Do we need to assume that Azumaya algebras are projective of finite type? The 2019 Stack...

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Do we need to assume that Azumaya algebras are projective of finite type?



The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Azumaya algebrasIf Hom(M,-) is stable under base change, is then M f.g. projective?Classification of separable algebras over a commutative ringSeparability of $A subseteq C$ implies separability of $B subseteq C$, where $A subseteq B subseteq C$Finite-dimensional algebras which do not satisfy Wedderburn's principal theoremArtinian rings that are not Artin algebrasWhen are commutative, finite-dimensional complex algebras isomorphic?Separable algebras over a non-commutative ringRings such that there are finitely many modules of a given lengthThe multiplication of rank for finite projective modules












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$begingroup$


There are several equivalent definitions of Azumaya algebras over commutative rings, but they always seem to include a finiteness/projectiveness assumption.



For instance, an Azumaya algebra over $R$ is a $R$-algebra $A$ that is a finite projective $R$-module such that the natural "sandwich" map
$$Aotimes_R A^{op}to operatorname{End}_R(A)$$
is an isomorphism.



I totally understand why we would want Azumaya algebras to be projective of finite type, but is it necessary to explicitely assume it ?



Are there examples of non-projective algebras with the above property ?










share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    There are several equivalent definitions of Azumaya algebras over commutative rings, but they always seem to include a finiteness/projectiveness assumption.



    For instance, an Azumaya algebra over $R$ is a $R$-algebra $A$ that is a finite projective $R$-module such that the natural "sandwich" map
    $$Aotimes_R A^{op}to operatorname{End}_R(A)$$
    is an isomorphism.



    I totally understand why we would want Azumaya algebras to be projective of finite type, but is it necessary to explicitely assume it ?



    Are there examples of non-projective algebras with the above property ?










    share|cite|improve this question











    $endgroup$















      1












      1








      1


      1



      $begingroup$


      There are several equivalent definitions of Azumaya algebras over commutative rings, but they always seem to include a finiteness/projectiveness assumption.



      For instance, an Azumaya algebra over $R$ is a $R$-algebra $A$ that is a finite projective $R$-module such that the natural "sandwich" map
      $$Aotimes_R A^{op}to operatorname{End}_R(A)$$
      is an isomorphism.



      I totally understand why we would want Azumaya algebras to be projective of finite type, but is it necessary to explicitely assume it ?



      Are there examples of non-projective algebras with the above property ?










      share|cite|improve this question











      $endgroup$




      There are several equivalent definitions of Azumaya algebras over commutative rings, but they always seem to include a finiteness/projectiveness assumption.



      For instance, an Azumaya algebra over $R$ is a $R$-algebra $A$ that is a finite projective $R$-module such that the natural "sandwich" map
      $$Aotimes_R A^{op}to operatorname{End}_R(A)$$
      is an isomorphism.



      I totally understand why we would want Azumaya algebras to be projective of finite type, but is it necessary to explicitely assume it ?



      Are there examples of non-projective algebras with the above property ?







      abstract-algebra ring-theory commutative-algebra






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 22 at 22:52









      Andrews

      1,2962423




      1,2962423










      asked Mar 22 at 22:16









      Captain LamaCaptain Lama

      10.1k1030




      10.1k1030






















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