Separable algebra

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

: $K\; \backslash to\; A$

is called separable if the multiplication map

:

& a \otimes b &\mapsto & ab

\end{array}

admits a section

: $\backslash sigma:\; A\; \backslash to\; A\; \backslash otimes\_K\; A$

that is a homomorphism of A-A-bimodules.

If the ring $K$ is commutative and $K\; \backslash to\; A$ 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

: $p\; :=\; \backslash sigma(1)\; =\; \backslash sum\; a\_i\; \backslash otimes\; b\_i\; \backslash in\; A\; \backslash otimes\_K\; A$

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

: $\backslash sum\; a\_i\; b\_i\; =\; 1$

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

: $\backslash sum\; a\; a\_i\; \backslash otimes\; b\_i\; =\; \backslash sum\; a\_i\; \backslash otimes\; b\_i\; a.$

Such an element p is called a separability idempotent, since regarded as an element of the algebra $A\; \backslash otimes\; A^\{\backslash rm\; op\}$ it satisfies $p^2\; =\; p$.

For any commutative ring R, the (non-commutative) ring of n-by-n matrices $M\_n(R)$ is a separable R-algebra. For any $1\; \backslash le\; j\; \backslash le\; n$, a separability idempotent is given by $\backslash sum\_\{i=1\}^n\; e\_\{ij\}\; \backslash otimes\; e\_\{ji\}$, where $e\_\{ij\}$ 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.

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 $p(x)\; =\; (x\; -\; a)\; \backslash sum\_\{i=0\}^\{n-1\}\; b\_i\; x^i$, then a separability idempotent is given by $\backslash sum\_\{i=0\}^\{n-1\}\; a^i\; \backslash otimes\_K\; \backslash frac\{b\_i\}\{p\text{'}(a)\}$. The tensorands are dual bases for the trace map: if $\backslash sigma\_1,\backslash ldots,\backslash sigma\_\{n\}$ are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by $Tr(x)\; =\; \backslash sum\_\{i=1\}^\{n\}\; \backslash sigma\_i(x)$. 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 $A\backslash otimes\_K\; L$ is semisimple.

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 $\backslash frac\{1\}\{o(G)\}\; \backslash sum\_\{g\; \backslash in\; G\}\; g\; \backslash otimes\; g^\{-1\}$.

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.{{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 $A^e$ 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 $\backslash mu\; :\; A\; \backslash otimes\_K\; A\; \backslash rightarrow\; A$ 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 $A\; \backslash rightarrow\; A\; \backslash otimes\_K\; A$ given by $a\; \backslash mapsto\; a\; \backslash otimes\; 1$. 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).{{sfn|Ford|2017|loc=Theorem 4.4.1|ps=none}}

Equivalently, the relative Hochschild cohomology groups $H^n(R,S;M)$ 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.

A separable algebra is said to be strongly separable if there exists a separability idempotent that is symmetric, meaning

: $e\; =\; \backslash sum\_\{i=1\}^n\; x\_i\; \backslash otimes\; y\_i\; =\; \backslash sum\_\{i=1\}^n\; y\_i\; \backslash otimes\; x\_i$

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.{{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}}.}}

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}}

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.

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Category:Algebras