A note on Galois theory for bialgebroids

双代数胚的Hopf双模

Hopf bimodules for bialgebroids

本文在双代数胚框架下定义四类Hopf模及双边双余边Hopf模（Hopf双模），通过Hopf-Galois余模证明基本定理，并构造两种辫子幺半范畴等价于Yetter-Drinfeld模范畴。

Dual Bialgebroids for Depth Two Ring Extensions

We introduce a general notion of depth two for ring homomorphism N --> M, and derive Morita equivalence of the step one and three centralizers, R = C_M(N) and C = End_{N-M}(M ø_N M), via dual bimodules and step two centralizers A = End_NM_N and B = (M ø_N M)^N, in a Jones tower above N --> M. Lu's bialgebroids End_k A' and A' ø_k {A'}^op over a k-algebra A' are generalized to left and right bialgebroids A and B with B the R-dual bialgebroid of A. We introduce Galois-type actions of A on M and B on End_NM when M_N is a balanced module. In the case of Frobenius extensions M | N, we prove an endo…

Galois theory for Hopf algebroids

An extension B\subset A of algebras over a commutative ring k is an H-extension for an L-bialgebroid H if A is an H-comodule algebra and B is the subalgebra of its coinvariants. It is H-Galois if the canonical map A\otimes_B A\to A\otimes_L H is an isomorphism or, equivalently, if the canonical coring (A\otimes_L H:A) is a Galois coring. In the case of a Hopf algebroid H=(H_L,H_R,S) any H_R-extension is shown to be also an H_L-extension. If the antipode is bijective then also the notions of H_R-Galois extensions and of H_L-Galois extensions are proven to coincide. Results about bijective entwi…

Hopf algebroids and Galois extensions

To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := \End_BA_B$ and $T := (A ø_B A)^B$ over the centralizer $R$ using the depth two theory in math.RA/0108067. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra \cite{KT,Doi} to depth two extensions using coring theory math.RA/0002105. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem 5.1 in math.QA/9505024 for smash products with special modules over the Drinfel'd double, the Miy…

Semisimple Hopf algebras and their depth two Hopf subalgebras

We prove that a depth two Hopf subalgebra K of a semisimple Hopf algebra H is normal (where the ground field $k$ is algebraically closed of characteristic zero). This means on the one hand that a Hopf subalgebra is normal when inducing (then restricting) modules several times as opposed to one time creates no new simple constituents. This point of view was taken in the paper http://arxiv.org/abs/math/0409346 which established a normality result in case H and K are finite group algebras. On the other hand this means that K is normal in H when H | K is a Galois extension with respect to action o…

A Note on a Standard Model for Galois Rings

In this work, we present a standard model for Galois rings based on the standard model of their residual fields, that is, a sequence of Galois rings starting with ${\mathbb Z}_{p^r} that coves all the Galois rings with that characteristic ring and such that there is an algorithm producing each member of the sequence whose input is the size of the required ring.

Hopf代数胚的Hopf双Galois理论

Hopf BiGalois Theory for Hopf Algebroids

该研究发展了不要求对极映射的Hopf代数胚的Hopf双Galois理论，利用Chemla的结果解决了右Galois映射非良定义的障碍，构造了Ehresmann Hopf代数胚并将其应用于量子结构示例。

Morita Contexts for Corings and Equivalences

In this note we study Morita contexts and Galois extensions for corings. For a coring $\QTR{cal}{C}$ over a (not necessarily commutative) ground ring $A$ we give equivalent conditions for $\QTR{cal}{M}^{\QTR{cal}{C}}$ to satisfy the weak. resp. the strong structure theorem. We also characterize the so called \QTR{em}{cleft}$C$\QTR{em}{-Galois extensions} over commutative rings. Our approach is similar to that of Y. Doi and A. Masuoka in their work on (cleft) $H$-Galois extensions.

Galois理论

Galois Theory

本研究是爱丁堡大学2021-2023年本科Galois理论课程讲义，以代数基础知识为前提，讲解域扩张与Galois群理论至基本定理，还包含尺规作图、可解多项式相关证明及有限域分类。

On Galois comodules

Generalising the notion of Galois corings, Galois comodules were introduced as comodules $P$ over an $A$-coring $\cC$ for which $P_A$ is finitely generated and projective and the evaluation map $μ_\cC:\Hom^\cC(P,\cC)\ot_SP\to \cC$ is an isomorphism (of corings) where $S=\End^\cC(P)$. It was observed that for such comodules the functors $\Hom_A(P,-)\ot_SP$ and $-\ot_A\cC$ from the category of right $A$-modules to the category of right $\cC$-comodules are isomorphic. In this note we call modules $P$ with this property {\em Galois comodules} without requiring $P_A$ to be finitely generated and pr…

Projective modules over Frobenius algebras and Hopf comodule algebras

This note presents some results on projective modules and the Grothendieck groups K_0 and G_0 for Frobenius algebras and for certain Hopf Galois extensions. Our principal technical tools are the Higman trace for Frobenius algebras and a product formula for Hattori-Stallings ranks of projectives over Hopf Galois extensions.

Difference Galois Theory For The "Applied" Mathematician

The lecture notes below correspond to the course given by the author in occasion of the VIASM school on Number Theory (18-24 June 2018, Hanoi). We have chosen to omit the proofs that are already presented in details in many references in the literature, although they were explained during the lectures, and we have devoted more space to statements useful in the applications, in particular to differential transcendence.

有限域的伽罗华理论

Galois theory for finite fields

本文阐述了有限域的伽罗华理论，通过描述中间域与次数n的除数之间的对应关系，证明了伽罗华对应定理，并介绍了通用域的伽罗华理论基本定理。

关于环扩展中莫里塔等价的注记

Note on Morita equivalence in ring extensions

本文研究环扩展中莫里塔等价的性质，证明若干扩展类别的莫里塔不变性，并给出非莫里塔不变的环扩展类别的例子。