symmetric monoidal (∞,1)-category of spectra
The concept of separable algebras is a strengthening of the concept of semisimple algebras, and a generalization of the concept of separable field extensions.
There are a several equivalent characterizations of separable algebras. For all of these we fix a field $k$. In what follows, all $k$-algebras will be assumed associative and unital.
First, a $k$-algebra $A$ is defined to be separable if for every field extension $K$ of $k$, the tensor product algebra $A \otimes_k K$ is semisimple.
Second, a $k$-algebra $A$ is separable if and only if it is flat when considered as a right module of $A^e = A \otimes_k A^{op}$ in the obvious (but perhaps not quite standard) way.
Third, a $k$-algebra $A$ is separable if and only if it is projective when considered as a left module of $A^e$ in the usual way.
Fourth, a $k$-algebra $A$ is separable if and only if the $A^e$-module morphism
has a right inverse, that is a $A^e$-module morphism
with $m f = 1_A$.
It is easy to see that the third and fourth definitions are equivalent. We have an epimorphism of $A^e$-modules
If $f$ as above exists, this splits, so $A$ is a summand of a free $A^e$-module, namely $A^e$ itself, so $A$ is projective as an $A^e$-module. Conversely, if $A$ is projective, any epimorphism to $A$ splits.
We can also state the fourth characterization in a more grungy way in terms of the element $p = f(1)$. Namely, a $k$-algebra $A$ is separable if and only if there exists an element
such that
and
for all $a \in A$. Such an element $p$ is called a separability idempotent, since it satisfies $p^2 = p$. While grungy, finding a separability idempotent is a practical way to prove that an algebra is separable.
There is a classification theorem for separable algebras: separable algebras are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field $k$.
A perfect field is one for which every extension is separable. Examples include fields of characteristic zero, or finite fields, or algebraically closed fields, or extensions of perfect fields. If $k$ is a perfect field, separable algebras are the same as finite products of matrix algebras over finite-dimensional division algebras over field $k$. In other words, if $k$ is a perfect field, there is no difference between a separable algebra over $k$ and a finite-dimensional semisimple algebra over $k$.
A result of Eilenberg & Nakayama (1955) states that any separable algebra over a field $k$ can be given the structure of a symmetric Frobenius algebra. Since the underlying vector space of a Frobenius algebra is isomorphic to its dual, any Frobenius algebra is necessarily finite dimensional, and so the same is true for separable algebras. For more details, see Eilenberg & Nakayama (1955), Endo & Watanabe (1967), in particular Thm. 4.2..
A separable algebra is said to be strongly separable if there exists a separability idempotent $p$ that is symmetric, meaning
An algebra is strongly separable if and only if it can be made into a special Frobenius algebra. When this can be done, it can be done in a unique way.
There is an equivalent characterization of strongly separable algebras which makes this fact clearer. Any element $a$ of an associative unital algebra gives a left multiplication map
When $A$ is finite-dimensional, there is a bilinear pairing $g: A \times A \to k$ defined by
An algebra $A$ is strongly separable if and only if $g$ is nondegenerate, i.e., if there is a vector space isomorphism $A \to A^*$ given by
In this case, there is just one way to make $A$ into a special Frobenius algebra, namely by defining the counit to be
Here are some examples of strongly separable algebras:
the algebra of $n \times n$ matrices with entries in a field $k$ is always separable, but it is strongly separable if and only if $n$ is not divisible by the characteristic of $k$.
the group algebra $k[G]$ of a finite group is strongly separable if and only if the order of $G$ is not divisible by the characteristic of $k$. (It is also semisimple precisely in this case, so it is separable precisely in this case.)
For more details, see Aguiar’s note below.
More generally, if $k$ is any unital commutative ring, we can define a separable $k$-algebra to be an algebra $A$ such that $A$ is projective as a module over $A^e = A \otimes_k A^{op}$.
As in the case of algebras over a field, an algebra $A$ over a commutative ring $k$ is separable if and only if the $A^e$-module epimorphism
splits, and this in turn is equivalent to the existence of a separability idempotent.
If a separable algebra $A$ is also projective as a module over $k$, it must be finitely generated as a $k$-module. For more details see DeMeyer-Ingraham.
If the center of a separable algebra $A$ over a commutative ring $k$ is $k$, $A$ is called an Azumaya algebra.
The ring extension $S$ over $R$ is said to be a separable extension if all short exact sequences of $S$-$S$-bimodules that are split as $S$-$R$-bimodules also split as $S$-$S$-bimodules. This is equivalent to the statement that the relative Hochschild cohomology $HH^n(S,R;M) = 0$ for all $n\gt 0$ and all coefficient bimodules $M$.
Commutative separable algebras play an important role in algebraic geometry, where the concept of étale cover is a way of implementing the idea of covering space, but any étale cover of the spectrum of a field corresponds to a commutative separable algebra over that field. Lieven Le Bruyn has written “in categorical terms, studying the monoidal category of commutative separable $k$-algebras is the same as studying the étale site of $k$”. This idea is partially captured by the so-called fundamental theorem of Grothendieck Galois theory: the category of commutative separable algebras over a field $k$ is contravariantly equivalent to the category of continuous actions on finite sets of the profinite fundamental group of $k$. As a group, the profinite fundamental group of $k$ is the Galois group $Gal(k_s|k)$ where $k_s$ is a separable closure of $k$, that is, a maximal separable field extension of $k$.
Separable algebras play a major role in the Galois theory of extensions of algebras. There are further generalizations, leading to separable functors…
Samuel Eilenberg, Tadasi Nakayama, On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. 9 (1955) 1-16 [euclid]
K. Hirata, K. Sugano, On semisimple and separable extensions of noncommutative rings, J. Math. Soc. Japan 18 (1966), 360-373.
Shizuo Endo, Yutaka Watanabe, On separable algebras over a commutative ring, Osaka J. Math. 4(2) (1967) 233-242 [euclid]
F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics 181, Springer, Berlin (1971)
Marcelo Aguiar, A note on strongly separable algebras, Boletín de la Academia Nacional de Ciencias (Córdoba, Argentina), special issue in honor of Orlando Villamayor, 65 (2000) 51-60 [pdf]
See also:
An explicit proof of the Grothendieck Galois theory statement that the category of commutative separable algebras over a field $K$ is anti-equivalent to the category of continuous actions on finite sets of the profinite fundamental group of $K$:
Last revised on May 30, 2023 at 14:19:37. See the history of this page for a list of all contributions to it.