I am studying the proof of Lemma 7.2. on page 108 in Dolgachev's "Lectures on invariant theory". It states (everything is done over the field $k = \overline{k}$):
Let $X$ be a normal affine variety (for example, nonsingluar) and $G$ is a connected affine algebraic group. Let $L$ be a line bundle on $G \times X$. Then there exist line bundles $L_1 \in Pic(G)$, $L_2 \in Pic(X)$ such that $L \cong pr_1^*(L_1) \otimes pr_2^*(L_2)$ (where $pr_i$ is the $i$-th projection of $G \times X$).
The proof uses the following facts about algebraic groups:
- $G$ containes a Zariski open set $U$, which is isomorphic to $(\mathbb{A}^1 \setminus \{0\})^N$.
- $pr_2^* \colon Pic(X) \to Pic((\mathbb{A}^1 \setminus \{0\}) \times X)$ is an isomorphism.
Then it is clear that $L$ restricted to $U \times X$ is isomorphic to $pr_2^*(L_2)$ for some $L_2 \in Pic(X)$.
The following is said afterwards: Let $D$ be a Cartier divisor on $G \times X$ representing $L$.
Question 1: Why does such a $D$ exist? $X$ is not projective, so I don't quite see why $D$ should exist.
The proof continues as follows:
Then the preceding isomorphism implies that there exists a Cartier Divisor $D_2$ on $X$ such that $D' = D-pr_2^*(D_2)|_{U \times X} = 0$.
Question 2: Why exactly does $D_2$ exist?
Now, take any irreducibly component $D_i'$ of $D'$ and consider the projection $D_i' \to G$. The image is contained in $G \setminus U =: Z$. By the theorem on the dimension of fibres, the fibres of this projection have dimension equal to $\dim X$. Then the last part which I don't understand follows:
This easily implies that $D_i' = pr_1^*(D_i)$, where $D_i \subset Z$.
Question 3: Why?
Thanks in advance for any help.
