The Stacks project

Lemma 27.12.4. Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $T$ be a scheme. Let $(d, \mathcal{L}, \psi )$ and $(d', \mathcal{L}', \psi ')$ be two triples over $T$. The following are equivalent:

  1. Let $n = \text{lcm}(d, d')$. Write $n = ad = a'd'$. There exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi |_{S^{(n)}}$ and $\psi '|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \to \Gamma _*(Y, (\mathcal{L}')^{\otimes n})$.

  2. The triples $(d, \mathcal{L}, \psi )$ and $(d', \mathcal{L}', \psi ')$ are equivalent.

  3. For some positive integer $n = ad = a'd'$ there exists an isomorphism $\beta : \mathcal{L}^{\otimes a} \to (\mathcal{L}')^{\otimes a'}$ with the property that $\beta \circ \psi |_{S^{(n)}}$ and $\psi '|_{S^{(n)}}$ agree as graded ring maps $S^{(n)} \to \Gamma _*(Y, (\mathcal{L}')^{\otimes n})$.

  4. The morphisms $\varphi : T \to X$ and $\varphi ' : T \to X$ associated to $(d, \mathcal{L}, \psi )$ and $(d', \mathcal{L}', \psi ')$ are equal.

Proof. Clearly (1) implies (2) and (2) implies (3) by restricting to more divisible degrees and powers of invertible sheaves. Also (3) implies (4) by the uniqueness statement in Lemma 27.12.1. Thus we have to prove that (4) implies (1). Assume (4), in other words $\varphi = \varphi '$. Note that this implies that we may write $\mathcal{L} = \varphi ^*\mathcal{O}_ X(d)$ and $\mathcal{L}' = \varphi ^*\mathcal{O}_ X(d')$. Moreover, via these identifications we have that the graded ring maps $\psi $ and $\psi '$ correspond to the restriction of the canonical graded ring map

\[ S \longrightarrow \bigoplus \nolimits _{n \geq 0} \Gamma (X, \mathcal{O}_ X(n)) \]

to $S^{(d)}$ and $S^{(d')}$ composed with pullback by $\varphi $ (by Lemma 27.12.1 again). Hence taking $\beta $ to be the isomorphism

\[ (\varphi ^*\mathcal{O}_ X(d))^{\otimes a} = \varphi ^*\mathcal{O}_ X(n) = (\varphi ^*\mathcal{O}_ X(d'))^{\otimes a'} \]

works. $\square$

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