I have a few questions concerning twisted sheaves as defined in Hartshorne, II.15.
1) Let $X = \text{Proj}(S)$ for a graded ring $S$. I do understand how Hartshorne defines the sheaves $\mathcal O_X(n)$ for $n \in \mathbb Z$. Then for any sheaf of $\mathcal O_X$-modules $\mathcal F$ and any $n \in \mathbb Z$ he defines $\mathcal F(n) := \mathcal F \otimes_{\mathcal O_X} \mathcal O_X(n)$. I understand that as well.
Now let $Y$ be any projective scheme over a noetherian ring $A$, i.e. a scheme $Y$ together with a morphism $\psi : Y \to \text{Spec}(A)$ such that there is a closed immersion $i : Y \to \mathbb P_A^r$ for some $r \in \mathbb N$ with $\psi = gi$, where $g$ is the natural morphism $\mathbb P_A^r = \mathbb P_{\mathbb Z} \times_{\mathbb Z} \text{Spec}(A) \to \text{Spec}(A)$. In this situation Hartshorne uses (e.g. in II.5.17) the notation $\mathcal F(n)$ for a coherent $\mathcal O_Y$-module $\mathcal F$. My question is:
How exactly is $\mathcal F(n)$ defined in this context? (My idea: I have to choose a very ample sheaf $\mathcal O(1)$ on $Y$ (as defined in II.5.17) and define $\mathcal F(n) := \mathcal F \otimes_{\mathcal O_Y} \mathcal O(1)^{\otimes n}$. This makes sense even for negative $n$ since $\mathcal O(1)$ is invertibel.)
2) Two questions about the proof of Hartshorne's theorem II.5.17. In the very first sentence, Hartshorne chooses for a very ample invertible sheaf $\mathcal O(1)$ on $X$ a closed immersion $i : X \to \mathbb P_A^r$ such that $i^*(\mathcal O(1)) = \mathcal O_X(1)$.
Why can we talk about the inverse image of $\mathcal O(1)$ although $\mathcal O(1)$ is defined on $X$?
Why can the immersion $i$ be chosen to be closed?
3) Let $X$ be a projective scheme and $\mathcal F$ any sheaf of $\mathcal O_X$-modules. Is it true that $\Gamma(X,\mathcal F(n)) = 0$ for $n<0$?
If not, is it true for $n<<0$?
Thanks in advance!
Best Answer
1): You are correct.
2): When Hartshorne writes "$i^*(\mathcal{O}(1))=\mathcal{O}_X(1)$", what he really means is "$i^*(\mathcal{O}_{\Bbb P^r_A}(1))=\mathcal{O}_X(1)$. Hartshorne frequently leaves off the subscript denoting projective space when dealing with twisting sheaves - this is occasionally unfortunate. The reason $i$ can be chosen to be closed is because any $A$-map out of a projective $A$-scheme is closed: projective schemes are proper and $\Bbb P^r_A\to \operatorname{Spec} A$ is separated (for instance, the proof of theorem II.4.9), and by corollary II.4.8 we get $X\to \Bbb P^r_A$ is proper, thus universally closed, thus closed.
3): No. Let $i:\{p\}\to \Bbb P^r_A$ be the closed immersion of a closed point with residue field $k$. Then the global sections of $i_*\mathcal{O}_{\{p\}}(n)$ are exactly $k$ for any $n$.