Separable algebra


In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

Definition and First Properties

A ring homomorphism
is called separable if the multiplication map
admits a section
by means of a homomorphism σ of A-A-bimodules. Such a section σ is determined by its value
σ. The condition that σ is a section of μ is equivalent to
and the condition to be an homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A:
Such an element p is called a separability idempotent, since it satisfies.

Examples

For any commutative ring R, the ring of n-by-n matrices is a separable R-algebra. For any, a separability idempotent is given by, where denotes the elementary matrix which is 0 except for the entry in position, which is 1. In particular, this shows that separability idempotents need not be unique.

Separable algebras over a field

If is a field extension, then L is separable as an associative K-algebra if and only if the extension of fields is separable.
If L/K has a primitive element with irreducible polynomial, then a separability idempotent is given by. The tensorands are dual bases for the trace map: if are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by. The trace map and its dual bases make explicit L as a Frobenius algebra over K.
More generally, separable algebras over a field K can be classified as follows: they 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. In particular: Every separable algebra is itself finite-dimensional. If K is a perfect field --- for example a field of characteristic zero, or a finite field, or an algebraically closed field --- then every extension of K is separable so that separable K-algebras are 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.
It can be shown by a generalized theorem of Maschke that an associative K-algebra A is separable if for every field extension the algebra is semisimple.

Group rings

If K is commutative ring and G is a finite group such that the order of G is invertible in K, then the group ring K is a separable K-algebra. A separability idempotent is given by.

Equivalent characterizations of separability

There are a several equivalent definitions of separable algebras. A K-algebra A is separable if and only if it is projective when considered as a left module of in the usual way. Moreover, an algebra A is separable if and only if it is flat when considered as a right module of in the usual way.
Separable extensions can also be characterized by means of split extensions: A is separable over K if all short exact sequences of A-A-bimodules that are split as A-K-bimodules also split as A-A-bimodules. Indeed, this condition is necessary since the multiplication mapping arising in the definition above is a A-A-bimodule epimorphism, which is split as an A-K-bimodule map by the right inverse mapping given by. The converse can be proven by a judicious use of the separability idempotent.
Equivalently, the relative Hochschild cohomology groups of in any coefficient bimodule M is zero for n > 0. Examples of separable extensions are many including first separable algebras where R = separable algebra and S = 1 times the ground field. Any ring R with elements a and b satisfying ab = 1, but ba different from 1, is a separable extension over the subring S generated by 1 and bRa.

Relation to Frobenius algebras

A separable algebra is said to be strongly separable if there exists a separability idempotent that is symmetric, meaning
An algebra is strongly separable if and only if its trace form is nondegenerate, thus making the algebra into a particular kind of Frobenius algebra called a symmetric algebra.
If K is commutative, A is a finitely generated projective separable K-module, then A is a symmetric Frobenius algebra.

Relation to formally unramified and formally étale extensions

Any separable extension A / K of commutative rings is formally unramified. The converse holds if A is a finitely generated K-algebra. A separable flat K-algebra A is formally étale.

Further results

A theorem in the area is that of J. Cuadra that a separable Hopf-Galois extension R | S has finitely generated natural S-module R. A fundamental fact about a separable extension R | S is that it is left or right semisimple extension: a short exact sequence of left or right R-modules that is split as S-modules, is split as R-modules. In terms of G. Hochschild's relative homological algebra, one says that all R-modules are relative -projective. Usually relative properties of subrings or ring extensions, such as the notion of separable extension, serve to promote theorems that say that the over-ring shares a property of the subring. For example, a separable extension R of a semisimple algebra S has R semisimple, which follows from the preceding discussion.
There is the celebrated Jans theorem that a finite group algebra A over a field of
characteristic p is of finite representation type if and only if its Sylow p-subgroup is cyclic: the clearest proof is to note this fact for p-groups, then note that the group algebra is a separable extension of its Sylow p-subgroup algebra B as the index is coprime to the characteristic. The separability condition above will imply every finitely generated A-module M is isomorphic to a direct summand
in its restricted, induced module. But if B has finite representation type, the restricted module
is uniquely a direct sum of multiples of finitely many indecomposables, which induce to a finite number of constituent indecomposable modules of which M is a direct sum. Hence A is of finite representation type if B is. The converse is proven by a similar argument noting that every subgroup algebra B is a B-bimodule direct summand of a group algebra A.