Separable algebra

<h2 id="definition-and-first-properties">Definition and first properties</h2>
A <a href="/facts/Homomorphism/0gyqdEse">homomorphism</a> of (unital, but not necessarily <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a>) <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a>

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 <a href="/facts/Section_(category_theory)/Ee1wexPw">section</a>

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

that is a homomorphism of A-A-<a href="/facts/Bimodule/NskmNGQ1">bimodules</a>. 
If the ring 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is commutative and 
 
 
 
 K
 →
 A
 
 
 {\displaystyle K\to A}
 
 maps 
 
 
 
 K
 
 
 {\displaystyle K}
 
 into the <a href="/facts/Center_(ring_theory)/kJcejgkj">center</a> 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 <a href="/facts/Idempotent/khbSpexv">idempotent</a>, 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}
 
.

<h2 id="examples">Examples</h2>
For any commutative ring R, the (<a href="/facts/Non-commutative_ring/3jmll7q7">non-commutative</a>) ring of n-by-n <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> 
 
 
 
 
 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 <a href="/facts/Elementary_matrix/coFCElwg">elementary matrix</a> 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.

<h3>Separable algebras over a field</h3>
A <a href="/facts/Field_extension/L7yIDtyK">field extension</a> L/K of finite <a href="/facts/Degree_of_a_field_extension/hY5LgHYk">degree</a> is a <a href="/facts/Separable_extension/9KBvXmz8">separable extension</a> if and only if L is separable as an associative K-algebra. If L/K has a <a href="/facts/Primitive_element_(field_theory)/xegn8Bpl">primitive element</a> 
 
 
 
 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 <a href="/facts/Algebraic_closure/yjewG1mM">algebraic closure</a> 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 <a href="/facts/Frobenius_algebra/xnxIFzQ7">Frobenius algebra</a> over K.
More generally, separable algebras over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K can be classified as follows: they are the same as finite products of <a href="/facts/Matrix_algebra/oj5HBr8B">matrix algebras</a> over <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Division_algebra/xp5Bepx3">division algebras</a> whose centers are finite-dimensional <a href="/facts/Separable_extension/9KBvXmz8">separable field extensions</a> of the field K. In particular: Every separable algebra is itself finite-dimensional. If K is a <a href="/facts/Perfect_field/0JJvr8Xx">perfect field</a> – for example a field of <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> zero, or a <a href="/facts/Finite_field/UkMGJo3m">finite field</a>, or an <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed field</a> – 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 <a href="/facts/Semisimple_algebra/JIXNLzck">semisimple algebra</a> 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.

<h3>Group rings</h3>
If K is commutative ring and G is a <a href="/facts/Finite_group/vv2tvogB">finite group</a> such that the <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a> of G is invertible in K, then the <a href="/facts/Group_ring/TJlkt5Te">group algebra</a> K[G] is a separable K-algebra.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> 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}}
 
.

<h2 id="equivalent-characterizations-of-separability">Equivalent characterizations of separability</h2>
There are several equivalent definitions of separable algebras. A K-algebra A is separable if and only if it is <a href="/facts/Projective_module/lHymqhNC">projective</a> when considered as a left <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> of 
 
 
 
 
 A
 
 e
 
 
 
 
 {\displaystyle A^{e}}
 
 in the usual way.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> Moreover, an algebra A is separable if and only if it is <a href="/facts/Flat_module/1WFrnJP3">flat</a> 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 <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequences</a> of A-A-bimodules that are <a href="/facts/Split_exact_sequence/fBswKvU9">split</a> 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 <a href="/facts/Maschke%27s_theorem/uFMFeyfA">Maschke's theorem</a>, applying its components within and without the splitting maps).<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Equivalently, the relative <a href="/facts/Hochschild_cohomology/0phwDIyX">Hochschild cohomology</a> 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.

<h2 id="relation-to-frobenius-algebras">Relation to Frobenius algebras</h2>
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 <a href="/facts/Frobenius_algebra/xnxIFzQ7">Frobenius algebra</a> called a symmetric algebra (not to be confused with the <a href="/facts/Symmetric_algebra/A8FthY0N">symmetric algebra</a> arising as the quotient of the <a href="/facts/Tensor_algebra/H7CKKryf">tensor algebra</a>).
If K is commutative, A is a <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> <a href="/facts/Projective_module/lHymqhNC">projective</a> separable K-module, then A is a symmetric Frobenius algebra.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="relation-to-formally-unramified-and-formally-tale-extensions">Relation to formally unramified and formally étale extensions</h2>
Any separable extension A / K of commutative rings is <a href="/facts/Formally_unramified/GNxTBHTG">formally unramified</a>. The converse holds if A is a finitely generated K-algebra.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> A separable <a href="/facts/Flat_module/1WFrnJP3">flat</a> (commutative) K-algebra A is <a href="/facts/Formally_%C3%A9tale/TLdIqRJe">formally étale</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h2 id="further-results">Further results</h2>
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 <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a>: 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 <a href="/facts/Coprime/bnCxCXxs">coprime</a> to the characteristic. The separability condition above will imply every finitely generated A-module M is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> 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 <a href="/facts/Indecomposable_module/N9B1gpzI">indecomposables</a>, 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.

<h2 id="citations">Citations</h2>

<ul><li>DeMeyer, F.; Ingraham, E. (1971). Separable algebras over commutative rings. Lecture Notes in Mathematics. Vol. 181. Berlin-Heidelberg-New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-05371-2. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0215.36602">0215.36602</a>.</li>
<li><a href="/facts/Samuel_Eilenberg/vs6aksKX">Samuel Eilenberg</a> and Tadasi Nakayama, <a href="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</a>, Nagoya Math. J. Volume 9 (1955), 1–16.</li>
<li>Endo, Shizuo; Watanabe, Yutaka (1967), <a href="http://projecteuclid.org/euclid.ojm/1200691953">"On separable algebras over a commutative ring"</a>, Osaka Journal of Mathematics, 4: 233–242, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0227211">0227211</a></li>
<li>Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4704-3770-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3618889">3618889</a></li>
<li>Hirata, H.; Sugano, K. (1966), "On semisimple and separable extensions of noncommutative rings", J. Math. Soc. Jpn., 18: 360–373</li>
<li>Kadison, Lars (1999), New examples of Frobenius extensions, University Lecture Series, vol. 14, Providence, RI: American Mathematical Society, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fulect%2F014">10.1090/ulect/014</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-1962-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1690111">1690111</a></li>
<li><a href="/facts/Irving_Reiner/rAXzMfas">Reiner, I.</a> (2003), Maximal Orders, London Mathematical Society Monographs. New Series, vol. 28, <a href="/facts/Oxford_University_Press/CYyTzzWG">Oxford University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-852673-3, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1024.16008">1024.16008</a></li>
<li><a href="/facts/Charles_Weibel/ERYbVCB9">Weibel, Charles A.</a> (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-55987-4. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1269324">1269324</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/36131259">36131259</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Ford 2017, §4.2 - Ford, Timothy J. (2017), Separable algebras, Providence, RI: American Mathematical Society, ISBN 978-1-4704-3770-1, MR 3618889 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3618889" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=3618889</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="https://zbmath.org/?format=complete&q=an:1024.16008" target="_blank">https://zbmath.org/?format=complete&q=an:1024.16008</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3618889" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=3618889</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="http://projecteuclid.org/euclid.ojm/1200691953" target="_blank">http://projecteuclid.org/euclid.ojm/1200691953</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3618889" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=3618889</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3618889" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=3618889</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Separable algebra open-in-new

Separable algebra