Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
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
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. –  user18119 Nov 3 '12 at 10:51

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.