Separable algebra

Not to be confused with an étale 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.

## Contents

* 1 Definition and first properties
* 2 Examples
* 2.1 Separable algebras over a field
* 2.2 Group rings
* 3 Equivalent characterizations of separability
* 4 Relation to Frobenius algebras
* 5 Relation to formally unramified and formally étale extensions
* 6 Further results
* 7 Citations
* 8 References

## Definition and first properties

A homomorphism of (unital, but not necessarily commutative) rings

K->A

is called separable if the multiplication map

μ:A⊗KA->Aa⊗b↦ab

admits a section

σ:A->A⊗KA

that is a homomorphism of A-A-bimodules.

If the ring K is commutative and K->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:=σ(1)=∑ai⊗bi∈A⊗KA

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

∑aibi=1

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

∑aai⊗bi=∑ai⊗bia.

Such an element p is called a separability idempotent, since regarded as an element of the algebra A⊗Aop it satisfies p2=p.

## Examples

For any commutative ring R, the (non-commutative) ring of n-by-n matrices Mn(R) is a separable R-algebra. For any 1≤j≤n, a separability idempotent is given by ∑i=1neij⊗eji, where eij 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=0n−1bixi, then a separability idempotent is given by ∑i=0n−1ai⊗Kbip′(a). The tensorands are dual bases for the trace map: if σ1,…,σn 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=1nσi(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⊗KL 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.[1] A separability idempotent is given by 1o(G)∑g∈Gg⊗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 Ae 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 Ae 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⊗KA->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⊗KA given by a↦a⊗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 Hn(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=∑i=1nxi⊗yi=∑i=1nyi⊗xi

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

1. ↑ Ford 2017, §4.2
2. ↑ Reiner 2003, p. 102 sfn error: no target: CITEREFReiner2003 (help)
3. ↑ Ford 2017, Theorem 4.4.1
4. ↑ Endo & Watanabe 1967, Theorem 4.2. If A is commutative, the proof is simpler, see Kadison 1999, Lemma 5.11 harvnb error: no target: CITEREFKadison1999 (help).
5. ↑ Ford 2017, Corollary 4.7.2, Theorem 8.3.6
6. ↑ Ford 2017, Corollary 4.7.3

## References

* DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics. 181. Berlin-Heidelberg-New York: Springer-Verlag. ISBN 978-3-540-05371-2.
* 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.
* Endo, Shizuo; Watanabe, Yutaka (1967), "On separable algebras over a commutative ring", Osaka Journal of Mathematics 4: 233–242, http://projecteuclid.org/euclid.ojm/1200691953
* Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1
* Hirata, H.; Sugano, K. (1966), "On semisimple and separable extensions of noncommutative rings", J. Math. Soc. Jpn. 18: 360–373
* New examples of Frobenius extensions, University Lecture Series, 14, Providence, RI: American Mathematical Society, 1999, doi:10.1090/ulect/014, ISBN 0-8218-1962-3
* Maximal Orders, London Mathematical Society Monographs. New Series, 28, Oxford University Press, 2003, ISBN 0-19-852673-3
* Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. OCLC 36131259.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Separable algebra. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Separable_algebra&oldid=4155755"