramification breaks associated with . We apply these results to biquadratic Galois extensions in characteristic 2, and to totally and weakly ramified Galois -extensions in characteristic . We also apply our results to the non-classical situation where is a finite primitive purely inseparable extension of arbitrary exponent that is acted on, via a higher derivation (but in many different ways), by the divided power -Hopf algebra.
Key words and phrases:Ramification, Galois module structure, Hopf-Galois theory
2010 Mathematics Subject Classification:11S15, 20C11, 16T05, 11R33
Let be a local field with residue field of characteristic , and let be a finite Galois extension of with Galois group . We write , for the valuation rings of , , respectively, and , for their maximal ideals. Then is a module over the integral group ring . By Noether’s criterion [Noe32], it is a free module if and only if the extension is at most tamely ramified. In order to study integral Galois module structure for wildly ramified extensions, H.W. Leopoldt [Leo59] introduced the associated order
of in the group algebra . Over the last 50 years, many authors have investigated, in various situations, when is free as a module over , or, more generally, when a fractional ideal of is free as a module over its associated order in ; see for instance [Jac64, BF72, Ber72, Mar74, Tay85, CM94, Miy98, Bon02, Aib03, dST07]. For a comprehensive overview of this area, and a far more extensive bibliography, we refer the reader to the survey [Tho10].
Our goal here is to give a systematic presentation of a new approach to such questions of integral Galois module structure, in a somewhat generalized sense. This approach is restricted to totally ramified extensions of local fields , whose degree is a power of the residue characteristic , and which admit an action by a -algebra of dimension . An -scaffold on consists of certain special elements in which act on suitable elements of in a way which is tightly linked to the valuation on . The most obvious setting where scaffolds may occur is that described above, where is a Galois extension with Galois group and . Our approach is not, however, limited to that situation. We will show in §5 how it can be applied to a divided power Hopf algebra acting in many different ways on an inseparable field extension. Other inseparable examples have been given by Koch [Koc14, Koc15]. Our approach could also be used for different Hopf Galois structures on a given separable (but not necessarily normal) field extension, as described by Greither and Pareigis [GP87].
When admits an -scaffold, is a free module over , in analogy to the Normal Basis Theorem of Galois theory. We can then consider any fractional ideal of as a module over its associated order in ,
and ask whether it is a free module. It is in this sense that our work is concerned with “generalized” integral Galois module structure. An -scaffold comes with a “precision” parameter, and the existence of a scaffold of high enough precision will enable us to extract a considerable amount of information about the -module ; not only can we determine if it is free, but (following [dST07] for extensions of degree ), we can also find the minimal number of generators required when it is not free, and obtain the embedding dimension of . An important feature of our approach is that all this information depends on purely numerical data, namely certain parameters attached to the scaffold (playing the role of ramification breaks) and the exponent of the ideal under consideration. Given the existence of a scaffold with specified parameters , our results are therefore, in some sense, universal: they are independent of the characteristic ( or ) of the fields involved, and, in the Galois case, independent of the precise structure of the Galois group. In particular, our results make no distinction between abelian and non-abelian extensions. Moreover, we obtain exactly the same results for, say, inseparable extensions as for Galois extensions, provided that the parameters coincide.
The intuition underlying our notion of a scaffold can be explained somewhat informally as follows. Let , denote normalized valuations such that . Given any positive integers for such that , there are elements such that . Since the valuations, , of the monomials
provide a complete set of residues modulo and, since is totally ramified of degree , these monomials provide a convenient -basis for . The action of on is clearly determined by its action on the monomials . So if there were for such that each acts on the monomial basis element of as if it were the differential operator (with the treated as independent variables), namely
then the monomials in the (with exponents at most ) would furnish a convenient basis for whose effect on the would be easy to follow. As a consequence, the determination of the associated order of a particular ideal , and of the structure of this ideal as a module over its associated order, would be reduced to purely numerical considerations. This remains true if (1) is loosened to the congruence
for a sufficiently large “precision” . The , together with the , constitute an -scaffold on . Our formal definition of an -scaffold (Definition 2.3) is a generalization of this situation. When the equality (1) holds, our scaffold has precision .
We now explain the background to this work. In the papers [Eld09, BE13, BE14], the first- and third-named authors began to develop the theory of scaffolds in the setting of Galois extensions. There, the parameters of these Galois scaffolds are just the ramification breaks of the extension . These scaffolds all have precision , apart from those on cyclic extensions of degree in [BE13]. The main result of [Eld09] is the existence of a Galois scaffold for a certain class of arbitrarily large elementary abelian extensions in characteristic (the “near one-dimensional extensions”). The Galois module structure of the valuation ring in such extensions is investigated in [BE14], where a necessary and sufficient condition (in terms of the ) is given for to be free over . This condition turns out to be equivalent to that given by Miyata [Miy98] (and reformulated in [Byo08]) for a class of cyclic Kummer extensions in characteristic 0. The striking observation that the same numerical condition holds for two apparently unrelated families of extensions, differing both in Galois group and in characteristic, suggests that the methods used in [BE14] to study Galois module structure for near one-dimensional extensions might be applied more widely. The present paper develops the machinery to substantiate this idea, while [BE13] indicates the limitations of our approach by demonstrating that most extensions will not admit a scaffold. In any case, our method is necessarily restricted to totally ramified extensions of -power degree, since if admits an -scaffold, then it possesses a “valuation criterion”: there is an integer such that any element of of valuation is a free generator of over (see Proposition 2.12). This property, which can be viewed as a strong version of the Normal Basis Theorem, has been studied in a number of papers [BE07, Tho08, Eld10, Byo11, dSFT12], and can only hold when is totally ramified and of -power degree (see [dSFT12, Proposition 1.2] for the Galois case).
When the residue field of is perfect, we know from [Eld09] that Galois scaffolds exist for all totally ramified biquadratic extensions in characteristic , and for all totally and weakly ramified -extensions in characteristic . To illustrate the sort of explicit information our methods can yield, we examine these two classes of extensions in detail (see Theorems 4.1 and 4.2). However, in this paper we are not primarily concerned with the problem of actually constructing -scaffolds. In a separate paper [BE], we give a criterion for a totally ramified Galois -extension to have a Galois scaffold of a given precision. This enables us to give an explicit construction for a class of extensions in characteristic which admit Galois scaffolds. These have elementary abelian Galois groups of arbitrarily large rank, and are the analogs in characteristic of the near one-dimensional extensions in characteristic constructed in [Eld09]. They include the totally ramified biquadratic extensions and the totally and weakly ramified -extensions satisfying some additional hypotheses. Under these hypotheses, our Galois module results for biquadratic and weakly ramified extensions in characteristic will also hold in characteristic .
Our work is somewhat similar in spirit to that of Bondarko [Bon00, Bon02, Bon06], who considers the existence of ideals free over their associated orders in the context of totally ramified extensions of -power degree. (Unlike us, Bondarko only considers Galois extensions.) Bondarko introduces the class of semistable extensions. Any such extension contains at least one ideal free over its associated order, and all such ideals can be determined from numerical data. Moreover, any abelian extension containing an ideal free over its associated order, and satisfying certain additional assumptions, must be semistable. Abelian semistable extensions can be completely characterized in terms of the Kummer theory of (one-dimensional) formal groups. It would be of interest to understand the precise relationship between Bondarko’s approach and our own, and we intend to return to this question in future work.
Finally, regarding the hypotheses needed on the ground field , we note that our main results on -scaffolds do not require the residue field of to be perfect. However, in order to construct scaffolds on particular families of Galois extensions (as we do in [Eld09, BE13, BE14, BE]), this hypothesis is essential. The hypothesis is also convenient when discussing higher ramification groups, since the standard exposition [Ser79] of higher ramification theory makes use of it at various points. We will therefore not include the condition that has perfect residue field among the running hypotheses of this paper, but will impose it from time to time when considering examples.
1.1. Outline of the paper
In §2 we define the notion of an -scaffold on and obtain some of its properties. A detailed discussion of the relationship between the -scaffolds considered here and the Galois scaffolds of our earlier papers is relegated to an Appendix at the end of the paper. Our main results, Theorems 3.1 and 3.6, relating -scaffolds to generalized integral Galois module structure, will be stated and proved in §3. In §4, we give some applications of our approach to Galois extensions, discussing in detail biquadratic extensions and weakly ramified extensions. Finally, in §5, we consider -scaffolds on inseparable extensions , where is a divided power Hopf algebra.
In this section, we consider a totally ramified extension of local fields, together with a -algebra which has a -linear action on . We assume that the residue field of has characteristic . The characteristic of may be either or . We do not require to be perfect. We assume that has degree , and that .
Before giving the definition of an -scaffold on , we require some notation. We set and , and we identify each with its vector of base- coefficients where
This indexing of the base- digits as , where increasing values of correspond to decreasing powers of , is natural in the context of Galois scaffolds, where the are the ramification breaks (in increasing order), and we need to consider expressions of the form defined in (4) below. We will almost always write in this way.
We further endow with a partial order that is based upon the usual multi-index partial order on , writing (or ) if and only if for . For the convenience of the reader, we record some facts.
Let and write and where , . Then if and only if and there are no carries in the base- addition of and . Furthermore, the following are equivalent:
Assume . Then clearly . Let , and write with . Since , we have . When we perform the addition we get with no carries. On the other hand, assume that and there are no carries in the base- addition of and . As we have , so that for some . Since there are no carries, for . Thus . Therefore for each , so that . The equivalence of (i)–(iv) is then clear. ∎
Associated to an -scaffold on will be a sequence of integer shift parameters, which are required to be relatively prime to . Using these integers, we define a function by
We write for the residue function . The coprimality assumption on the ensures that is bijective. The function , defined by , is therefore also bijective. We denote its inverse by . Abusing notation, we will also write for where , and so regard as a function .
(i) is determined by the residues ;
(ii) if for all then for ;
(iii) if and then ;
(iv) for all ;
(v) for all .
We are now prepared for the definition.
Definition 2.3 (-scaffold on ).
Let , and be as above, and let . Then an -scaffold on of precision with shift parameters consists of
elements for , such that and whenever .
elements for , such that , and such that, for each and for each , there exists a unit making the following congruence modulo hold:
An -scaffold of precision consists of the above data where the congruence in (ii) is replaced by equality.
Condition (ii) in Definition 2.3 should be interpreted as saying that the effect of on is approximated either by a single term or by . The precision determines the accuracy of this approximation, with a precision of meaning that the “approximation” is exact. In more detail, the approximation works as follows. Since is associated with an increase of valuation of , we express the effect of on the basis in terms of the basis . Thus we have
Then (ii) says that when , a condition which is independent of , and each diagonal coefficient is congruent mod to either or a unit of , according to a criterion involving as well as . We observe that the matrix of exponents is constant on each of the diagonals (from top left to bottom right) and the main diagonal resides within a band of diagonals where the exponent is . How this band straddles the main diagonal depends on the residue class .
In all the examples of -scaffolds known to date, we can take all the units in Definition 2.3(ii) to be . Moreover, we can assume , for some fixed uniformizing parameter of , whenever . The extra generality allowed in Definition 2.3 does not significantly add to the complexity of our arguments, and is included since the flexibility it provides may be useful in future applications.
The reader should keep in mind the following situation.
Definition 2.6 (Galois scaffold).
Suppose that is a Galois extension with Galois group . We will call a -scaffold on a Galois scaffold if the residue field is perfect and the shift parameters of the scaffold are the (lower) ramification breaks of , counted with multiplicity in the following sense: we set where is the th ramification group. In particular, the existence of a Galois scaffold means that the ramification breaks are prime to .
Example 2.8 (Galois extensions of degree ).
We show that a totally ramified Galois extension of degree admits a Galois scaffold in almost all cases. There is a unique ramification break , which in characteristic may be any positive integer relatively prime to . In characteristic we have
see [Ser79, IV,§2, Prop. 11 and Ex. 3].)
If we exclude the exceptional case in characteristic then , and we can obtain a Galois scaffold as follows. Let , where is any generator of , let be a uniformizing parameter of , and let with . Then and are given by and . In particular, . For each , put . Then the elements satisfy condition (i) of Definition 2.3. Also, , and unless . But precisely when , in which case , , and . If has characteristic then , so and we have a Galois scaffold of precision . Now suppose that has characteristic . Expanding , we have . Hence
Thus when , so we have a Galois scaffold of precision .
If we replace the element in an -scaffold by , where is some uniformizing parameter of , then we obtain a new scaffold with the same precision , but with the shift parameter replaced by . Suppose that is a Galois extension with ramification breaks . If there exists a Galois scaffold on (whose shift parameters are, by definition, the ), we can adjust the by powers of to obtain a -scaffold whose shift parameters are any integers with ; this new scaffold will in general not be a Galois scaffold, since its shift parameters will not coincide with the ramification breaks. We do not know whether it is possible to have a -scaffold on a Galois extension with shift parameters that do not satisfy the congruences . We do know from [BE13], however, that if is a -extension in characteristic , and there exists a -scaffold on with precision and some shift parameters , , then there will also exist a Galois scaffold on (with the ramification breaks , as its shift parameters) of precision .
In an earlier version of this paper, we called the “tolerance” of the scaffold, and this terminology is used by Koch in [Koc14]. We thank the referee for suggesting the more satisfactory word “precision”.
For each , let be the set of monomials in the (not necessarily commuting) elements such that, for each , the exponents associated with in the monomial sum to . We write for the distinguished element
When is commutative, we have .
Suppose that we have an -scaffold as in Definition 2.3. Then it follows inductively that if , and then there is a unit such that, modulo , we have
Thus we have
From this we may check that is a -basis of and that is a free -module of rank (cf. Proposition 2.12 below). Moreover, for all , and all , so that the algebra is commutative in this case. In general, there are two potential sources of noncommutativity in , namely the “error terms” which are implied by the congruences of Definition 2.3(ii), and the units .
The Normal Basis Theorem ensures, in the Galois case, that is a free -module of rank . We now show that a similar assertion holds whenever we have an -scaffold. Furthermore, satisfies the stronger condition of having a “valuation criterion” for its -module generator.
Let have an -scaffold of precision . Then is a -basis of . Moreover, let be any integer that satisfies , and let with . Then is a free -module on the generator . Additionally, for each , the ring is an -order in .
Since is bijective, the condition determines uniquely mod . We have for and . From (8), for and for each we have . Thus for each . Since is surjective, these valuations represent all residue classes mod . As is totally ramified, it follows that is a -basis for . Thus , and, comparing dimensions, is a free -module on the generator . Moreover, the must be linearly independent over . Since , it follows that the form a -basis of . As is a free -module and spans over , it is immediate that is an -order in . ∎
Suppose we have a Galois scaffold on an abelian extension . By the Hasse-Arf Theorem [Ser79, V, §7], the ramification breaks in the upper numbering are integers. Translating to the lower numbering, we obtain the congruences . Thus we have and . In particular, we can then take in Proposition 2.12 to be . The same will hold if is a nonabelian Galois extension which satisfies the conclusion of the Hasse-Arf Theorem.
3. Integral -module structure
3.1. Statement of the main results
Fix and as in §2. Assume that there is an -scaffold on of precision as in Definition 2.3. Thus we have shift parameters and the associated functions and , as well as elements with for each . By Proposition 2.12, we also have a -basis of . We choose once and for all a uniformizing parameter of .
Now let , and consider the fractional -ideal as a module over its associated order
in . If for some then . It follows that , and that and are isomorphic as modules over this order. Thus only matters up to congruence mod .
Both the order , and the structure of over , depend only on the residue class . Let . Note that , and that is an -basis for . We now fix a specific choice of in Proposition 2.12 (where was only determined mod ) by stipulating
Thus we have .
For each we define
In particular, since . We also define
Using Lemma 2.1, we have
In particular, for all . Note that whether or not the upper bound is achieved depends only on the residue classes , not the integers themselves. In any case, it is important to realize that both and , as well as and , depend on and on , although we do not indicate this dependence explicitly in our notation.
For , we normalize the in (6), and set
The first of our main results explains how the existence of an -scaffold of high enough precision allows us to give an explicit description of , and to determine whether or not is free over , using only the numerical invariants and .
Suppose that . Then is an -basis of . If for all , then is free over with .
Now suppose that the stronger condition holds. Then is free over if and only if for all . Moreover, when is free over , we have for any with .
Since was chosen so that , the stronger condition holds for all ideals if the -scaffold has precision .
Example 3.3 (Galois extensions of degree ).
For a totally ramified Galois extension of degree , the Galois module structure, both of the valuation ring and of its fractional ideals , has been studied extensively. We briefly review the existing results and relate them to Theorem 3.1.
For the valuation ring itself, we have , so the number in Theorem 3.1 is just the least positive residue of mod . For of characteristic , Bertrandias and Ferton [BF72] show that is free over its associated order if and only if divides , provided that is not too close to its maximal value. (See [BBF72] for the excluded cases.) Now , and one can verify that our condition in this case is equivalent to . We therefore recover the result of Bertandias and Ferton whenever we have a Galois scaffold with ; by Example 2.8, this occurs when
In characteristic , Aiba [Aib03] gives a different condition for to be free, but his condition can be shown to be equivalent to ; de Smit and Thomas [dST07] also obtain . Since there is a Galois scaffold with , these results follow from our Theorem 3.1, exactly as in characteristic (but with no upper bound on ).
We now consider arbitrary ideals . In characteristic , Ferton [Fer73] determines which ideals are free over their associated orders, giving her result in terms of the continued fraction expansion of . A corresponding result in characteristic is given by Huynh [Huy14], who gives a different criterion but proves it is equivalent to Ferton’s. Our condition, for all , must therefore be equivalent to Ferton’s continued fraction criterion. This equivalence is verified in [Mar14] (which also contains some partial results relating our Theorem 3.6 below to continued fractions). Given this equivalence, and assuming (14) in the characteristic case, the results of Ferton and Huynh follow from our Theorem 3.1.
The following example considers another situation where the technical details associated with Theorem 3.1 are easy to digest.
Example 3.4 ().
Suppose that is totally ramified extension of degree (for arbitrary ) which admits an -scaffold with precision such that for each . We consider the valuation ring (so ). Write with . Using (4), we see that . Thus and . In particular, for all with , so that for all . Moreover, . Thus by Theorem 3.1(i), is free over , and has the particularly simple form:
We make one further remark, concerning the precision in Theorem 3.1.
In some cases it is possible to relax the assumptions on in Theorem 3.1 at the expense of stronger assumptions on the in Definition 2.3. For example, in [BE13, Theorem 1.1] we give a freeness criterion, which is equivalent to that in Theorem 3.1(ii), for the valuation ring of a cyclic extension of degree in characteristic admitting a different sort of “scaffold”. From the perspective of Definition 2.3, this is a Galois scaffold of precision , but this value is not used in the proof of the result. In fact, although the residue class satisfied by the ramification breaks could be any class mod relatively prime to , the proof of the result requires only that the “scaffold” have precision . In contrast, we would need to assume that to guarantee that Theorem 3.1 applies for all possible values of the ramification breaks. The result in [BE13] depends on the fact that the “scaffold” there satisfies the additional relations and .
The second of our main results, Theorem 3.6, adapts the techniques of [dST07] (see in particular Theorem 4) to extract some further information from the numerical data and . For , , we write if and . Let
Note that and , since there are no relevant in these cases. Thus we always have and . Again, the dependence on and on the is suppressed from the notation.
Let be as in Theorem 3.1, with the strong condition . Then the minimal number of generators of the -module is . Also, is a (not necessarily commutative) local ring with residue field , and, writing for its unique maximal ideal, the embedding dimension of is .
Since is a free -module by Proposition 2.12, the minimal number of generators of over is one precisely when is free over .
We keep the notation of the previous subsection. In particular, admits an -scaffold with precision and with shift parameters , giving rise to the functions and . We fix and study the ideal as a module over its associated order . Recall that is the unique integer satisfying (11).