Given an $A$-module $M$, then we have an $\mathcal O_{\operatorname{Spec}A}$-modules $M^\sim$. If $M^\sim$ is locally of finite presentation, is $M$ an $A$-module with finite presentation?


I assume your question is the following:

Given an quasi-coherent $\mathscr{O}_{\text{Spec } A}$-module $\widetilde{M}$, associated to an $A$-module $M$. If $\widetilde{M}$ is of finite presentation then $M$ is of finite presentation.

Well then the answer is yes. Your question corresponds to (3) of https://stacks.math.columbia.edu/tag/00EO.

Further explanation: We can find finitely many $f_i\in A$, such that $\cup D(f_i) = \text{Spec } A$. And since $\widetilde{M}$ is quasi coherent we have that $\widetilde{M}_{|D(f_i)} = \widetilde{M_{f_i}}$. Thus if $\widetilde{M}$ is of finite presentation we have that $M_{f_i}$ is a finitely presented $A_{f_i}$ module for all $i$ (note that the $D(f_i)$ form a basis of the topology). Thus with the Lemma above we can conclude that $M$ is of finite presentation.

  • $\begingroup$ But I say $M^\sim$ is locally of finite presentation in my question. $\endgroup$ – Born to be proud Sep 14 '18 at 12:02
  • $\begingroup$ For me a module over a ringed space $X$ is of finite presentation if for all $x\in X$ there is a neighbourhood such that the restriction of the module to that neighbourhood is of finite presentation. I think thats the same, so I don't assume that anything is quasi compact (well here Spec $A$ is quasi compact) or has finite covering with this property or is noetherian. Or do you mean something elese with "locally" of finite presentation ? $\endgroup$ – Can Yaylali Sep 14 '18 at 12:05
  • $\begingroup$ You are right. Thank you for your answer. $\endgroup$ – Born to be proud Sep 14 '18 at 12:09
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    $\begingroup$ The meaning of "locally of finite presentation" I discuss is the same as the meaning of "finite presentation you discuss. $\endgroup$ – Born to be proud Sep 14 '18 at 12:41
  • $\begingroup$ Haha, you have given a circular definition. $\endgroup$ – Born to be proud Sep 14 '18 at 12:51

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