Hopf Order Research Articles

Galois scaffolds for cyclic $$p^n$$-extensions in characteristic p

Let K be a local field of characteristic p and let L/K be a totally ramified Galois extension such that \({{\,\mathrm{Gal}\,}}(L/K)\cong C_{p^n}\). In this paper we find sufficient conditions for L/K to admit a Galois scaffold, as defined in Byott et al (Ann Inst Fourier 68:965–1010, 2018). This leads to sufficient conditions for the ring of integers \({\mathcal {O}}_L\) to be free of rank 1 over its associated order \({\mathfrak {A}}\), and to stricter conditions which imply that \({\mathfrak {A}}\) is a Hopf order in the group ring \(K[C_{p^n}]\).

Orders of Nikshych's Hopf algebra

Let p be an odd prime number and K a number field having a primitive p th root of unity \zeta_p . We prove that Nikshych's non group-theoretical Hopf algebra H_p , which is defined over \mathbb Q(\zeta_p) , admits a Hopf order over the ring of integers \mathcal O_K if and only if there is an ideal I of \mathcal O_K such that I^{2(p-1)} = (p) . This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over \mathcal O_K exists, it is unique and we describe it explicitly.

Primitively generated Hopf orders in characteristic p

ABSTRACTLet 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.

Hopf powers and orders for some bismash products

The exponent of a finite group G can be viewed as a Hopf algebraic invariant of the group algebra H = kG : it is the least integer n for which the nth Hopf power endomorphism [ n ] of H is trivial. The exponent of a group scheme G as studied by Gabriel and Tate and Oort can be defined in the same way using the coordinate Hopf algebra H = O ( G ) . The power map and the corresponding notion of exponent have been studied for a general finite-dimensional Hopf algebra beginning with work of Kashina . Several positive results, suggested by analogy to the group case, were proved by Kashina and by Etingof and Gelaki. Given these positive results, there was some hope that the Hopf order of an individual element of a Hopf algebra might also be a well-behaved notion, with some properties analogous to well-known facts on the orders of elements of a finite group. In fact we prove that such analogous properties do hold for Hopf algebras satisfying the usual rule for iterated powers; for example, such a Hopf algebra H has an element of order n if and only if n divides the exponent of H. However, in general such properties are not true. We will give examples where the behavior of Hopf powers, Hopf orders, and related notions is rather strange, unexpected, and seemingly hard to predict. We will see this using computer algebra calculations in Drinfeld doubles of finite groups, and more generally in bismash products constructed from factorizable groups.

Maximal tame extensions over Hopf orders in rings of integers of [formula omitted]-adic number fields

Let K/ k be a cyclic totally ramified Kummer extension of degree p n with Galois group G. Let O and o be the rings of integers in K and k, respectively. For n=1, F. Bertrandias and M.-J. Ferton determined the ring End oG ( O) of oG -endomorphisms of O and L.N. Childs constructed the maximal order S in O whose ring of oG -endomorphisms is a Hopf order. A Hopf order whose linear dual is a Larson order in kG is called a dual Larson order. In this paper, the generators of a dual Larson order are given. We determine End oG ( O) and obtain a maximal dual Larson order contained in End oG ( O) . We construct a Hopf Galois extension S in O , whose ring H( S) of oG -endomorphisms is a dual Larson order. We affirm an order S′ in O with a dual Larson order H( S′) is a tame H( S′)-extension and show that S is maximal in such orders S′.

Cyclic Hopf orders defined by isogenies of formal groups

Using degree 2 polynomial formal groups, we construct Hopf algebras over valuation rings of local number fields K that are orders in KG, G cyclic of order p n . The construction does not yield all Greither orders when n = 2, but does yield new classes of Hopf orders for all n ≥= 3, and shows that a general Hopf order in KG, G cyclic of order p n , involves at least n ( n +1)/2 parameters.

Hopf filtrations and Larson-type orders in Hopf algebras

The theory of orders and maximal orders over Dedekind domains or valuation-rings is an important ingredient in the study of arithmetical properties of algebras, in particular central simple algebras over fields. In [2], Larson introduced certain Hopf orders in a group algebra KG, where G is a finite group. The essential tool is a Zassenhaus valuation of G. Because of the multiplicative nature of that theory, those so-called Larson orders cannot be defined in other Hopf algebras without essential modification of the theory. In this paper we present the idea to replace the valuation filtration by a more general filtration, obtaining a bijective correspondence between the “Hopf filtrations” thus obtained and Hopf orders appearing as the degree zero part in such filtration. Just like in the group algebra case it is then possible to relate a Hopf order to an arithmetical object, i.e., a function ξ :H → R+ ∪ {∞}. These functions exhibit specific properties comparable to the ones expected for a valuation-like order function, i.e., so that −ξ should rightfully be called a “Hopf valuation.” We point out that −ξ can be expressed as a function defined on subspaces in H so that it may be thought of as a generalized place (pseudo-place) in the sense of [4]. In Section 2 we introduce the deformation of a Hopf (valuation) filtration and establish how it can be used to construct new Hopf orders. This explains in a general Hopf algebra framework how Larson-type orders come into being. We provide several examples, e.g., for Sweedler’s 4-dimensional Hopf algebra, Taft algebras, etc. in Section 4. In a forthcoming paper we apply the theory of Hopf orders to the theory of orders in H -module algebras.

Integral Hopf–Galois Structures on Degree p2 Extensions of p-adic Fields

Let L/K be a totally ramified, normal extension of p-adic fields of degree p2. We investigate the behavior of the valuation ring OL in the various Hopf–Galois structures on L/K. Specifically, we determine when OL is Hopf–Galois with respect to a Hopf order in the corresponding Hopf algebra. When this occurs, OL is necessarily a free module over this Hopf order. We also determine which Hopf orders can arise in this way. For cyclic extensions L/K of degree p2, L. N. Childs has shown, under certain restrictions on the ramification numbers, that if OL is Hopf–Galois with respect to a Hopf order in one of the Hopf–Galois structures on L/K, then the same is true in all p Hopf–Galois structures on L/K. We show that this no longer holds if the ramification conditions are relaxed, or if elementary abelian extensions of degree of p2 are considered. We illustrate our results with a special family of Kummer extensions, and with certain extensions arising from Lubin–Tate formal groups.

Local Leopoldt's problem for rings of integers in Abelian $p$-extensions of complete discrete valuation fields

Using the standard duality we construct a linear embedding of an associated module for a pair of ideals in an extension of a Dedekind ring into a tensor square of its fraction field. Using this map we investigate properties of the coefficient-wise multiplication on associated orders and modules of ideals. This technique allows to study the question of determining when the ring of integers is free over its associated order. We answer this question for an Abelian totally wildly ramified p -extension of complete discrete valuation fields whose different is generated by an element of the base field. We also determine when the ring of integers is free over a Hopf order as a Galois module.

An infinite family of elliptic curves and Galois module structure

Let E be an elliptic curve dened over a number eld F with everywhere good reduction. By dividing F -rational torsion points with respect to the group law of E M. Taylor dened certain Kummer orders and studied their Galois module structure. His results led to the conjecture that these Kummer orders are free over an explicitly given Hopf order. In this paper we prove that the conjecture does not hold for innitely many elliptic curves which are dened over quadratic imaginary number elds k and endowed with a k-rational 2torsion point.

Tame Realisable Classes over Hopf Orders

LetObe the ring of algebraic integers in a number fieldKand letGbe a finite abelian group. McCulloh has characterised the classes in the locally free classgroup Cl(OG) which are realisable by rings of integers of tame normal extensions ofKwith Galois groupG. We extend this to certain extensions which in general need be neither tame nor normal by replacingOGwith a Hopf order A and introducing the strongly A-tame orders as an analogue of tame rings of integers. We also describe the classes realised by principal homogeneous spaces over the dual of A.

The valuative condition and R-Hopf algebra orders in KC p 3

Let K be a finite extension of Q p endowed with the p-adic valuation and let R be its ring of integers. In this paper we give a complete classification of R-Hopf algebra orders in the group ring KC p 3 . For an arbitrary Hopf order H , we show that either the group scheme SpH or the group scheme SpH *, with H * the linear dual, can be viewed as the middle term of the Baer product of a distinguished extension with a generically trivial extension. Using this characterization we then compute algebra generators for either H or H *.

Hopf Orders and a Generalization of a Theorem of L. R. McCulloh

Let K be an algebraic number field with ring of integers O and let G be an elementary abelian group of order l k . Let A be a Hopf order in KG and let B be its dual. An order X in a Galois G-extension of K is a semilocal principal homogeneous space over B if X l is a principal homogeneous space over B l and X is integrally closed away from l. We define a map ψ from the group of such X to the locally free classgroup Cl( A ). Assuming that A admits C ≅ F l k ×⊆ Aut( G), we describe the image of ψ in terms of a Stickelberger ideal in Z C. This generalizes a result of L. R. McCulloh on the classes in Cl( O G) realized by tame rings of integers.

Cleft Extensions of Hopf Algebras, II

Let K be a p-adic field, let G and U be groups of order p, and let H) (respectively A) be a Hopf order in the group algebra KG (respectively in the algebra of maps U→K). We use the algebraic machinery of [2] to determine the cleft Hopf algebra extensions of A by H, and investigate which of these cleft extensions are Hopf orders in a group algebra.

Taming Wild Extensions with Hopf Algebras

Let $K \subset L$ be a Galois extension of number fields with abelian Galois group $G$ and rings of integers $R \subset S$, and let $\mathcal {A}$ be the order of $S$ in $KG$. If $\mathcal {A}$ is a Hopf $R$-algebra with operations induced from $KG$, then $S$ is locally isomorphic to $\mathcal {A}$ as $\mathcal {A}$-module. Criteria are found for $\mathcal {A}$ to be a Hopf algebra when $K = {\mathbf {Q}}$ or when $L/K$ is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over $R$ in $L$ which are tame or Galois $H$-extensions, $H$ a Hopf order in $KG$, using a generalization of the discriminant.

Taming wild extensions with Hopf algebras

Let K ⊂ L K \subset L be a Galois extension of number fields with abelian Galois group G G and rings of integers R ⊂ S R \subset S , and let A \mathcal {A} be the order of S S in K G KG . If A \mathcal {A} is a Hopf R R -algebra with operations induced from K G KG , then S S is locally isomorphic to A \mathcal {A} as A \mathcal {A} -module. Criteria are found for A \mathcal {A} to be a Hopf algebra when K = Q K = {\mathbf {Q}} or when L / K L/K is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over R R in L L which are tame or Galois H H -extensions, H H a Hopf order in K G KG , using a generalization of the discriminant.