Non isomorphic pure Galois-Eisenstein rings

Non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras

We give a degree 8 separable extension having two non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras.

The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions

Let $L/F$ be a Galois extension of fields with Galois group isomorphic to the quaternion group of order $ 8 $. We describe all of the Hopf-Galois structures admitted by $ L/F $, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebras. In the case that $ F $ has characteristic zero we also determine which of these Hopf algebras are isomorphic as $ F $-algebras and explicitly compute their Wedderburn-Artin decompositions.

Hopf-Galois extensions and isomorphisms of small categories

We associate two linear categories with two objects to a module over the subalgebra of coinvariants of a Hopf-Galois extension, and prove that they are isomorphic. The structure Theorem for cleft extensions, and the Militaru \cStefan lifting Theorem can be obtained using these isomorphisms.

Hopf-Galois Structures of Squarefree Degree

Let $n$ be a squarefree natural number, and let $G$, $Γ$ be two groups of order $n$. We determine the number of Hopf-Galois structures of type $G$ admitted by a Galois extension of fields with Galois group isomorphic to $Γ$. We give some examples, including a full treatment of the case where $n$ is the product of three primes.

Hopf Galois结构，阶为$p^nq$的群的斜brace：循环Sylow子群情形

Hopf Galois structures, skew braces for groups of size $p^nq$: The cyclic Sylow subgroup case

本文研究了阶为$p^nq$且Sylow-$p$-子群为循环群的群上的Hopf-Galois结构，枚举了Galois $G$-扩张上类型为$N$的结构，计算了对应的斜brace数量，并在$q<p$时给出了完整分类。

Infinite Comatrix Corings

We characterize the corings whose category of comodules has a generating set of small projective comodules in terms of the (non commutative) descent theory. In order to extricate the structure of these corings, we give a generalization of the notions of comatrix coring and Galois comodule which avoid finiteness conditions. A sufficient condition for a coring to be isomorphic to an infinite comatrix coring is found. We deduce in particular that any coalgebra over a field and the coring associated to a group-graded ring are isomorphic to adequate infinite comatrix corings. We also characterize w…