H-separable Extension Research Articles

Uniquely separable extensions

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only if the H-depth is 1 (H-separable extension). Similarly, a split extension has a bimodule projection; this idempotent is full if and only if the ring extension has depth 1 (centrally projective extension). Separable and split extensions have separability idempotents and bimodule projections in 1–1 correspondence via an endomorphism ring theorem in Section 3. If the separable idempotent is unique, then the separable extension is called uniquely separable. A Frobenius extension with invertible E-index is uniquely separable if the centralizer equals the center of the over-ring. It is also shown that a uniquely separable extension of semisimple complex algebras with invertible E-index has depth 1. Earlier group-theoretic results are recovered and related to depth 1. The dual notion, uniquely split extension, only occurs trivially for finite group algebra extensions over complex numbers.

Odd H-depth and H-separable extensions

Abstract Let C n(A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if C n+1(A,B) is isomorphic to a direct summand of a multiple of C n(A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B. In this paper the H-depth of semisimple subalgebra pairs is derived from the transpose inclusion matrix, and for QF extensions it is derived from the odd depth of the endomorphism ring extension. For general extensions characterizations of H-depth are possible using the H-equivalence generalization of Morita theory.

Simplicial Hochschild cochains as an Amitsur complex 1

It is shown how the cochain complex of the relative Hochschild A-valued cochains of a depth two extension A|B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End BAB over the centralizer R = AB with grouplike element 1S. This specializes to finite dimensional algebras, Hopf-Galois extensions and H-separable extensions.

Simplicial Hochschild cochains as an Amitsur complex

It is shown that the cochain complex of relative Hochschild A-valued cochains of a depth two extension A | B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End {}_BA_B over the centralizer R = A^B with grouplike element 1_S, which itself is isomorphic to the Cartier complex of S with coefficients in the (S,S)-bicomodule R^e. This specializes to finite dimensional algebras, H-separable extensions and Hopf-Galois extensions.

Depth Two, Normality, and a Trace Ideal Condition for Frobenius Extensions

A ring extension A ‖ B is depth two if its tensor-square satisfies a projectivity condition w.r.t. the bimodules A A B and B A A . In this case the structures (A ⊗ B A) B and End B A B are bialgebroids over the centralizer C A (B) and there is a certain Galois theory associated to the extension and its endomorphism ring. We specialize the notion of depth two to induced representations of semisimple algebras and character theory of finite groups. We show that depth two subgroups over the complex numbers are normal subgroups. As a converse, we observe that normal Hopf subalgebras over a field are depth two extensions. A generalized Miyashita–Ulbrich action on the centralizer of a ring extension is introduced, and applied to a study of depth two and separable extensions, which yields new characterizations of separable and H-separable extensions. With a view to the problem of when separable extensions are Frobenius, we supply a trace ideal condition for when a ring extension is Frobenius.

Hopf algebroids and Galois extensions

To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := End\,_BA_B$ and $T := (A \o_B A)^B$ over the centralizer $R$ using the depth two theory in [18,Kadison-Szlachányi]. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra [21,8] to depth two extensions using coring theory [3]. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem \cite[23,5.1] for smash products with special modules over the Drinfel'd double, the Miyashita-Ulbrich action, the fact that $R$ is a commutative algebra in the pre-braided category of Yetter-Drinfel'd modules [28] and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double [24]. In our last section, an exposition of results of Sugano [29,30] leads us to a Galois correspondence between sub-Hopf algebroids of $S$ over simple subalgebras of the centralizer with finite projective intermediate simple subrings of a finite projective H-separable extension of simple rings $A \supseteq B$.

On H-separable extensions of QF-3 rings

Let a ring A be an H-separable extension of a subring B of A, that is, A ⊗ B A is an A-A-direct summand of a finite direct sum of copies of A. If furthermore, (a) B is a left B-direct summand of A (or (b) A is left B-finitely generated projective), and if B is a right (resp. left) artinian QF-3 ring, then A is also a right (resp. left) artinian QF-3 ring. Consequently, if A is an H-separable extension of a serial ring B with one of the conditions (a), (b), then A is also a serial ring. In particular H-separable extension of a uni-serial ring is always uni-serial.

Hopf algebroids and H-separable extensions

Since an H-separable extension A|B is of depth two, we associate to it dual bialgebroids S:= End B A B and T:= (A ⊗B A) B over the centralizer R as in Kadison-Szlachanyi. We show that S has an antipode τ and is a Hopf algebroid. T op is also Hopf algebroid under the condition that the centralizer R is an Azumaya algebra over the center Z of A. For depth two extension A|B, we show that End A A ⊗ B A ≅ T α End B A.

Skew group rings which are azumaya

If S is a ring with 1, and G is a finite group acting faithfully as automorphisms of S, then it is well known that the skew group ring S ∗ G is a separable extension of S if and only if there exists a central element in S with trace one. A ring is an Azumaya algebra if it is separable over its center. The case of when the group ring S[G] is Azumaya was studied by De Meyer and Janusz in [2] ; and lately the case of twisted group rings was studied by Szeto and Wong in [5] ; but the techniques used on those cases cannot be applied to the skew group rings precisely because the elements of the ring S do not commute with the elements of the group G. The purpose of this paper is to give conditions on the action of the group in order to make the skew group ring an Azumaya algebra. A ring A is said to be separable over a subring B if the (A − A)-module homomorphism of A⊗BA onto A defined by a⊗b 7−→ ab splits, and A is called H-separable over B if A ⊗B A is isomorphic as (A − A)-bimodule to a direct summand of A for some n ≥ 1. Clearly H-separable extension are separable. Let S be a ring with 1 and G be a finite group acting as automorphisms of S with fixed ring S; that we will denote R. The skew group ring S ∗G is the free left S-module with basis G, where multiplication is defined according

On H-separable extensions of two sided simple rings II

On H-separable extensions of two sided simple rings II