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.
Proj construction and fibered products
1
$\begingroup$
algebraic-geometry
commutative-algebra
schemes
projective-schemes
1 Answers
1
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.
-
5@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