# Questions tagged [separable-algebras]

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

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### Separable algebras and separably closed local rings (a.k.a strictly Henselian local rings)

Let $A$ be a local ring. Say a monic $f\in A[x]$ is unramifiable if it admits a simple root in its universal splitting algebra (equivalently, it admits a simple root in any ring over which it splits). ...

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### Separable nonassociative algebras

In his paper "Structure and Representation of Nonassociative Algebras", Schafer notes that an arbitrary nonassociative algebra over a field is separable "if and only if the ...

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71 views

### Separability of $\mathbb{C}[x,y_1,\ldots,y_r]$ over $\mathbb{C} + (h,y_1,\ldots,y_r)$

The answer to this MO question says the following:
Lemma 1. Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified if and only if $h$ ...

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### Maximal orders separable over their centre

Let $\mathcal{A}$ be a central simple $K$-algebra, where $K$ is an algebraic number field. It is known that $\mathcal{A}$ is separable over $K$ (following the definition of DeMeyer and Ingraham's book)...

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### Flatness of certain subrings

The following question appears, more or less, here:
Let $k$ be an algebraically closed field of characteristic zero and let $S$ be a commutative $k$-algebra
(I do not mind to further assume that $S$ ...

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168 views

### Separable non-flat simple ring extension

Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that:
(1) $R$ and $S$ are integral domains.
(2) $Q(R)=Q(S)$, namely, their fields of fractions are equal.
(3) $S=R[w]$, for some $w \...

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391 views

### Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras

For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
...

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98 views

### A certain property of integral domains $A \subseteq B$ with $Q(A) \cap B= A$

I have asked the following question in MSE:
Let $k$ be a field of characteristic zero. Let $A \subseteq B$ be $k$-algebras which are also (commutative) integral domains with fields of fractions $Q(A) \...

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552 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 ...

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### 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 $K$-...

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151 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....

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### 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?

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249 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 ...

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120 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 \...

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### 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 ...

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221 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 ...

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482 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 ...

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241 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 ...

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231 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 ...

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421 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 ...

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991 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 ...

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### 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 ...

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### 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.
...

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375 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 $...

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371 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 ...