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How to show, that $Proj \, A[x_0,...,x_n] = Proj \, \mathbb{Z}[x_0,...,x_n] \times_\mathbb{Z} Spec \, A$? It is used in Hartshorne, Algebraic geometry, section 2.7.

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Show that you have an isomorphism on suitable open subsets, and that the isomorphisms glue. The standard ones on $\mathbb{P}^n_a$ should suffice. Use that $\mathbb{Z}[x_0, \ldots, x_n] \otimes_\mathbb{Z} A \cong A[x_0,..., \ldots, x_n].$ Maybe you could prove the isomorphism by using the universal property of projective spaces too, but that might be overkill / not clean at all.

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    @user46336: the equality on$Proj$you want is proved in the book "Algebraic geometry and arithmetic curves", 3.1.9 for all projective schemes.2012-11-03