Alan Koch

Monogenic bialgebras over finite fields and rings of Witt vectors

Abstract We classify a certain collection of bialgebras that are generated by a single element over a finite field k , giving a parameterization using positive integers. We then put necessary and sufficient conditions on this pair so that the bialgebra lifts uniquely to the ring W ( k ) of Witt vectors with coefficients in k , and finally provide a formula for the number of such lifts.

Monogenic Hopf algebras and local Galois module theory

Let K be an unramified extension of Qp, and denote the ring of integers of K by R=OK. Let H be an R-Hopf algebra with monogenic dual H∗. We realize H∗ as the kernel of an isogeny of one-dimensional formal groups. This allows us to give a complete list of fields L for which L/K is H⊗K-Hopf Galois and S=OL is a free H-module.

Endomorphisms of Monogenic Hopf Algebras

Let k be a finite or algebraically closed field of characteristic p > 0, and let H be a k-Hopf algebra. We obtain a classification of End (H), the k-Hopf algebra endomorphisms of H, obtaining a simple Dieudonné module description for each map. For H a Hopf algebra over the ring of Witt vectors W(k) we use finite Honda systems to describe End (H), and obtain concrete results for several special cases, including the case where k = F p as well as the case where H has rank p.

Breuil–Kisin Modules and Hopf Orders in Cyclic Group Rings

For K, a finite extension of ℚ p with ring of integers R, we show how Breuil–Kisin modules can be used to determine Hopf orders in K-Hopf algebras of p-power dimension. We find all cyclic Breuil–Kisin modules and use them to compute all of the Hopf orders in the group ring KΓ where Γ is cyclic of order p or p 2. We also give a Laurent series interpretation of the Breuil–Kisin modules that give these Hopf orders.

p -ADIC ORDER BOUNDED GROUP VALUATIONS ON ABELIAN GROUPS

For a fixed integer e and prime p we construct the p-adic order bounded group valuations for a given abelian group G. These valuations give Hopf orders inside the group ring KG where K is an extension of p with ramification index e. The orders are given explicitly when G is a p-group of order p or p2. An example is given when G is not abelian. 2000 Mathematics Subject Classification. Primary 16W30. Secondary 20K01, 20K27, 20E15. Let p > 0 be prime. Let R be a discrete valuation ring with uniformizing parameter π and with quotient field K, an extension of p. Let e be the absolute ramification index of K/ p. Any finite group G gives rise to group rings RG and KG – these have the structures of an R-Hopf algebra and a K-Hopf algebra respectively. Clearly we have RG ⊂ KG. One of the objectives in local Galois module theory is to find finitely generated projective R-Hopf algebras H such that H ⊗R K ∼= KG. Such Hopf algebras are called Hopf orders in KG (more precisely, R-Hopf orders in KG). There are several reasons why we might want to find such Hopf orders. For example, in the case where G is cyclic order n a classification of R-Hopf orders would yield a classification of group schemes over R with generic fibre μn. Additionally, if L is an extension of K with ring of integers S, and L/K is Galois with group G then S has a normal integral basis over R if and only if the associated order A = {α ∈ KG | α (S) ⊆ S} is an R-Hopf order in KG [3]. While much work has been done in constructing Hopf orders in the case where G is cyclic of order pn for n ≤ 3 – see, for example, [4], [6], [10], and [11] – for many other groups the orders are unknown. In 1976 Larson [8] showed a correspondence between certain Hopf orders in KG and functions G → ≥0 ∪ {∞} satisfying certain properties, where ≥0 is the set of nonnegative integers. These functions are called p-adic order bounded group valuations, and their corresponding orders are called Larson orders. In general Larson orders do not exhaust all of the Hopf orders, yet they remain worthy of study for two reasons. First, they are the only class of Hopf orders constructed in the case where G is nonabelian. Second, they can be useful in constructing other orders – as an example of this the classification of orders in KG, where G is the cyclic group of order p2 was started by Greither in [6] using extensions of Larson orders https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089507003680 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 07 Oct 2019 at 17:25:09, subject to the Cambridge Core terms of use, available at 270 ALAN KOCH AND AUDREY MALAGON of cyclic groups of order p and the work was completed by Underwood in [10] by considering the duals of these “Greither orders.” In this work we shall focus on the case where G is an abelian p-group. It should be pointed out that in the abelian case there are other classes of orders which have been constructed – in addition to the Greither orders above Childs et al. have constructed triangular Hopf orders [5] and Hopf orders via formal groups [4]. We construct the p-adic order bounded group valuations on G, providing explicit calculations in the special cases where G is cyclic and where G is an elementary abelian group. We will also give the corresponding Larson order. While the applications are to local Galois module theory, the calculations are entirely group-theoretic: the approach starts with the construction of a sequence of nested subgroups of G satisfying certain relations. The results in the elementary abelian case will be needed in an upcoming work by the first author [7] and hopefully will be of use in the classification of all Hopf orders in KG for G an elementary abelian p-group. The first section introduces the concept of a p-adic order bounded group valuation. Following this we investigate the case where G is a cyclic group of order p, providing a very easy (and well-known) classification of the corresponding Hopf orders. Then we turn our attention to arbitrary finite abelian groups. This is the point where we introduce the nested sequence of subgroups that a p-adic order bounded group valuation determines, and how we may start with certain nested sequences to construct valuations. Next, we focus on the two special cases mentioned above. Finally, we discuss the difficulties that arise when we try to extend these ideas to the nonabelian case, yet we provide an example in the case that p2 divides the order of G and |G| < p3. Throughout this paper p will denote a fixed prime, K is an extension of p with ramification index e, and we will set e′ = e/ (p − 1) . Furthermore, v will denote the unique extension of the p-adic valuation on p with the property that v (e) = p. While it is common to express the operation in an abelian group additively, we will always use multiplicative notation since it creates less confusion when working with group rings. The authors would like to thank the referee for his useful comments and suggestions in the preparation of the paper. 1. Background. We start with the definition of a p-adic order bounded group valuation. As in the introduction, we use the symbol ≥0 to denote the nonnegative integers. DEFINITION 1.1. Let G be a finite group with identity 1. A p-adic order bounded group valuation is a function ξ : G → ≥0 ∪ {∞} such that, for all g, h ∈ G: GV1. ξ (1) = ∞ and ξ (g) < ∞ if g = 1 GV2. ξ (gh) ≥ min{ξ (g), ξ (h)} GV3. ξ ([g, h]) ≥ ξ (g) + ξ (h) where [g, h] is the commutator of g and h GV4. ξ (g) = 0 if |g| is not a power of p, and ξ (g) ≤ e φ(|g|) for |g| = ps, s ≥ 1 GV5. ξ (gp) ≥ pξ (g). https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089507003680 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 07 Oct 2019 at 17:25:09, subject to the Cambridge Core terms of use, available at p-ADIC ORDER BOUNDED GROUP VALUATIONS 271 Notice that this definition depends not only on p but on the field K – more precisely, the valuation v. It would be more precise to define the above as a “p-adic group valuation with order bounded by e/φ (|g|)”, however we will refer to such a function simply as a p-adic order bounded group valuation on G, or a “p-adic obgv” for short. The motivation for studying p-adic obgv’s is as follows. Let R be the ring of integers of K. Let π be a uniformizing parameter of R. We say an R-Hopf algebra H is a Hopf order in KG if H is finitely generated and projective as an R-module and H ⊗R K ∼= KG. Clearly RG is a simple example of a Hopf order, and in fact every Hopf order in KG contains RG [1, 5.2]. Given a p-adic order bounded group valuation ξ , it is easy to construct a Hopf order. Indeed, define Hξ to be the R-algebra generated by {(g − 1)π−ξ (g)}, where g runs through all of the nontrivial elements of G. Then Hξ has a Hopf algebra structure given by the restriction of the Hopf algebra structure maps on KG, i.e. ( (g − 1)π−ξ (g)) = 1 πξ (g) ( (g) − (1)) = 1 πξ (g) (g ⊗ g − 1 ⊗ 1) ε ( (g − 1)π−ξ (g)) = 1 πξ (g) (ε(g) − ε(1)) = 1 πξ (g) (1 − 1) = 0 λ ( (g − 1)π−ξ (g)) = 1 πξ (g) (λ (g) − λ(1)) = 1 πξ (g) (g−1 − 1). The reader can verify that (Hξ ) ⊂ Hξ ⊗ Hξ and that λ(Hξ ) ⊂ Hξ and hence Hξ is an R-Hopf algebra. By construction Hξ ⊂ KG, and since for all g ∈ G we have g = ((g − 1)π−ξ (g))πξ (g) + 1 and thus RG ⊂ Hξ . It can be shown [1, 18.1] that Hξ is a finitely generated R-module. The “finitely generated” part is a nontrivial argument: note that in what is presented above the only p-adic obgv property we use is that ξ (1) = ∞. Thus Hξ is a Hopf order in KG. These Hopf orders were originally constructed by R. Larson in [8] and are called Larson orders. EXAMPLE 1.2. For any finite group G we can define a map ξ : G → ≥0 ∪ {∞} by ξ (g) = {∞ g = 1 0 g = 1 . It is clear that this map is a p-adic order bounded group valuation. We shall call it the trivial valuation. In this case the corresponding Hopf algebra is generated by {(g − 1) | g ∈ G} and we can see that Hξ = RG. In the case that p |G| this is the only p-adic obgv. Of course, if G is abelian then the commutator is trivial, hence any map ξ : G → ≥0 ∪ {∞} satisfies GV3. Furthermore, if G is an elementary abelian p-group, then gp = 1 for all g and hence GV5 is also satisfied. 2. p-adic OBGV’s on Cp. Let Cp denote the cyclic group of order p. Recall we are viewing this cyclic group multiplicatively; hence 1 will be used to denote the identity https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089507003680 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 07 Oct 2019 at 17:25:09, subject to the Cambridge Core terms of use, available at 272 ALAN KOCH AND AUDREY MALAGON element. We will see it is easy to find all p-adic obgv’s on this group. While the results here would also follow from the work in the next section, the simplicity of the G = Cp case makes it a useful first example. This case is stated as an example in [2, p. 3] and is certainly well-known. We start with a necessary condition. PROPOSITION 2.1. Let ξ be a p-adic obgv on Cp. Then ξ (g) = ξ (h) for all nontrivial g, h ∈ Cp. Proof. Let g, h be nontrivial elements in Cp. Then both g and h are generators of Cp, hence there exist integers m, n ∈ such that gm = h and hn = g. By GV2 we have ξ (h) = ξ (gm) = ξ (g · g · · · · · g) ≥ min{ξ (g), ξ (g), . . . , ξ (g)} = ξ (g) as well as ξ (g) = ξ (hn) = ξ (h · h · · · · · h) ≥ min{ξ (h), ξ (h), . . . , ξ (h)} = ξ (h) and hence ξ (g) = ξ (h). For any p-adic obgv ξ we let ξ (g) denote the range of ξ and |ξ (g)| will be the number of elements in ξ (g). In other words, ξ (g) is the number of distinct values achieved by ξ on G. The elements of ξ (g) will frequently be referred to as “values.” The above proposition shows that a p-adic obgv on Cp has at most one finite value, and so we get: COROLLARY 2.2. Let ξ be a p-adic obgv on Cp. Then |ξ (Cp)| = 2. Now let G be any group, and let H ≤ G. Then any p-adic obgv on G

Primitively generated Hopf orders in characteristic p

ABSTRACT Let R be a characteristic p discrete valuation ring with field of fractions K. Let H be a commutative, cocommutative K-Hopf algebra of p-power rank which is generated as a K-algebra by primitive elements. We construct all of the R-Hopf orders of H in K; each Hopf order corresponds to a solution to a single matrix equation. For R complete, we greatly simplify the matrix equation and give explicit examples of Hopf orders in some rank p2 K-Hopf algebras.

Condition dependent effects on sex allocation and reproductive effort in sequential hermaphrodites.

Theory predicts the optimal timing of sex change will be the age or size at which half of an individuals expected fitness comes through reproduction as a male and half through reproduction as a female. In this way, sex allocation across the lifetime of a sequential hermaphrodite parallels the sex allocation of an outbreeding species exhibiting a 1∶1 ratio of sons to daughters. However, the expectation of a 1∶1 sex ratio is sensitive to variation in individual condition. If individuals within a population vary in condition, high-condition individuals are predicted to make increased allocations to the sex with the higher variance in reproductive success. An oft-cited example of this effect is seen in red deer, Cervus elaphus, in which mothers of high condition are more likely to produce sons, while those in low condition are more likely to produce daughters. Here, we show that individual condition is predicted to similarly affect the pattern of sex allocation, and thus the allocation of reproductive effort, in sequential hermaphrodites. High-condition sex-changers are expected to obtain more than half of their fitness in the high-payoff second sex and, as a result, are expected to reduce the allocation of reproductive effort in the initial sex. While the sex ratio in populations of sequential hermaphrodites is always skewed towards an excess of the initial sex, condition dependence is predicted to increase this effect.