Let $k$ be a field, and let $A$ be a commutative $k$-algebra.

Assume that $A$ is a noetherian ring, and let $I\subseteq A$ be a proper ideal.

Consider the ideal $I\otimes_k A \subseteq A\otimes_k A$. Is this ideal, considered as an $A\otimes_k A$-module, a noetherian module? is it coherent?

  • $\begingroup$ Oh sorry, I read it as "Is this an ideal considered as..." but you wrote "Is this ideal, considered as..." Sorry for the confusion. My question then changes to: what strategy are you trying to pursue to show it is Noetherian? What've you tried so far? $\endgroup$ – rschwieb Jun 30 '14 at 14:21
  • $\begingroup$ I am mainly trying to understand how do sub-modules of it look like? I am hoping that they are all of the form $I\otimes_k J$ where $I$ and $J$ are ideals in $A$. $\endgroup$ – user160906 Jun 30 '14 at 14:27
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    $\begingroup$ Well, not many of us are mindreaders, so yes, include that hypothesis and spare us from the "of course but that's not what I meant" and properly constrain your question. Good luck, and regards. $\endgroup$ – rschwieb Jun 30 '14 at 14:50
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    $\begingroup$ @rschwieb, I don't understand your comment, nor the downvote to the question. It is customary in commutative algebra that when one says ideal, one means proper ideal. Thus, the question as stated makes sense. Also, your "solution" does not make much sense, because if the op thought that the enveloping algebra is noetherian then the answer to both of his question will be obviously yes (because his ideal is f.g, and f.g ideals over noetherian rings are noetherian and coherent). $\endgroup$ – the L Jun 30 '14 at 16:46
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    $\begingroup$ Dear @theL : ...nor the downvote: I don't understand the downvote either. ...when one says ideal, one means proper ideal: I have never seen anyone use that convention, so apparently it's rare enough to merit clarification. ...your "solution" does not make much sense I did not offer a solution. Are you talking about the $\Bbb C\otimes _\Bbb Q\Bbb C$ comment? Obviously this was made with regard to the terminology misunderstanding that is now resolved. Hope this helps to understand the comment. Regards $\endgroup$ – rschwieb Jun 30 '14 at 16:55

The answer is typically no. E.g. if $I$ is a principal ideal, and say $A$ is a domain, then $I$ is isomorphic to $A$ as an $A$-module, so $I\otimes_kA$ is isomorphic to $A\otimes_k A$ as an $A\otimes_k A$-module. Typically $A\otimes_k A$ will not be Noetherian (and I don't think it will be coherent in general either, although I've not thought much about that, so don't have a counterexample at hand).


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