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)

${\displaystyle K\to A}$

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 commutative 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.

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 ${\displaystyle a}$ with irreducible polynomial ${\textstyle p(x)=(x-a)\sum _{i=0}^{n-1}b_{i}x^{i}}$, then a separability idempotent is given by ${\textstyle \sum _{i=0}^{n-1}a^{i}\otimes _{K}{\frac {b_{i}}{p'(a)}}}$. The tensorands are dual bases for the trace map: if ${\textstyle \sigma _{1},\ldots ,\sigma _{n}}$ are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by ${\textstyle Tr(x)=\sum _{i=1}^{n}\sigma _{i}(x)}$. 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 ${\textstyle L/K}$ the algebra ${\textstyle A\otimes _{K}L}$ 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 ring K[G] is a separable K-algebra.^[1] A separability idempotent is given by ${\textstyle {\frac {1}{o(G)}}\sum _{g\in G}g\otimes g^{-1}}$.

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 ${\displaystyle A^{e}}$ 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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