Separability in algebra and category theory

  title={Separability in algebra and category theory},
  author={R. Wisbauer},
Separable field extensions are essentially known since the 19th century and their formal definition was given by Ernst Steinitz in 1910. In this survey we first recall this notion and equivalent characterisations. Then we outline how these were extended to more general structures, leading to separable algebras (over rings), Frobenius algebras, (non associative) Azumaya algebras, coalgebras, Hopf algebras, and eventually to separable functors. The purpose of the talk is to demonstrate that the… Expand
Algebraic and coalgebraic structures are often handled independently. In this survey we want to show that they both show up naturally when approaching them from a categorical point of view. Azumaya,Expand
Object-unital groupoid graded rings, crossed products and separability
Abstract We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal)Expand


Algebras Versus Coalgebras
  • R. Wisbauer
  • Computer Science, Mathematics
  • Appl. Categorical Struct.
  • 2008
This survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 1970s, to look at the interplay between algebraic and coalgebraic notions. Expand
Coalgebraic structures in module theory
Although coalgebras and coalgebraic structures are well-known for a long time it is only in recent years that they are getting new attention from people working in algebra and module theory. TheExpand
Hopf Semialgebras
In this paper, we introduce and investigate bisemialgebras and Hopf semialgebras over commutative semirings. We generalize to the semialgebraic context several results on bialgebras and Hopf algebrasExpand
Separable algebras over commutative rings
Introduction. The main objects of study in this paper are the commutative separable algebras over a commutative ring. Noncommutative separable algebras have been studied in [2]. Commutative separableExpand
Strongly separable pairings of rings
The theory of adjoint functors has been used by Morita to develop a theory of Frobenius and quasi-Frobenius extensions subsuming the work of Kasch, Miller, Nakayama, and others. We use adjointExpand
The Brauer group of a commutative ring
Introduction. This paper contains the foundations of a general theory of separable algebras over arbitrary commutative rings. Of the various equivalent conditions for separability in the classicalExpand
A bialgebraic approach to automata and formal language theory
It is shown that K -linear automaton morphisms can be used as the sole rule of inference in a complete proof system for automaton equivalence and there is an adjunction between the categories of “algebraic” automata and the category of deterministic automata. Expand
Adjoint pairs of functors and Frobenius extensions
Throughout this paper A and B are assumed to be associative rings which possess identity elements 1A and IB respectively. The category of all left (resp. right) A-modules will be denoted by A9JeExpand
Let A be an algebra over a field k. If M is an A–bimodule, we let M and MA denote respectively the k–spaces of invariants and coinvariants of M , and φM : M A → MA be the natural map. In this note weExpand
Hopf Galois theory: A survey
We consider a Hopf Galois extension B A, for A a comodule algebra over the Hopf algebra H with coinvariant algebra B . After giving a number of examples, we discuss Galois extensions with additionalExpand