# 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 homomorphism of (unital, but not necessarily commutative) rings
    $K\to A$
is called separable if the multiplication map
    ${\begin{array}{rccc}\mu :&A\otimes _{K}A&\to &A\\&a\otimes b&\mapsto &ab\end{array}}$
admits a section
    $\sigma :A\to A\otimes _{K}A$
that is a homomorphism of A-A-bimodules.
If the ring $K$ is commutative and $K\to A$ maps $K$ into the center of $A$, we call $A$ a separable algebra over $K$.
It is useful to describe separability in terms of the element
    $p:=\sigma (1)=\sum a_{i}\otimes b_{i}\in A\otimes _{K}A$
The reason is that a section σ is determined by this element. The condition that σ is a section of μ is equivalent to
    $\sum a_{i}b_{i}=1$
and the condition that σ is a homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A:
    $\sum aa_{i}\otimes b_{i}=\sum a_{i}\otimes b_{i}a.$
Such an element p is called a separability idempotent, since regarded as an element of the algebra $A\otimes A^{\rm {op}}$ 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 $1\leq j\leq n$, a separability idempotent is given by ∑$_{i=1}$$^n$e$_{ij}$⊗e$_{ji}${\textstyle \sum _{i=1}^{n}e_{ij}\otimes e_{ji}}, where $e_{ij}$ denotes the elementary matrix which is 0 except for the entry in the (i, j) entry, which is 1. In particular, this shows that separability idempotents need not be unique.

### Separable algebras over a field

A field extension L/K of finite degree is a separable extension if and only if L is separable as an associative K-algebra. If L/K has a primitive element $a$ with irreducible polynomial p(x)=(x−a)∑$_{i=0}$$^{n−1}$b$_i$x$^i${\textstyle p(x)=(x-a)\sum _{i=0}^{n-1}b_{i}x^{i}}, then a separability idempotent is given by ∑$_{i=0}$$^{n−1}$a$^i$⊗$_K$ b$_i$/p$^′$(a) {\textstyle \sum _{i=0}^{n-1}a^{i}\otimes _{K}{\frac {b_{i}}{p'(a)}}}. The tensorands are dual bases for the trace map: if σ$_1$, …, σ$_n${\textstyle \sigma _{1}, \ldots , \sigma _{n}} 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)=∑$_{i=1}$$^n$σ$_i$(x){\textstyle Tr(x)=\sum _{i=1}^{n}\sigma _{i}(x)}. The trace map and its dual bases make explicit L as a Frobenius algebra over K.
More generally, separable algebras over a field K can be classified as follows: they are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field K. In particular: Every separable algebra is itself finite-dimensional. If K is a perfect field – for example a field of characteristic zero, or a finite field, or an algebraically closed field – then every extension of K is separable, so that separable K-algebras are finite products of matrix algebras over finite-dimensional division algebras over field K. In other words, if K is a perfect field, there is no difference between a separable algebra over K and a finite-dimensional semisimple algebra over K. It can be shown by a generalized theorem of Maschke that an associative K-algebra A is separable if for every field extension L/K{\textstyle L/K} the algebra A⊗$_K$L{\textstyle 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 algebra K[G] is a separable K-algebra. A separability idempotent is given by 1/o(G) ∑$_{g∈G}$g⊗g$^{−1}${\textstyle {\frac {1}{o(G)}}\sum _{g\in G}g\otimes g^{-1}}.

## 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 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 μ:A⊗$_K$A→A{\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 A→A⊗$_K$A{\textstyle 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 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
    $e=\sum _{i=1}^{n}x_{i}\otimes y_{i}=\sum _{i=1}^{n}y_{i}\otimes x_{i}$
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.

## 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.

## Citations

* ↑ Ford 2017, §4.2
* ↑ Reiner 2003, p. 102
* ↑ Ford 2017, Theorem 4.4.1
* ↑ Endo & Watanabe 1967, Theorem 4.2. If A is commutative, the proof is simpler, see Kadison 1999, Lemma 5.11.
* ↑ Ford 2017, Corollary 4.7.2, Theorem 8.3.6
* ↑ Ford 2017, Corollary 4.7.3