# Questions tagged [separable-algebras]

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

17
questions

**3**

votes

**1**answer

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

**0**answers

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

**1**answer

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**

vote

**0**answers

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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**0**answers

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

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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 ...