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I can't understand a step in this proposition from Gathmann's notes. Link: http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002.pdf

How does he go from the fact that $J$ contains an element not vanishing at $P$ to concluding that the zero locus of the ideal is contained in $Z(f)$?

Also, how can you add $I(X)$ and $J$ when $I(X)$ is an ideal of the ring of polynomials whereas $J$ is an ideal of $A(X)$ which is the quotient of the polynomial ring by $I(X)$. Could you please explain the reasoning?

Thank you.

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$I(X)+J$ is just the preimage of $J$ under the projection $k[x_1,...,x_n]\to A(X)$.

For the conclusion note that he shows first that $P\in X_f$ implies $P\notin V(J)$ (as there is a function in $J$, which does not vanish on $P$). This is equivalent to $P\in V(J)$ implying $P\notin X_f$, i.e. $P$ vanishing on $f$.

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  • $\begingroup$ I understand it now. Thank you very much! $\endgroup$
    – Jehu314
    May 12, 2018 at 3:22

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