论文标题
在$ρ$ -Conjugate Hopf-Galois结构上
On $ ρ$-conjugate Hopf-Galois structures
论文作者
论文摘要
hopf-galois结构由galois $ l/k $的Galois组$ g $与某些子组的$ \ mathrm {permrm {perm}（g）$相对应。我们使用此类子组集的自然分区来获取一种方法来分区相应的hopf-galois结构集，我们将其称为$ρ$ -Conjugation。我们研究了这种结构的特性，特别着重于Galois对应关系的Hopf-Galois类似物，与偏斜左括号的连接以及对本地或全球领域扩展中整体模块结构问题的应用。特别是，我们表明，给定的hopf-galois结构的不同$ρ$ - 缀合物的数量取决于相应的偏斜左撑杆，并且如果$ h，h'$是hopf代数提供$ρ$ρ$ - conjugate-conjugate hopf-galois hop-galois结构当它在$ h $中的相关顺序上，仅当它在$ h'$中免费的相关顺序免费。我们展示了与文献中现有结构相互作用引起的各种例子。
The Hopf-Galois structures admitted by a Galois extension of fields $ L/K $ with Galois group $ G $ correspond bijectively with certain subgroups of $ \mathrm{Perm}(G) $. We use a natural partition of the set of such subgroups to obtain a method for partitioning the set of corresponding Hopf-Galois structures, which we term $ ρ$-conjugation. We study properties of this construction, with particular emphasis on the Hopf-Galois analogue of the Galois correspondence, the connection with skew left braces, and applications to questions of integral module structure in extensions of local or global fields. In particular, we show that the number of distinct $ ρ$-conjugates of a given Hopf-Galois structure is determined by the corresponding skew left brace, and that if $ H, H' $ are Hopf algebras giving $ ρ$-conjugate Hopf-Galois structures on a Galois extension of local or global fields $ L/K $ then an ambiguous ideal $ \mathfrak{B} $ of $ L $ is free over its associated order in $ H $ if and only if it is free over its associated order in $ H' $. We exhibit a variety of examples arising from interactions with existing constructions in the literature.
