The Stacks project

Lemma 99.11.4. In Situation 99.11.1 let $\sigma : B \to X$ be a section. Assume that $\mathcal{O}_ T \to f_{T, *}\mathcal{O}_{X_ T}$ is an isomorphism for all $T$ over $B$. Then

\[ 0 \to \mathop{\mathrm{Pic}}\nolimits (T) \to \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathrm{Pic}_{X/B}(T) \to 0 \]

is a split exact sequence with splitting given by $\sigma _ T^* : \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathop{\mathrm{Pic}}\nolimits (T)$.

Proof. Denote $K(T) = \mathop{\mathrm{Ker}}(\sigma _ T^* : \mathop{\mathrm{Pic}}\nolimits (X_ T) \to \mathop{\mathrm{Pic}}\nolimits (T))$. Since $\sigma $ is a section of $f$ we see that $\mathop{\mathrm{Pic}}\nolimits (X_ T)$ is the direct sum of $\mathop{\mathrm{Pic}}\nolimits (T)$ and $K(T)$. Thus by Lemma 99.11.3 we see that $K(T) \subset \mathrm{Pic}_{X/B}(T)$ for all $T$. Moreover, it is clear from the construction that $\mathrm{Pic}_{X/B}$ is the sheafification of the presheaf $K$. To finish the proof it suffices to show that $K$ satisfies the sheaf condition for fppf coverings which we do in the next paragraph.

Let $\{ T_ i \to T\} $ be an fppf covering. Let $\mathcal{L}_ i$ be elements of $K(T_ i)$ which map to the same elements of $K(T_ i \times _ T T_ j)$ for all $i$ and $j$. Choose an isomorphism $\alpha _ i : \mathcal{O}_{T_ i} \to \sigma _{T_ i}^*\mathcal{L}_ i$ for all $i$. Choose an isomorphism

\[ \varphi _{ij} : \mathcal{L}_ i|_{X_{T_ i \times _ T T_ j}} \longrightarrow \mathcal{L}_ j|_{X_{T_ i \times _ T T_ j}} \]

If the map

\[ \alpha _ j|_{T_ i \times _ T T_ j} \circ \sigma _{T_ i \times _ T T_ j}^*\varphi _{ij} \circ \alpha _ i|_{T_ i \times _ T T_ j} : \mathcal{O}_{T_ i \times _ T T_ j} \to \mathcal{O}_{T_ i \times _ T T_ j} \]

is not equal to multiplication by $1$ but some $u_{ij}$, then we can scale $\varphi _{ij}$ by $u_{ij}^{-1}$ to correct this. Having done this, consider the self map

\[ \varphi _{ki}|_{X_{T_ i \times _ T T_ j \times _ T T_ k}} \circ \varphi _{jk}|_{X_{T_ i \times _ T T_ j \times _ T T_ k}} \circ \varphi _{ij}|_{X_{T_ i \times _ T T_ j \times _ T T_ k}} \quad \text{on}\quad \mathcal{L}_ i|_{X_{T_ i \times _ T T_ j \times _ T T_ k}} \]

which is given by multiplication by some section $f_{ijk}$ of the structure sheaf of $X_{T_ i \times _ T T_ j \times _ T T_ k}$. By our choice of $\varphi _{ij}$ we see that the pullback of this map by $\sigma $ is equal to multiplication by $1$. By our assumption on functions on $X$, we see that $f_{ijk} = 1$. Thus we obtain a descent datum for the fppf covering $\{ X_{T_ i} \to X\} $. By Descent on Spaces, Proposition 74.4.1 there is an invertible $\mathcal{O}_{X_ T}$-module $\mathcal{L}$ and an isomorphism $\alpha : \mathcal{O}_ T \to \sigma _ T^*\mathcal{L}$ whose pullback to $X_{T_ i}$ recovers $(\mathcal{L}_ i, \alpha _ i)$ (small detail omitted). Thus $\mathcal{L}$ defines an object of $K(T)$ as desired. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0D28. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 99.11.4 as pdf