In mathematics, a separable algebra is a kind of
semisimple algebra
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensio ...
. It is a generalization to
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
s of the notion of a
separable field extension In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomi ...
.
Definition and First Properties
A
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preservi ...
(of unital, but not necessarily
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
s)
:
is called ''separable'' (or a ''separable extension'') if the multiplication map
:
admits a
section
Section, Sectioning or Sectioned may refer to:
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* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
:
by means of a homomorphism σ of ''A''-''A''-
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
s. Such a section σ is determined by its value
:
σ(1). The condition that σ is a section of μ is equivalent to
:
and the condition to be an homomorphism of ''A''-''A''-bimodules is equivalent to the following requirement for any ''a'' in ''A'':
:
Such an element ''p'' is called a ''separability idempotent'', since it satisfies
.
Examples
For any commutative ring ''R'', the (non-commutative) ring of ''n''-by-''n''
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
is a separable ''R''-algebra. For any
, a separability idempotent is given by
, where
denotes the
elementary matrix In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multip ...
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
If
is a
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
, then ''L'' is separable as an associative ''K''-algebra if and only if the extension of fields is
separable.
If ''L''/''K'' has a
primitive element with irreducible polynomial
, then a separability idempotent is given by
. The tensorands are dual bases for the trace map: if
are the distinct ''K''-monomorphisms of ''L'' into an algebraic closure of ''K'', the trace mapping Tr of ''L'' into ''K'' is defined by
. The trace map and its dual bases make explicit ''L'' as a
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice dualit ...
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
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
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 In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k'' is ...
--- 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
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensio ...
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
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
the algebra
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
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
''K''
'G''is a separable ''K''-algebra. A separability idempotent is given by
.
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
in the usual way. Moreover, an algebra ''A'' is separable if and only if it is
flat
Flat or flats may refer to:
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Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
when considered as a right module of
in the usual way.
Separable extensions can also be characterized by means of split extensions: ''A'' is separable over ''K'' if all
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
s 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
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
given by
. The converse can be proven by a judicious use of the separability idempotent (similarly to the proof of
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
, applying its components within and without the splitting maps).
Equivalently, the relative
Hochschild cohomology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, ...
groups
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 = 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
:
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
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice dualit ...
called a symmetric algebra (not to be confused with the
symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
arising as the quotient of the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
).
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
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
(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.
References
*
*
Samuel Eilenberg
Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra.
Early life and education
He was born in Warsaw, Kingdom of Poland to a ...
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.
*
*
* .
*
*
* {{Weibel IHA
Algebras