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)

:$K \to A$

is called ''separable'' (or a ''separable extension'') if the multiplication map

:$\mu : A \otimes_K A \to A, a \otimes b \mapsto ab$

admits a section

:$\sigma: A \to A \otimes_K A$

by means of a homomorphism σ of ''A''-''A''- bimodules. Such a section σ is determined by its value

:$p := \sigma(1) = \sum a_i \otimes b_i$

σ(1). The condition that σ is a section of μ is equivalent to

:$\sum a_i b_i = 1$

and the condition to be an homomorphism of ''A''-''A''-bimodules is equivalent to the following requirement for any ''a'' in ''A'':

:$\sum a a_i \otimes b_i = \sum a_i \otimes b_i a.$
Such an element ''p'' is called a ''separability idempotent'', since it satisfies $p^2 = p$.

Examples

For any commutative ring ''R'', the (non-commutative) ring of ''n''-by-''n'' matrices $M_n(R)$ is a separable ''R''-algebra. For any $1 \le j \le n$, a separability idempotent is given by $\sum_^n e_ \otimes e_$, where $e_$ 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 $\scriptstyle L/K$ 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 $a$ with irreducible polynomial $p(x) = (x - a) \sum_^ b_i x^i$, then a separability idempotent is given by $\sum_^ a^i \otimes_K \frac$. The tensorands are dual bases for the trace map: if $\sigma_1,\ldots,\sigma_$ are the distinct ''K''-monomorphisms of ''L'' into an algebraic closure of ''K'', the trace mapping Tr of ''L'' into ''K'' is defined by $Tr(x) = \sum_^ \sigma_i(x)$. 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 $\scriptstyle L/K$ the algebra $\scriptstyle A\otimes_K L$ 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 $\frac \sum_ g \otimes g^$.

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 $A^e$ in the usual way. Moreover, an algebra ''A'' is separable if and only if it is flat when considered as a right module of $A^e$ 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 $\mu : A \otimes_K A \rightarrow A$ 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 $A \rightarrow A \otimes_K A$ given by $a \mapsto a \otimes 1$. 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 $H^n(R,S;M)$ 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

:$e = \sum_^n x_i \otimes y_i = \sum_^n y_i \otimes x_i$

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.

References

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* Samuel Eilenberg and Tadasi Nakayama
On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings
Nagoya Math. J. Volume 9 (1955), 1–16.
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* {{Weibel IHA

Algebras