1
$\begingroup$

(This was also simultaneously asked on math stack exchange: https://math.stackexchange.com/questions/4857715/vakils-generalization-of-qcqs-lemma)

In the most recent notes of Vakil, this is problem 15.4.Y (which I think is just Hartshorne Lemma 5.14):

Let $X$ be qcqs, $\mathscr{L}$ invertible sheaf on $X$, $s\in \Gamma(X, \mathscr{L})$, and $\mathscr{F}$ quasi coherent on $X$. Viewing $\bigoplus_{n\geq 0} \Gamma(X, \mathscr{L^{\otimes n}})$ as a graded ring with $s$ in degree 1 and $\bigoplus_{n\geq 0} \Gamma(X,\mathscr{F}\otimes_{\mathscr{O}_X} \mathscr{L}^{\otimes n})$ as a graded $\bigoplus_{n\geq 0} \Gamma(X, \mathscr{L^{\otimes n}})$-module, show that there is a natural morphism

$$\left(\left(\bigoplus_{n\geq 0} \Gamma(X,\mathscr{F}\otimes_{\mathscr{O}_X} \mathscr{L}^{\otimes n})\right)_s\right)_0 \to \Gamma(X_s, \mathscr{F})$$

which is an isomorphism under these assumptions.

I am stuck on defining the morphism above. I am guessing this morphism should exist without the assumptions on $X$ and $\mathscr{F}$. My guess is that we take something on the left, say $f/s^d$, where $f\in \Gamma(X,\mathscr{F}\otimes_{\mathscr{O}_X} \mathscr{L}^{\otimes d})$ and appropriately view $1/s^d$ as an element of $\Gamma(X, \mathscr{Hom}_{\mathscr{O}_X}(\mathscr{L}^{\otimes d}, \mathscr{O}_X))=\Gamma(X,(\mathscr{L}^{\otimes d})^{*})$ so that the morphism becomes $f/s^d \mapsto f\otimes 1/s^d$. I am stuck on the step of "appropriately viewing $1/s^d$" (which might not even be possible). Does this approach work, or is it too hopeful to have such a global description of this map?

Improve this question
$\endgroup$
3
  • $\begingroup$ By the definition of $X_s$, the section $s: O_X \to L$ becomes an isomorphism when restricted to $X_s$. This means that the restriction $s|_{X_s}$ has an inverse $t: L|_{X_s} \to O_{X_s}$. Now for any $f/s^d$, where $f:O_X \to F \otimes L^d$, you can restrict to $f|_{X_s}: O_{X_s} \to F|_[X_s} \otimes (L|_{X_s})^d$ and postcompose with $id_F \otimes t^d: F|_{X_s} \otimes (L|_{X_s})^d \to F|_{X_s}$ to obtain a section in $\Gamma(X_s, F)$. $\endgroup$
    – afh
    Commented Feb 6 at 16:29
  • $\begingroup$ (Sorry, broken diagram above is supposed to read $f|_{X_s}: O_{X_s} \to F|_{X_s} \otimes (L|_{X_s})^d$). $\endgroup$
    – afh
    Commented Feb 6 at 16:42
  • 2
    $\begingroup$ It is bad form to ask duplicates without links/people volunteer their time to answer questions $\endgroup$
    – kodlu
    Commented Feb 6 at 17:49

0

Reset to default

You must log in to answer this question.