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:

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 $S^{(n)} \to \Gamma _*(Y, (\mathcal{L}')^{\otimes n})$.

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

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 $S^{(n)} \to \Gamma _*(Y, (\mathcal{L}')^{\otimes n})$.

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

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