The Stacks project

27.20 Twisting by invertible sheaves and relative Proj

Let $S$ be a scheme. Let $\mathcal{A} = \bigoplus _{d \geq 0} \mathcal{A}_ d$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra. Let $\mathcal{L}$ be an invertible sheaf on $S$. In this situation we obtain another quasi-coherent graded $\mathcal{O}_ S$-algebra, namely

\[ \mathcal{B} = \bigoplus \nolimits _{d \geq 0} \mathcal{A}_ d \otimes _{\mathcal{O}_ S} \mathcal{L}^{\otimes d} \]

It turns out that $\mathcal{A}$ and $\mathcal{B}$ have isomorphic relative homogeneous spectra.

Lemma 27.20.1. With notation $S$, $\mathcal{A}$, $\mathcal{L}$ and $\mathcal{B}$ as above. There is a canonical isomorphism

\[ \xymatrix{ P = \underline{\text{Proj}}_ S(\mathcal{A}) \ar[rr]_ g \ar[rd]_\pi & & \underline{\text{Proj}}_ S(\mathcal{B}) = P' \ar[ld]^{\pi '} \\ & S & } \]

with the following properties

  1. There are isomorphisms $\theta _ n : g^*\mathcal{O}_{P'}(n) \to \mathcal{O}_ P(n) \otimes \pi ^*\mathcal{L}^{\otimes n}$ which fit together to give an isomorphism of $\mathbf{Z}$-graded algebras

    \[ \theta : g^*\left( \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_{P'}(n) \right) \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \mathcal{O}_ P(n) \otimes \pi ^*\mathcal{L}^{\otimes n} \]
  2. For every open $V \subset S$ the diagrams

    \[ \xymatrix{ \mathcal{A}_ n(V) \otimes \mathcal{L}^{\otimes n}(V) \ar[r]_{multiply} \ar[d]^{\psi \otimes \pi ^*} & \mathcal{B}_ n(V) \ar[dd]^\psi \\ \Gamma (\pi ^{-1}V, \mathcal{O}_ P(n)) \otimes \Gamma (\pi ^{-1}V, \pi ^*\mathcal{L}^{\otimes n}) \ar[d]^{multiply} \\ \Gamma (\pi ^{-1}V, \mathcal{O}_ P(n) \otimes \pi ^*\mathcal{L}^{\otimes n}) & \Gamma (\pi '^{-1}V, \mathcal{O}_{P'}(n)) \ar[l]_-{\theta _ n} } \]

    are commutative.

  3. Add more here as necessary.

Proof. This is the identity map when $\mathcal{L} \cong \mathcal{O}_ S$. In general choose an open covering of $S$ such that $\mathcal{L}$ is trivialized over the pieces and glue the corresponding maps. Details omitted. $\square$

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