{{Distinguish|text = 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.
== Definition and first properties == A homomorphism of (unital, but not necessarily commutative) rings : <math>K \to A</math> is called ''separable'' if the multiplication map : <math>\begin{array}{rccc} \mu :& A \otimes_K A &\to& A \\ & a \otimes b &\mapsto & ab \end{array}</math> admits a section : <math>\sigma: A \to A \otimes_K A</math> that is a homomorphism of ''A''-''A''-bimodules.
If the ring <math>K</math> is commutative and <math>K \to A</math> maps <math>K</math> into the center of <math>A</math>, we call <math>A</math> a ''separable algebra over'' <math>K</math>.
It is useful to describe separability in terms of the element : <math>p := \sigma(1) = \sum a_i \otimes b_i \in A \otimes_K A</math>
The reason is that a section ''σ'' is determined by this element. The condition that ''σ'' is a section of ''μ'' is equivalent to : <math>\sum a_i b_i = 1</math> and the condition that σ is a homomorphism of ''A''-''A''-bimodules is equivalent to the following requirement for any ''a'' in ''A'': : <math>\sum a a_i \otimes b_i = \sum a_i \otimes b_i a.</math> Such an element ''p'' is called a ''separability idempotent'', since regarded as an element of the algebra <math>A \otimes A^{\rm op}</math> it satisfies <math>p^2 = p</math>.
== Examples == For any commutative ring ''R'', the (non-commutative) ring of ''n''-by-''n'' matrices <math>M_n(R)</math> is a separable ''R''-algebra. For any <math>1 \le j \le n</math>, a separability idempotent is given by <math display="inline">\sum_{i=1}^n e_{ij} \otimes e_{ji}</math>, where <math>e_{ij}</math> denotes the elementary matrix which is 0 except for the entry in the {{nowrap|(''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 <math> a</math> with irreducible polynomial <math display="inline"> p(x) = (x - a) \sum_{i=0}^{n-1} b_i x^i</math>, then a separability idempotent is given by <math display="inline"> \sum_{i=0}^{n-1} a^i \otimes_K \frac{b_i}{p'(a)}</math>. The tensorands are dual bases for the trace map: if <math display="inline"> \sigma_1,\ldots,\sigma_{n} </math> are the distinct ''K''-monomorphisms of ''L'' into an algebraic closure of ''K'', the trace mapping Tr of ''L'' into ''K'' is defined by <math display="inline">Tr(x) = \sum_{i=1}^{n} \sigma_i(x)</math>. 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 <math display="inline">L/K</math> the algebra <math display="inline">A\otimes_K L</math> 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.{{sfn|Ford|2017|loc=§4.2|ps=none}} A separability idempotent is given by <math display="inline"> \frac{1}{o(G)} \sum_{g \in G} g \otimes g^{-1}</math>.
== 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 <math>A^e</math> in the usual way.{{sfn|Reiner|2003|p=102|ps=none}} Moreover, an algebra ''A'' is separable if and only if it is flat when considered as a right module of <math>A^e</math> 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 <math display="inline">\mu : A \otimes_K A \rightarrow A </math> 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 <math display="inline"> A \rightarrow A \otimes_K A</math> given by <math> a \mapsto a \otimes 1 </math>. 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).{{sfn|Ford|2017|loc=Theorem 4.4.1|ps=none}}
Equivalently, the relative Hochschild cohomology groups <math> H^n(R,S;M)</math> of {{nowrap|(''R'', ''S'')}} in any coefficient bimodule ''M'' is zero for {{nowrap|''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 {{nowrap|1=''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 : <math> e = \sum_{i=1}^n x_i \otimes y_i = \sum_{i=1}^n y_i \otimes x_i</math>
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.{{sfn|Endo|Watanabe|1967|loc=Theorem 4.2|ps=. If ''A'' is commutative, the proof is simpler, see {{harvnb|Kadison|1999|loc=Lemma 5.11}}.}}
== Relation to formally unramified and formally étale extensions ==
Any separable extension {{nowrap|''A'' / ''K''}} of commutative rings is formally unramified. The converse holds if ''A'' is a finitely generated ''K''-algebra.{{sfn|Ford|2017|loc=Corollary 4.7.2, Theorem 8.3.6|ps=none}} A separable flat (commutative) ''K''-algebra ''A'' is formally étale.{{sfn|Ford|2017|loc=Corollary 4.7.3|ps=none}}
== Further results ==
A theorem in the area is that of J. Cuadra that a separable Hopf–Galois extension {{nowrap|''R'' {{!}} ''S''}} has finitely generated natural {{nowrap|''S''-module}} ''R''. A fundamental fact about a separable extension {{nowrap|''R'' {{!}} ''S''}} is that it is left or right semisimple extension: a short exact sequence of left or right {{nowrap|''R''-modules}} that is split as {{nowrap|''S''-modules}}, is split as {{nowrap|''R''-modules}}. In terms of G. Hochschild's relative homological algebra, one says that all {{nowrap|''R''-modules}} are relative {{nowrap|(''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 {{nowrap|''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 {{nowrap|''B''-bimodule}} direct summand of a group algebra ''A''.
== Citations == {{reflist}}
== References == * {{cite book |last1=DeMeyer |first1=F. |last2=Ingraham |first2=E. |title=Separable algebras over commutative rings |series=Lecture Notes in Mathematics | volume=181 | publisher=Springer-Verlag |location=Berlin-Heidelberg-New York | year=1971 | isbn=978-3-540-05371-2 | zbl=0215.36602 }} * Samuel Eilenberg and Tadasi Nakayama, [https://web.archive.org/web/20110120012931/http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.nmj%2F1118799677 On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings], Nagoya Math. J. Volume 9 (1955), 1–16. * {{citation |last1=Endo |first1=Shizuo |last2=Watanabe |first2=Yutaka |title=On separable algebras over a commutative ring |journal=Osaka Journal of Mathematics |volume=4 |year=1967 |pages=233–242 |mr=0227211 |url=http://projecteuclid.org/euclid.ojm/1200691953 }} * {{citation |last1=Ford |first1=Timothy J. |title=Separable algebras |publisher=American Mathematical Society|location=Providence, RI |year=2017 |isbn=978-1-4704-3770-1 |mr=3618889 }} * {{citation |last1=Hirata |first1=H. |last2=Sugano |first2=K. |title=On semisimple and separable extensions of noncommutative rings |journal=J. Math. Soc. Jpn. |volume=18 |year=1966 |pages=360–373 }} * {{citation |last1=Kadison |first1=Lars |author-link=Lars Kadison |title=New examples of Frobenius extensions |series=University Lecture Series |volume=14 |publisher=American Mathematical Society |location=Providence, RI |year=1999 |isbn=0-8218-1962-3 |mr=1690111 |doi=10.1090/ulect/014 }} * {{citation |last1=Reiner |first1=I. |authorlink=Irving Reiner |title=Maximal Orders |series=London Mathematical Society Monographs. New Series |volume=28 |publisher=Oxford University Press |year=2003 |isbn=0-19-852673-3 |zbl=1024.16008 }} * {{Weibel IHA}}
Category:Algebras