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Stefan Kohl ♦
Stefan Kohl ♦
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on On exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

ifIf $A$ is a Noetherian ring, $M$ is a finitely generated module,  $I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then then we know that $\hat{A}\otimes_{A}M\cong\hat{M}$.

Also onin Atiyah&Macdonald, there is a remark on Page 109 that the functor $M \mapsto \hat{M}$ is not exact without assuming $M$ finitely generated. but the functor $M \mapsto \hat{A}\otimes_{A}M$ is always exact.

How to prove this assertion?

And what is an example of the breakdown of exactness of $M \mapsto \hat{M}$ when $M$ is not finitely generated?

(This is not a homework problem)

on exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

if $A$ is a Noetherian ring, $M$ a finitely generated module,$I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know $\hat{A}\otimes_{A}M\cong\hat{M}$.

Also on Atiyah&Macdonald, there is a remark on Page 109 that the functor $M \mapsto \hat{M}$ is not exact without assuming $M$ finitely generated. but the functor $M \mapsto \hat{A}\otimes_{A}M$ is always exact.

How to prove this assertion?

And what is an example of the breakdown of exactness of $M \mapsto \hat{M}$ when $M$ is not finitely generated?

(This is not a homework problem)

On exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

If $A$ is a Noetherian ring, $M$ is a finitely generated module,  $I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know that $\hat{A}\otimes_{A}M\cong\hat{M}$.

Also in Atiyah&Macdonald, there is a remark on Page 109 that the functor $M \mapsto \hat{M}$ is not exact without assuming $M$ finitely generated. but the functor $M \mapsto \hat{A}\otimes_{A}M$ is always exact.

How to prove this assertion?

And what is an example of the breakdown of exactness of $M \mapsto \hat{M}$ when $M$ is not finitely generated?

(This is not a homework problem)

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Alex
Alex
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on exactness of the functors $M \mapsto \hat{M}$ and $M \mapsto \hat{A}\otimes_{A}M$

if $A$ is a Noetherian ring, $M$ a finitely generated module,$I$ is an ideal of $A$, and $\hat{A}$ is the $I-adic$ completion of $A$, then we know $\hat{A}\otimes_{A}M\cong\hat{M}$.

Also on Atiyah&Macdonald, there is a remark on Page 109 that the functor $M \mapsto \hat{M}$ is not exact without assuming $M$ finitely generated. but the functor $M \mapsto \hat{A}\otimes_{A}M$ is always exact.

How to prove this assertion?

And what is an example of the breakdown of exactness of $M \mapsto \hat{M}$ when $M$ is not finitely generated?

(This is not a homework problem)