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We define the projective $n$-space over any ring $A$, $$\mathbb{P}_A^n:={\rm Proj}~A[x_0,\cdots,x_n]$$.

my first question is that $\mathbb{Z}[x_1,\cdots,x_n]\otimes_\mathbb{Z} A \cong A[x_1,\cdots,x_n]$? how define an isomorphism??

If it is an isomrphism, then using this can we prove that $\mathbb{P}_A^n \cong \mathbb{P}_\mathbb{Z}^n \times_{{\rm Spec}~Z} {\rm Spec}~A$?

I need to prove in detail. Help me.

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1 Answer 1

up vote 3 down vote accepted

Hint for the tensor product isomorphism: There is a natural bilinear map $\mathbb{Z}[x_1,\dots,x_n] \times A \rightarrow A[x_1,\dots,x_n]$ given by $(p(x_1,\dots,x_n),a) \mapsto a \cdot p(x_1,\dots,x_n)$.

Hints for the second part: Do you know how to show that $\text{Spec}(A[x_1,\dots,x_n])$ is isomorphic to $\text{Spec}(\mathbb{Z}[x_1,\dots,x_n]) \times_{\text{Spec}(\mathbb{Z})} \text{Spec}(A)$? (The tensor product isomorphism is very helpful for this...) Once you do that, you can establish your desired isomorphism by using the standard affine cover of projective space.

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