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?
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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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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
$endgroup$
add a comment |
$begingroup$
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 ?
abstract-algebra ring-theory commutative-algebra
$endgroup$
add a comment |
$begingroup$
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 ?
abstract-algebra ring-theory commutative-algebra
$endgroup$
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
abstract-algebra ring-theory commutative-algebra
edited Mar 22 at 22:52
Andrews
1,2962423
1,2962423
asked Mar 22 at 22:16
Captain LamaCaptain Lama
10.1k1030
10.1k1030
add a comment |
add a comment |
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