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This is related to my question herehere. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?

Our algebras here are associative, unital, and not necessarily commutative.

This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?

Our algebras here are associative, unital, and not necessarily commutative.

This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?

Our algebras here are associative, unital, and not necessarily commutative.

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Jakob W
Jakob W
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This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?

Our algebras here are associative, unital, and not necessarily commutative.

This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?

This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?

Our algebras here are associative, unital, and not necessarily commutative.

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Jakob W
Jakob W
  • 349
  • 1
  • 7

Nonnegatively graded algebra $A$ finitely generated as $k$-algebra iff $A_0$ finitely generated, $A_{>0}$ finitely generated as $A$-module?

This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated as a $k$-algebra and $A_{>0}$ is finitely generated as an $A$-module (i.e. as a left ideal of $A$)?