Questions tagged [separable-algebras]

For questions about separable algebras over commutative rings, separable ring extensions, Azumaya algebras, finite etale algebras.

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3
votes
1answer
287 views

Structure theorem for etale algebras over a more general ring than a field

I call etale a finite-type flat $R$-algebra $A$ such that $\Omega_A =0$ (I hope this is the standard definition). In the case where $R=k$ is a field, any such algebra $A$ decomposes as a finite ...
2
votes
0answers
78 views

Classical reductive group schemes vs. unitary groups of separable algebras with involution — reference request

Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...
3
votes
1answer
123 views

Operation of a p'-group on a set of p-power order and fix points

The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical....
1
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0answers
38 views

Maximal separable subalgebras of semisimple algebras

Is anything known about maximal separable subalgebras of semisimple algebras in finite-dimension? Are those subalgebras of unique dimension or isomorphic or conjugate?
2
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0answers
241 views

Proof of this ‘lemme connu’

In the proof of Corollary 10.12 of Exposé I of SGA 1 something like the following is asserted as a ‘known lemma’: Let $k$ be an infinite field and $B$ a finite $k$-algebra. If $B$ is not a product ...
2
votes
1answer
107 views

Separability of a subring and of a pre-image

Let $A \subseteq B \subseteq C$ be commutative rings. A known result, which can be found in De Meyer and Ingraham's book, says that separability of $A \subseteq C$ implies separability of $B \...
2
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0answers
74 views

Separability of the ring extension $A \subset A[T]/(h_1,\ldots,h_n)$

Is there a generalization of Wang's known result, Corollary 8, which says the following: $A \subset B=A[T]/(h(T))=A[w]=B$ is a separable ring extension if and only if $h'(w)$, the formal derivative of ...
5
votes
0answers
153 views

What is the relationship between Frobenius extensions and Separable extensions

Let $R\to S$ be an extension of possibly non-commutative rings. I am interested in the relationship between $R\to S$ being Frobenius and it being separable. If it is a Frobenius extension, then there ...
12
votes
1answer
378 views

Is a “smooth” finite-dimensional algebra separable modulo its radical?

Let $k$ be a field, and let us write the "unadorned" tensor $\otimes$ in place of $\otimes_k$. For a unital finite-dimensional $k$-algebra $A$, let $A^e = A \otimes A^{op}$ denote the enveloping ...
1
vote
0answers
214 views

Separability and smoothness

Let $A \subseteq B$ be commutative noetherian rings. I have found the following claim: "Separability implies smoothness" with the following explanation: "The natural thing is to prove that a separable ...
2
votes
1answer
178 views

Are the fibers of this morphism geometrically regular?

Let $A\rightarrow B$ be a local morphism of complete noetherian rings making $B$ a formally smooth $A$-algebra. Does the induced morphism $\textrm{Spec}(B)\to\textrm{Spec}(A)$ have geometrically ...
10
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1answer
348 views

Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?

Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My ...
0
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0answers
823 views

Finite separable extension of fields imply the number of intermediate subfield is finite

The proof of statements either uses Galois theory or Artin primitive element theorem.I would like to know whether there is a proof without using these.The reason to avoid using Galois theory is that ...
7
votes
0answers
229 views

Non Commutative Hyperspaces

Let $X$ be a compact metric space. Recall that the hyperspace $2^{X}$ is the set of all non empty compact subsets of $X$ with the Hausdorff metric. Assume that $\mathcal{C}$ is the category of all ...
4
votes
2answers
435 views

An R-algebra A is R-separable if and only if all derivations are inner.

Hello everybody. I'm readying about derivations. It is very very known fact that all derivations $\delta: A\rightarrow M$ (A R-algebra, M A-module) are inner when the algebra is R-separable. ...
2
votes
1answer
369 views

On the Separability of Certain Extensions of Fields

I asked this question a couple of weeks ago. The question arose while looking for a criterion of separability for extensions of fraction fields $K(A)\to K(B)$ induced by a faithfully flat morphism $...
7
votes
1answer
336 views

Hochschild H^1 (R,M) = 0 vs. H_1 (R,M) = 0 where R is a ring and M is an (R,R)-bimodule

Let $k$ be a commutative ring (with unity). Let $R$ be a $k$-algebra (with unity, but not of necessity commutative). Let $M$ be an $\left(R,R\right)$-bimodule where $k$ acts in the same way from the ...