nLab separable algebra

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The idea

The concept of separable algebras is a strengthening of the concept of semisimple algebras, and a generalization of the concept of separable field extensions.

Definition

There are a several equivalent characterizations of separable algebras. For all of these we fix a field kk. In what follows, all kk-algebras will be assumed associative and unital.

First, a kk-algebra AA is defined to be separable if for every field extension KK of kk, the tensor product algebra A kKA \otimes_k K is semisimple.

Second, a kk-algebra AA is separable if and only if it is flat when considered as a right module of A e=A kA opA^e = A \otimes_k A^{op} in the obvious (but perhaps not quite standard) way.

Third, a kk-algebra AA is separable if and only if it is projective when considered as a left module of A eA^e in the usual way.

Fourth, a kk-algebra AA is separable if and only if the A eA^e-module morphism

m: A e A ab ab \array{ m : & A^e &\to & A \\ & a \otimes b & \mapsto & a b }

has a right inverse, that is a A eA^e-module morphism

f:AA e f: A \to A^e

with mf=1 Am f = 1_A.

It is easy to see that the third and fourth definitions are equivalent. We have an epimorphism of A eA^e-modules

A emA0 A^e \stackrel{m}{\longrightarrow} A \to 0

If ff as above exists, this splits, so AA is a summand of a free A eA^e-module, namely A eA^e itself, so AA is projective as an A eA^e-module. Conversely, if AA is projective, any epimorphism to AA splits.

We can also state the fourth characterization in a more grungy way in terms of the element p=f(1)p = f(1). Namely, a kk-algebra AA is separable if and only if there exists an element

p= i=1 na ib iA e p = \sum_{i=1}^n a_i \otimes b_i \in A^e

such that

i=1 na ib i=1 \sum_{i=1}^n a_i b_i = 1

and

ap=pa a p = p a

for all aAa \in A. Such an element pp is called a separability idempotent, since it satisfies p 2=pp^2 = p. While grungy, finding a separability idempotent is a practical way to prove that an algebra is separable.

Properties

Classification

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 kk.

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 kk is a perfect field, separable algebras are the same as finite products of matrix algebras over finite-dimensional division algebras over field kk. In other words, if kk is a perfect field, there is no difference between a separable algebra over kk and a finite-dimensional semisimple algebra over kk.

Relation to Frobenius algebras

A result of Eilenberg & Nakayama (1955) states that any separable algebra over a field kk 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 pp that is symmetric, meaning

p= i=1 na ib i= i=1 nb ia i p = \sum_{i=1}^n a_i \otimes b_i = \sum_{i=1}^n b_i \otimes a_i

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 aa of an associative unital algebra gives a left multiplication map

L a: A A b ab \array{ L_a : &A &\to& A \\ &b &\mapsto& a b }

When AA is finite-dimensional, there is a bilinear pairing g:A×Akg: A \times A \to k defined by

g(a,b)=tr(L aL b) g(a,b) = tr(L_a L_b)

An algebra AA is strongly separable if and only if gg is nondegenerate, i.e., if there is a vector space isomorphism AA *A \to A^* given by

ag(a,) a \mapsto g(a, -)

In this case, there is just one way to make AA into a special Frobenius algebra, namely by defining the counit to be

ϵ(a)=tr(L a) \epsilon(a) = tr(L_a)

Here are some examples of strongly separable algebras:

  • the algebra of n×nn \times n matrices with entries in a field kk is always separable, but it is strongly separable if and only if nn is not divisible by the characteristic of kk.

  • the group algebra k[G]k[G] of a finite group is strongly separable if and only if the order of GG is not divisible by the characteristic of kk. (It is also semisimple precisely in this case, so it is separable precisely in this case.)

For more details, see Aguiar’s note below.

Over commutative rings

More generally, if kk is any unital commutative ring, we can define a separable kk-algebra to be an algebra AA such that AA is projective as a module over A e=A kA opA^e = A \otimes_k A^{op}.

As in the case of algebras over a field, an algebra AA over a commutative ring kk is separable if and only if the A eA^e-module epimorphism

m: A e A ab ab \array{ m \colon & A^e &\to & A \\ & a \otimes b & \mapsto & a b }

splits, and this in turn is equivalent to the existence of a separability idempotent.

If a separable algebra AA is also projective as a module over kk, it must be finitely generated as a kk-module. For more details see DeMeyer-Ingraham.

If the center of a separable algebra AA over a commutative ring kk is kk, AA is called an Azumaya algebra.

Separable extensions of noncommutative rings

The ring extension SS over RR is said to be a separable extension if all short exact sequences of SS-SS-bimodules that are split as SS-RR-bimodules also split as SS-SS-bimodules. This is equivalent to the statement that the relative Hochschild cohomology HH n(S,R;M)=0HH^n(S,R;M) = 0 for all n>0n\gt 0 and all coefficient bimodules MM.

In algebraic geometry

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 kk-algebras is the same as studying the étale site of kk”. This idea is partially captured by the so-called fundamental theorem of Grothendieck Galois theory: the category of commutative separable algebras over a field kk is contravariantly equivalent to the category of continuous actions on finite sets of the profinite fundamental group of kk. As a group, the profinite fundamental group of kk is the Galois group Gal(k s|k)Gal(k_s|k) where k sk_s is a separable closure of kk, that is, a maximal separable field extension of kk.

Separable algebras play a major role in the Galois theory of extensions of algebras. There are further generalizations, leading to separable functors

References

  • 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 KK is anti-equivalent to the category of continuous actions on finite sets of the profinite fundamental group of KK:

  • Federico G. Lastaria, On separable algebras in Grothendieck Galois theory, Le Mathematiche 51:3, 1996, link

Last revised on May 30, 2023 at 14:19:37. See the history of this page for a list of all contributions to it.