I am trying to understand projective morphisms (primarily from reading Ravi Vakil's Foundations of Algebraic Geometry notes Ch 17 and 18) and I have run across a very basic problem. Say I have a ring $A,$ a graded $A$ algebras $S$ which is finitely generated in degree 1, an $A$-scheme $Y$ with structure morphism $\pi:Y\rightarrow \text{spec} A,$ and the structure morphism $\beta: \text{Proj} S\rightarrow \text{spec} A.$

In general it is true that maps of $A$-schemes, $f: Y\rightarrow \text{Proj} S$ are in correspondence with maps of graded rings $t_1:S\rightarrow \Gamma_{\ast}(X,L)$ for some line bundle $L$ on $Y$ where $L$ is globally generated by $f(S_1)$. (This is 17.4.2).

However later in the notes 18.2.1, Ravi states that such maps should be in bijection with maps $t_2:S\rightarrow \oplus_{n\geq 0} \pi_{\ast}L^{\otimes n}$ where there is a similar condition on being surjective in degree 1 and in this case I am viewing $S$ as an $O_{\text{spec} A}$ module.

What I don't understand is how maps $t_1$ correspond to maps $t_2$ aka why maps on global sections correspond to maps of sheaves. Morally speaking $L = f_{\ast} O(1)$ so if we knew that $f_{\ast}f^{\ast} O(1)$ was quasicoherent on $\text{Proj} S$ then using the fact that $\pi_{\ast} L = \beta_{\ast}f_{\ast}f^{\ast} O(1)$ and the fact that $\beta$ is qcqs (it is proper and $\text{Proj} S$ is quasicompact) would imply that $\pi_{\ast} L$ is quasicoherent. However I am not sure how to prove this result or whether I am looking at things the right way. Any guidance would be appreciated and please feel free to ask me to clarify.


First of all, are you sure that all the details in your question are correct? I'm looking at Ravi's notes now, and it seems that Ex. 17.4.A is the closest approximation to the statement in your question, and the conditions are not quiet what you've stated, e.g. the image of $S_1$ should globally generate the line bundle $L$. (The kind of thing this forbids is taking $Y$ to be $\mathbb P^1$, taking $L$ to be $\mathcal O(-1)$, all of whose positive tensor powers have vanishing global sections, and taking $t_1$ (in your notation) to be the map which just kills $S_{> 0}$.)

In any case, leaving this aside for now, the point is the following:

if $M$ is any $A$-module, e.g. a graded piece of $S$, and $\mathcal N$ is any sheaf of $\mathcal O$-modules (where $\mathcal O$ , is the structure sheaf on Spec $A$), e.g. $\pi_* L$, then $$Hom_A\bigl(M,\Gamma(\mathrm{Spec} A, \mathcal N)\bigr) = Hom_{\mathcal O}(\widetilde{M},\mathcal N),$$ where $\widetilde{M}$ is the quasicoherent sheaf attached to $M$. This is the universal property of $\widetilde{M}$, and explains why $\tilde{M}$ is sometimes written as $\widetilde{M} := M\otimes_A \mathcal O$. Note that it doesn't require $\mathcal N$ to be quasi-coherent.

In any case, we see now that graded Homs from $S$ (as a quasicoherent $\mathcal O$-algebra, which I would prefer to denote by $\widetilde{S}$, in accordance with the preceding paragraph) to $\oplus \pi_* L^{\otimes n}$ are the same as Homs form $S$ (now just a graded $A$-algebra) to $\oplus \Gamma(\mathrm{Spec} A, \pi_* L^{\otimes n}),$ which equals $\oplus \Gamma(Y,L^{\otimes n})$ by definition.

This reconciles the discussion of section 17 with that of section 18.

  • $\begingroup$ That is clearly what I am missing. Thanks for the help! You are right about my incorrect statement - it is fixed now. The example you give is particularly enlightening. $\endgroup$ – Lalit Jain Jun 16 '11 at 0:18

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