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separable algebra

Contents

The idea

The concept of separable algebra is a strengthening of the concept of semisimple algebra, and a generalization of the concept of a separable field extension.

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 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.

Classification

There is a classification theorem for separable algebras: separable algebras are the same as finite products of matrix algebras over division algebras whose centers are finite dimensional separable field extensions of the field kk.

A perfect field is one for which every extension of 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 division algebras whose centers are finite-dimensional field extensions of the 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 and Nakayama 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:

  • Samuel Eilenberg and Tadasi Nakayama, On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. 9 (1955), 1-16. (web)

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 an 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 the field kk 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.

For more details, see Aguiar’s book 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 : & 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.

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 are important in algebraic geometry. The concept of étale cover in algebraic geometry is sort of a combination of covering space and separable algebra business. Lieven Le Bruyn has written “in categorical terms, studying the monoidal cat of commutative separable kk-algebras is the same as studying the étale site of kk”. This stuff would be nice to make precise…

Separable algebras play a major role in the Galois theory of extensions of algebras. Every separable kk-algebra is a filtered colimit of finite-dimensional separable kk-algebras???

There are further generalizations, leading to separable functors

  • nnLab: separable functor, separable field extension, separable coring

  • wikipedia separable algebra

  • 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. (web)

  • F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics 181, Springer, Berlin, 1971.

  • K. Hirata, K. Sugano, On semisimple and separable extensions of noncommutative rings, J. Math. Soc. Japan 18 (1966), 360-373.

An explicit proof of the Grothendieck Galois theory statement that the category of 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

Revised on May 21, 2012 20:32:24 by Zoran Škoda (193.51.104.33)