15.2.E. Suppose $X$ is a normal irreducible Noetherian scheme, and $\mathscr{L}$ is an invertible sheaf, and $s$ is an nonzero rational section of $\mathscr{L}$.
(a) Describe an isomorphism $\mathscr{O} ( \operatorname{div}s) \leftrightarrow \mathscr{L}$ . (You will use the normality hypothesis!) Hint: show that those open subsets for which $\mathscr{O} ( \operatorname{div}s) \cong \mathcal{O}_U$ form a base for the Zariski topology. For each such $U$, define $\phi |_U : \mathscr{O} ( \operatorname{div}s )( U )\to \mathscr{L} ( U )$ sending a rational function $t$ (with zeros and poles “constrained by div $s$”) to $st$. Show that $\phi |_U$ is an isomorphism (with the obvious inverse map, division by $s$). Argue that this map induces an isomorphism of sheaves $\phi:\mathscr{O}( \operatorname{div} s ) \stackrel{\sim}{\to} \mathscr{L} $.
(b) Let $\sigma$ be the map from $K(X)$ to the rational sections of $\mathscr{L}$ , where $\sigma( t )$ is the rational section of $\mathscr{O}_X(D)$ defined via (15.2.2.1) (as described in Remark 15.2.3). Show that the isomorphism of (a) can be chosen such that $\sigma( 1 ) = s$. (Hint: the map in part (a) sends 1 to $s$.)
Here are the portions of the text mentioned above:
15.2.2 Important Definition. Assume now that $X$ is irreducible (purely to avoid making (15.2.2.1) look uglier – but feel free to relax this, see exercise 15.2.B). Assume also that $X$ is normal – this will be a standing assumption for the rest of this section. Let $D$ be a Weil divisor, define $\mathscr{O}_X(D)$ by
$$\text{(15.2.2.1)}\qquad \Gamma(U,\mathscr{O}_X(D)):=\{t\in K(X)^{\times}:\operatorname{div}|_Ut+D|_Ut\geq 0\}\cup\{0\}.
$$
15.2.3. Remark. It will be helpful to note that$ \mathscr{O}_X( D )$ comes along with a canonical “rational section” corresponding to $1 \in K ( X )^{\times}$. (It is a rational section in the sense that it is a section over a dense open set, namely the complement of Supp $D$.)
I am confused with the map $\sigma$ in (b), since I don't find the explicit definition of it. And what is the $D$ in (b)? This problem is a big obstacle in my way. Thanks for everyone's help.
Best Answer
$O(div(s))$ has a rational section $1 \in O(div(s))(U)$ for some open set $U$. Define $\sigma : K(X)^* \rightarrow $rational sections of $L$ as follows : given $a \in K(X)^*$, $a$ corresponds to an element of $O_X(V)$ for some open set $V$. Then, the rational section is $a * 1 \in O(div(s))(U \cap V)$.
From part (a) we know that $O(div(s))$ and $L$ are isomorphic. Therefore, $a * 1$ also corresponds to an element of $L(U \cap V)$. We want to find an isomorphism that sends $a * 1$ to $s$.