Lemma 31.28.5 (0BXJ)—The Stacks project

Lemma 31.28.5. Let $R$ be a UFD. The Picard group of $\mathbf{P}^ n_ R$ is $\mathbf{Z}$. More precisely, there is an isomorphism

\[ \mathbf{Z} \longrightarrow \mathop{\mathrm{Pic}}\nolimits (\mathbf{P}^ n_ R),\quad m \longmapsto \mathcal{O}_{\mathbf{P}^ n_ R}(m) \]

In particular, the Picard group of $\mathbf{P}^ n_ k$ of projective space over a field $k$ is $\mathbf{Z}$.

Proof. By Lemma 31.28.4 the Picard groups of the opens $D_+(T_ i) \cong \mathbf{A}^ n_ R$ are trivial. Thus if $\mathcal{L}$ is an invertible module on $\mathbf{P}^ n_ R$, then it is given by a $1$-cocycle with values in the sheaf of invertible functors $\mathcal{O}^*$ for the open covering

\[ \mathbf{P}^ n_ R = \bigcup \nolimits _{i = 0, \ldots , n} D_+(T_ i) \]

Observe that for $i \not= j$ we have

\[ \mathcal{O}^*(D_+(T_ i) \cap D_+(T_ j)) = \mathcal{O}^*(D_+(T_ iT_ j)) = \left(R[T_0, \ldots , T_ n]_{(T_ iT_ j)}\right)^* = R^* \times (T_ i/T_ j)^\mathbf {Z} \]

Thus such a cocycle $(g_{ij})$ is given by units

\[ g_{ij} = u_{ij} (T_ i/T_ j)^{e_{ij}} \]

with $u_{ij} \in R^*$ and $e_{ij} \in \mathbf{Z}$ satisfying the cocycle condition. The cocycle condition over $D_+(T_ iT_ jT_ k)$ for $\# \{ i, j, k\} = 3$ tell us that

\[ u_{ik} = u_{ij} u_{jk} \quad \text{and}\quad e_{ik} = e_{ij} = e_{jk} \]

Whence $u_{ij} = u_{i1} u_{j1}^{-1}$ is a boundary. Thus all isomorphism classes of invertible modules are given by taking the cocycle with

\[ g_{ij} = (T_ i/T_ j)^ e \]

for some $e \in \mathbf{Z}$. Since $\mathcal{O}(n)$ has trivializing section $T_ i^ n$ over $D_+(T_ i)$ we see that the corresponding cocycle of $\mathcal{O}(n)$ is $(T_ i/T_ j)^ n$ and the proof is complete. $\square$

Comments (3)

Comment #8749 by Lukas Heger on December 12, 2023 at 04:24

Don't we need to assume that $R$ is Noetherian?

Comment #8764 by Lukas Heger on January 06, 2024 at 10:29

Maybe one can use \ref{https://stacks.math.columbia.edu/tag/0B8W} to reduce to the Noetherian case, but I'm not sure if every UFD is a colimit of Noetherian UFDs.

Comment #9306 by Stacks project on May 22, 2024 at 15:56

Good catch! OK, I rewrote the proof to make it work in the non-Noetherian case. More interesting was the case of Lemma 31.28.4. Finally, Lemma 31.31.3 was unfixable and I needed to assume the Noetherian assumption. See these changes.

Comments (3)

Post a comment