# Proj construction and fibered products

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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## 1 Answer

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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Yes, but it's very difficult to show, that they glue. – user46336 Nov 3 '12 at 10:37
What is your problem when trying to show that they glue? Maybe we can help you. – Dedalus Nov 3 '12 at 10:40
We cover $Proj \, A[x_0,...,x_n]$ with open sets, isomorphic to $Spec \, A[x_0,...,x_n]_{(f)}$. How to show, that sheafs are agreed on intersections? In general, $Proj$ construction is not clear intuitively. It's chore, and the geometric sense is not quite understandable. – user46336 Nov 3 '12 at 10:43
@user46336: the equality on Proj you want is proved in the book "Algebraic geometry and arithmetic curves", 3.1.9 for all projective schemes. – user18119 Nov 3 '12 at 10:51