Uniquely separable extensions

doi:10.1016/j.exmath.2020.01.003

Expositiones Mathematicae

Volume 38, Issue 2, June 2020, Pages 217-231

In memory of my father

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Abstract

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.

Section snippets

Introduction and preliminaries

The classical notion of separable algebra is one of a semisimple algebra that remains semisimple under every base field extension. The approach of Hochschild to Wedderburn’s theory of associative algebras in the Annals was cohomological, and characterized a separable $k$ -algebra $A$ by having a separability idempotent in $A^{e} = A \otimes_{k} A^{op}$ . A separable extension of noncommutative rings is characterized similarly by possessing separability elements in [13]: separable extensions are shown to be left and right

Full idempotent characterizations of H-separable and centrally projective ring extensions

In this section, we give characterizations of H-depth 1 and depth 1 ring extension in terms of well-known idempotents being full idempotents. Recall the equivalent form of Morita theory, which states that for a ring $A$ with idempotent $e \in A$ , $e A e$ is Morita equivalent to $A$ so long as $e$ is a full idempotent, i.e. $A e A = A$ : the Morita context bimodules are $e A$ and $A e$ [1]. Recall from the previous section the ring structure on $T = {(A \otimes_{B} A)}^{B}$ of a ring extension $A \supseteq B$ .

Proposition 2.1

A separable extension $R \supseteq S$ is H-separable if

Endomorphism ring theorem for uniquely separable extension

A uniquely separable extension is a ring extension $A | B$ , which is a separable extension having a unique separability idempotent. In this section we prove an endomorphism ring theorem for separable extensions $A | B$ showing that separability idempotents are in one-to-one correspondence with bimodule projections from $End A_{B}$ onto $A$ (embedded as left multiplication endomorphisms). For this reason, we define a uniquely split extension $A | B$ as a ring extension possessing a unique bimodule projection (or

Uniquely separable frobenius extensions

Recall that a Frobenius (ring) extension $A \supseteq B$ is characterized by having a (Frobenius) homomorphism $E : A \to B$ in $Hom (_{B} A_{B},_{B} B_{B})$ with elements (dual bases) $x_{i}, y_{i} \in A$ ( $i = 1, \dots, n$ ) such that ${id}_{A} = \sum_{i = 1}^{n} E (- x_{i}) y_{i} = \sum_{i = 1}^{m} x_{i} E (y_{i} -)$ . Equivalently, $A_{B}$ is finite projective and $A ≅ Hom (A_{B}, B_{B})$ as natural $B$ - $A$ -bimodules: see [15] for more details. For example, given a group $G$ and subgroup $H$ of finite index $n$ with right coset representatives $g_{1}, \dots, g_{n}$ , $K$ an arbitrary commutative ring, the group algebra $A = K G$ is a Frobenius

Group algebra extensions

Let $A = k G$ where $G$ is a finite group with subgroup $H < G$ , and let $B = k H \subseteq A$ where $k$ is a field containing the inverse of $| G : H | 1$ . Then $A \supseteq B$ is a split, separable Frobenius extension. If $g_{1}, \dots, g_{n} \in G$ is a right transversal of $H$ in $G$ , then $e = \frac{1}{| G : H |} \sum_{i = 1}^{n} {g_{i}}^{- 1} \otimes_{B} g_{i}$ is a separability element. The next theorem is a consequence of a theorem by Singh–Hanna in [32], which we show is also a consequence of Theorem 4.1.

Theorem 5.1

[32]

A separable finite group algebra extension $A \supseteq B$ has unique separability element $e \in A \otimes_{B} A$ if and only if

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Uniquely separable extensions

Abstract

Section snippets

Introduction and preliminaries

Full idempotent characterizations of H-separable and centrally projective ring extensions

Endomorphism ring theorem for uniquely separable extension

Uniquely separable frobenius extensions

Group algebra extensions

[32]

J. Algebra

J. Algebra

J. Algebra

J. Pure Appl. Algebra

Adv. Math.

J. Algebra

Adv. Math.

Group algebra extensions of depth one

Algebra Number Theory

Depth one extensions of semisimple algebras and Hopf subalgebras

Algebra Rep. Theory

Subgroups of depth three

Surv. Differ. Geom.

On subgroup depth

I.E.J.A.

Are biseparable extensions Frobenius?

K-Theory

The depth of subgroups of PSL(2,q)

J. Algebra

Some types of separable extensions of rings

Nagoya Math. J.

The depth of subgroups of $PSL (2, q)$