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

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 ?

abstract-algebra ring-theory commutative-algebra

share|cite|improve this question

edited Mar 22 at 22:52

Andrews

1,2962423

asked Mar 22 at 22:16

Captain LamaCaptain Lama

10.1k1030

$endgroup$

add a comment | 

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 ?

abstract-algebra ring-theory commutative-algebra

share|cite|improve this question

edited Mar 22 at 22:52

Andrews

1,2962423

asked Mar 22 at 22:16

Captain LamaCaptain Lama

10.1k1030

$endgroup$

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

share|cite|improve this question

edited Mar 22 at 22:52

Andrews

1,2962423

asked Mar 22 at 22:16

Captain LamaCaptain Lama

10.1k1030

$endgroup$

share|cite|improve this question

edited Mar 22 at 22:52

Andrews

1,2962423

asked Mar 22 at 22:16

Captain LamaCaptain Lama

10.1k1030

share|cite|improve this question

edited Mar 22 at 22:52

Andrews

1,2962423

asked Mar 22 at 22:16

Captain LamaCaptain Lama

10.1k1030

edited Mar 22 at 22:52

Andrews

1,2962423

edited Mar 22 at 22:52

Andrews

1,2962423

asked Mar 22 at 22:16

Captain LamaCaptain Lama

10.1k1030

asked Mar 22 at 22:16

Captain LamaCaptain Lama

10.1k1030