In the book "Algebraic geometry" by Andreas Gathmann, in page 26 Lemma 2.3.16 , why the ideal $J$ is prime?. i mean why $Z(J)$ is affine variety?.it says" we claim that the ringed space ... is isomorphic to affine variety $Z(J)$".
and by the way why $\pi^{-1}(P)$ is morphism? cause it can't send polynomials to polynomials .
https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002.pdf&ved=2ahUKEwizjqXCq8TfAhXFSRUIHYSWDnMQFjAAegQIBxAB&usg=AOvVaw3dsFU1a0Lna0NLsTrDJv9v
2026-03-25 09:27:27.1774430847
Affine variety of an algebraic set in ghathmann notebook
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