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 (of unital, but not necessarily commutative rings)
is called "separable" (or a "separable extension") 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(1). 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 formula_7.
Examples.
For any commutative ring "R", the (non-commutative) ring of "n"-by-"n" matrices formula_8 is a separable "R"-algebra. For any formula_9, a separability idempotent is given by formula_10, where formula_11 denotes the elementary matrix which is 0 except for the entry in position ("i", "j"), which is 1. In particular, this shows that separability idempotents need not be unique.
Separable algebras over a field.
If formula_12 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 formula_13 with irreducible polynomial formula_14, then a separability idempotent is given by formula_15. The tensorands are dual bases for the trace map: if formula_16 are the distinct "K"-monomorphisms of "L" into an algebraic closure of "K", the trace mapping Tr of "L" into "K" is defined by formula_17. 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 formula_12 the algebra formula_19 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"["G"] is a separable "K"-algebra. A separability idempotent is given by formula_20.
Equivalent characterizations of separability.
There are 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 formula_21 in the usual way. Moreover, an algebra "A" is separable if and only if it is flat when considered as a right module of formula_21 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 formula_23 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 formula_24 given by formula_25. The converse can be proven by a judicious use of the separability idempotent (similarly to the proof of Maschke's theorem, applying its components within and without the splitting maps).
Equivalently, the relative Hochschild cohomology groups formula_26 of (R,S) 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 (not to be confused with the symmetric algebra arising as the quotient of the tensor 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 (commutative) "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 (R,S)-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.