Separable algebra

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 a a_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 \le j \le n, a separability idempotent is given by \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) \sum_{i=0}^{n-1} b_i x^i, then a separability idempotent is given by \sum_{i=0}^{n-1} a^i \otimes_K \frac{b_i}{p'(a)}. The tensorands are dual bases for the trace map: if \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) = \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 the algebra 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 \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 \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 is a separable algebra and S = 1 times the ground field. Any ring R with elements a and b satisfying , 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

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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