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Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

(It is not true, by the two answers. Is it at least true that the Hochschild cohomology is zero if the outer automorphism group is finite?)

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this (at least for finite dimensional algebras)?

Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this (at least for finite dimensional algebras)?

Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

(It is not true, by the two answers. Is it at least true that the Hochschild cohomology is zero if the outer automorphism group is finite?)

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this (at least for finite dimensional algebras)?

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Source Link
Mare
Mare
  • 26.5k
  • 6
  • 25
  • 104

Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this (at least for finite dimensional algebras)?

Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this?

Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this (at least for finite dimensional algebras)?

Source Link
Mare
Mare
  • 26.5k
  • 6
  • 25
  • 104

Relation of the first Hochschild cohomology and the outer automorphism group

Let $R$ be a ring.

Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?

This is true for finite dimensional algebras but possibly true in a much more general sitation (maybe even for more general categories than module categories of rings). Is there a suitable reference for this?