2
$\begingroup$

I'm stuck on another problem (17.2.G) from Vakil's notes, and I'm wondering if somebody could get me started.

Specifically, we are given a scheme $X$, an invertible sheaf $\mathscr{L}$ on $X$ and a quasicoherent sheaf $\mathscr{S}$ of graded algebras on $X$. We define $\mathscr{S}' = \oplus_{n=0}^{\infty}{(\mathscr{S}_n\otimes \mathscr{L}^{\otimes n})}$. How do we get a natural map \begin{align*} \mathcal{Proj}\mathscr{S}'\leftarrow \mathcal{Proj}\mathscr{S}? \end{align*} The problem asks for more (in particular the map should be an isomorphism, where we might have to use that $\mathscr{S}$ is generated in degree 1), but I'm mainly stuck on just defining the map.

I've have tried writing down the transition maps explicitly for the case where $X=\mathbb{P}^1$, $\mathscr{L}=\mathcal{O}(1)$, and $\mathscr{S}=\pi_{*}\mathcal{O}_{\mathbb{P}^1\times\mathbb{A}^2}$, where $\pi$ is the projection $\mathbb{P}^{1}\times \mathbb{A}^2\rightarrow \mathbb{P}^1$, but I'm still missing something silly. (not homework)

$\endgroup$
  • 2
    $\begingroup$ Dear Dtseng, I don't think your example really fits the set-up, at least not in the way you were imagining. In fact $\pi_*\mathcal O_{\mathbb P^1 \times \mathbb P^1}$ is just $\mathcal O_{\mathbb P^1}$; in particular, it is not graded. $\endgroup$ – Matt E Jun 13 '14 at 21:34
  • $\begingroup$ Oops, you're right. I tried to fix the example. $\endgroup$ – DCT Jun 13 '14 at 23:36
4
$\begingroup$

Remember that relative Proj is constructed by choosing an affine open cover $\{U_i\}$ of $X$, say $U_i =$ Spec $A_i$, considering the graded rings $B_i = \oplus_n B_{i,n} = \oplus_n \Gamma(U_i,\mathcal S_n)$, forming the absolute Proj of each $B_i$, and then gluing these Proj's on the overlaps of the $U_i$.

So why don't you try choosing a cover $U_i$ which trivializes $\mathcal L$, and seeing what happens?

$\endgroup$
  • $\begingroup$ Oh, shoot, I think I see now. The fact we're taking elements of degree zero in the absolute Proj means all the twisting doesn't affect the structure sheaf. I think I was too worried about trying to get a map between the two sheaves of algebras. Thanks for your help! $\endgroup$ – DCT Jun 13 '14 at 23:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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