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I have been reading the book Algebraic Geometry by Robin Hartshorne and I have found the following proposition:

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For part b) the proof goes as follows:

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The thing is that, How can we ensure that $f \in \sqrt{a}$?. And I really don't follow proof that $\psi $ surjective. That goes as follows:

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So I hope someone can help me understanding this parts of this proof. Thanks in advance.

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  • $\begingroup$ Thank you, let me post the proof of the surjectivity :) can you help me with that? $\endgroup$
    – user162343
    May 26, 2016 at 22:16
  • $\begingroup$ Ready I have edited my post $\endgroup$
    – user162343
    May 26, 2016 at 22:26

1 Answer 1

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It's in the lemma 2.1. $V(\mathfrak{a}) \cap D(f) = \varnothing$ is equivalent to $V(\mathfrak{a}) \subset V(f)$, which according to lemma 2.1 means that $\sqrt{(f)} \subset \sqrt{\mathfrak{a}}$, so $f \in \sqrt{\mathfrak{a}}$.

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  • $\begingroup$ Thank you, let me post the proof of the surjectivity :) can you help me with that? $\endgroup$
    – user162343
    May 26, 2016 at 22:15
  • $\begingroup$ Ready :) I have edited my post $\endgroup$
    – user162343
    May 26, 2016 at 22:25
  • $\begingroup$ The idea is to use a "partition of unity" argument. I recommend looking at Ravi Vakil's course notes, it's much more clearly explained over there. $\endgroup$
    – xyzzyz
    May 26, 2016 at 23:02
  • $\begingroup$ Ok let me check just a second :) If I have questions, can I leave you a comment ? $\endgroup$
    – user162343
    May 26, 2016 at 23:23
  • $\begingroup$ In Hartshorne's proof, why can we cover a basic open set $D_f$ with open sets $V_i$? $\endgroup$
    – user162343
    May 27, 2016 at 0:04

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