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slinshady
slinshady
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I have encountered the following "Nakayama Lemma" recently:

Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal C_\bullet$ be a chain complex of $I$-(derived) complete $A$-modules, bounded above by zero. Then $\mathcal C$ is acyclic if $C\otimes^{\mathbf L}_A A/I$ is acyclic.

I have tried to prove it first in the case where $C$ consists of classically $I$-complete $A$-modules, but I encountered some problems. Here is what I tried so far:

Suppose, $\mathcal C$ is not acyclic and let $n$ be the greatest integer such that $H_n(C)\neq 0$. We can apply the usual Nakayama Lemma (for separated modules, Theorem 8.4 in Matsumuras Commutative Ring THeory) to the $I$-adic completion $H_n(\mathcal C)^\wedge$ to see that $H_n(\mathcal C)^\wedge \otimes_A A/I\neq 0$. Now I somehow want to use that $H_n(\mathcal C\otimes_A^{\mathbf L} A/I)=H_n(\mathcal C) \otimes_A A/I$ by our choice of $n$ but I have a mental blockage when it comes to relating it to $H_n(\mathcal C)^\wedge \otimes_A A/I$. How do I proceed? How would I proceed in the derived case? Are there any good references to read up on derived completions?

I have encountered the following "Nakayama Lemma" recently:

Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal C_\bullet$ be a chain complex of $I$-(derived) complete $A$-modules, bounded above by zero. Then $\mathcal C$ is acyclic if $C\otimes^{\mathbf L}_A A/I$ is acyclic.

I have tried to prove it first in the case where $C$ consists of classically $I$-complete $A$-modules, but I encountered some problems. Here is what I tried so far:

Suppose, $\mathcal C$ is not acyclic and let $n$ be the greatest integer such that $H_n(C)\neq 0$. We can apply the usual Nakayama Lemma (for separated modules) to the $I$-adic completion $H_n(\mathcal C)^\wedge$ to see that $H_n(\mathcal C)^\wedge \otimes_A A/I\neq 0$. Now I somehow want to use that $H_n(\mathcal C\otimes_A^{\mathbf L} A/I)=H_n(\mathcal C) \otimes_A A/I$ by our choice of $n$ but I have a mental blockage when it comes to relating it to $H_n(\mathcal C)^\wedge \otimes_A A/I$. How do I proceed? How would I proceed in the derived case? Are there any good references to read up on derived completions?

I have encountered the following "Nakayama Lemma" recently:

Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal C_\bullet$ be a chain complex of $I$-(derived) complete $A$-modules, bounded above by zero. Then $\mathcal C$ is acyclic if $C\otimes^{\mathbf L}_A A/I$ is acyclic.

I have tried to prove it first in the case where $C$ consists of classically $I$-complete $A$-modules, but I encountered some problems. Here is what I tried so far:

Suppose, $\mathcal C$ is not acyclic and let $n$ be the greatest integer such that $H_n(C)\neq 0$. We can apply the usual Nakayama Lemma (for separated modules, Theorem 8.4 in Matsumuras Commutative Ring THeory) to the $I$-adic completion $H_n(\mathcal C)^\wedge$ to see that $H_n(\mathcal C)^\wedge \otimes_A A/I\neq 0$. Now I somehow want to use that $H_n(\mathcal C\otimes_A^{\mathbf L} A/I)=H_n(\mathcal C) \otimes_A A/I$ by our choice of $n$ but I have a mental blockage when it comes to relating it to $H_n(\mathcal C)^\wedge \otimes_A A/I$. How do I proceed? How would I proceed in the derived case? Are there any good references to read up on derived completions?

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slinshady
slinshady
  • 309
  • 1
  • 8

Derived Nakayama for complete modules

I have encountered the following "Nakayama Lemma" recently:

Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal C_\bullet$ be a chain complex of $I$-(derived) complete $A$-modules, bounded above by zero. Then $\mathcal C$ is acyclic if $C\otimes^{\mathbf L}_A A/I$ is acyclic.

I have tried to prove it first in the case where $C$ consists of classically $I$-complete $A$-modules, but I encountered some problems. Here is what I tried so far:

Suppose, $\mathcal C$ is not acyclic and let $n$ be the greatest integer such that $H_n(C)\neq 0$. We can apply the usual Nakayama Lemma (for separated modules) to the $I$-adic completion $H_n(\mathcal C)^\wedge$ to see that $H_n(\mathcal C)^\wedge \otimes_A A/I\neq 0$. Now I somehow want to use that $H_n(\mathcal C\otimes_A^{\mathbf L} A/I)=H_n(\mathcal C) \otimes_A A/I$ by our choice of $n$ but I have a mental blockage when it comes to relating it to $H_n(\mathcal C)^\wedge \otimes_A A/I$. How do I proceed? How would I proceed in the derived case? Are there any good references to read up on derived completions?