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Definition and first properties

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

K → A {\displaystyle K\to A}

is called separable if the multiplication map

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

admits a section

σ : A → A ⊗ K A {\displaystyle \sigma :A\to A\otimes _{K}A}

that is a homomorphism of A-A-bimodules.

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

It is useful to describe separability in terms of the element

p := σ ( 1 ) = ∑ a i ⊗ b i ∈ A ⊗ K A {\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

∑ a i b i = 1 {\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:

∑ a a i ⊗ b i = ∑ a i ⊗ b i 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 A ⊗ A o p {\displaystyle A\otimes A^{\rm {op}}} it satisfies p 2 = p {\displaystyle p^{2}=p} .

Examples

For any commutative ring R, the (non-commutative) ring of n-by-n matrices M n ( R ) {\displaystyle M_{n}(R)} is a separable R-algebra. For any 1 ≤ j ≤ n {\displaystyle 1\leq j\leq n} , a separability idempotent is given by ∑ i = 1 n e i j ⊗ e j i {\textstyle \sum _{i=1}^{n}e_{ij}\otimes e_{ji}} , where e i j {\displaystyle 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 {\displaystyle 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 T r ( 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.¹ 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 {\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 A e {\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 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 ↦ a ⊗ 1 {\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 H n ( R , S ; M ) {\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

e = ∑ i = 1 n x i ⊗ y i = ∑ i = 1 n y i ⊗ x i {\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

• DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics. Vol. 181. Berlin-Heidelberg-New York: Springer-Verlag. ISBN 978-3-540-05371-2. Zbl 0215.36602.
• 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, MR 0227211
• Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1, MR 3618889
• Hirata, H.; Sugano, K. (1966), "On semisimple and separable extensions of noncommutative rings", J. Math. Soc. Jpn., 18: 360–373
• Kadison, Lars (1999), New examples of Frobenius extensions, University Lecture Series, vol. 14, Providence, RI: American Mathematical Society, doi:10.1090/ulect/014, ISBN 0-8218-1962-3, MR 1690111
• Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series, vol. 28, Oxford University Press, ISBN 0-19-852673-3, Zbl 1024.16008
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.

References

1. Ford 2017, §4.2 - Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1, MR 3618889 https://mathscinet.ams.org/mathscinet-getitem?mr=3618889 ↩

2. Reiner 2003, p. 102 - Reiner, I. (2003), Maximal Orders, London Mathematical Society Monographs. New Series, vol. 28, Oxford University Press, ISBN 0-19-852673-3, Zbl 1024.16008 https://zbmath.org/?format=complete&q=an:1024.16008 ↩

3. Ford 2017, Theorem 4.4.1 - Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1, MR 3618889 https://mathscinet.ams.org/mathscinet-getitem?mr=3618889 ↩

4. Endo & Watanabe 1967, Theorem 4.2. If A is commutative, the proof is simpler, see Kadison 1999, Lemma 5.11. - Endo, Shizuo; Watanabe, Yutaka (1967), "On separable algebras over a commutative ring", Osaka Journal of Mathematics, 4: 233–242, MR 0227211 http://projecteuclid.org/euclid.ojm/1200691953 ↩

5. Ford 2017, Corollary 4.7.2, Theorem 8.3.6 - Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1, MR 3618889 https://mathscinet.ams.org/mathscinet-getitem?mr=3618889 ↩

6. Ford 2017, Corollary 4.7.3 - Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1, MR 3618889 https://mathscinet.ams.org/mathscinet-getitem?mr=3618889 ↩