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)

${\displaystyle K\to A}$

is called separable if the multiplication map

${\displaystyle {\begin{array}{rccc}\mu$ :&A\otimes _{K}A&\to &A\\&a\otimes b&\mapsto &ab\end{array}}}

admits a section

${\displaystyle \sigma :A\to A\otimes _{K}A}$

that is a homomorphism of A-A-bimodules.

If the ring ${\displaystyle K}$ is commmutative and ${\displaystyle K\to A}$ maps ${\displaystyle K}$ into the center of ${\displaystyle A}$, we call ${\displaystyle A}$ a separable algebra over ${\displaystyle K}$.

It is useful to describe separability in terms of the element

${\displaystyle p:=\sigma (1)=\sum a_{i}\otimes b_{i}\in A\otimes _{K}A}$

The reason is that a section σ is determined by this element. The condition that σ is a section of μ is equivalent to

${\displaystyle \sum a_{i}b_{i}=1}$

and the condition that σ is a homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A:

${\displaystyle \sum aa_{i}\otimes b_{i}=\sum a_{i}\otimes b_{i}a.}$

Such an element p is called a separability idempotent, since regarded as an element of the algebra ${\displaystyle A\otimes A^{\rm {op}}}$ it satisfies ${\displaystyle p^{2}=p}$.

Examples

For any commutative ring R, the (non-commutative) ring of n-by-n matrices ${\displaystyle M_{n}(R)}$ is a separable R-algebra. For any ${\displaystyle 1\leq j\leq n}$, a separability idempotent is given by ${\textstyle \sum _{i=1}^{n}e_{ij}\otimes e_{ji}}$, where ${\displaystyle e_{ij}}$ 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.

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 ${\displaystyle A^{e}}$ in the usual way.^[2] Moreover, an algebra A is separable if and only if it is flat when considered as a right module of ${\displaystyle A^{e}}$ in the usual way.

Separable algebras can also be characterized by means of split extensions: A is separable over K if and only 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 ${\textstyle \mu :A\otimes _{K}A\rightarrow A}$ 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 ${\textstyle A\rightarrow A\otimes _{K}A}$ given by ${\displaystyle a\mapsto a\otimes 1}$. 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).^[3]

Equivalently, the relative Hochschild cohomology groups ${\displaystyle H^{n}(R,S;M)}$ 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 is a 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

${\displaystyle e=\sum _{i=1}^{n}x_{i}\otimes y_{i}=\sum _{i=1}^{n}y_{i}\otimes x_{i}}$

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.^[4]

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.^[5] A separable flat (commutative) K-algebra A is formally étale.^[6]

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.

Citations

References

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