Lemma 39.9.7. Let k be a field. Let A be an abelian variety over k. Let \mathcal{L} be an invertible \mathcal{O}_ A-module. Then
[n]^*\mathcal{L} \cong \mathcal{L}^{\otimes n(n + 1)/2} \otimes ([-1]^*\mathcal{L})^{\otimes n(n - 1)/2}
where [n] : A \to A sends x to x + x + \ldots + x with n summands and where [-1] : A \to A is the inverse of A.
Proof.
Consider the morphism A \to A \times _ k A \times _ k A, x \mapsto (x, x, -x) where -x = [-1](x). Pulling back the relation of Lemma 39.9.6 we obtain
\mathcal{L} \otimes \mathcal{L} \otimes \mathcal{L} \otimes [-1]^*\mathcal{L} \cong [2]^*\mathcal{L}
which proves the result for n = 2. By induction assume the result holds for 1, 2, \ldots , n. Then consider the morphism A \to A \times _ k A \times _ k A, x \mapsto (x, x, [n - 1]x). Pulling back the relation of Lemma 39.9.6 we obtain
[n + 1]^*\mathcal{L} \otimes \mathcal{L} \otimes \mathcal{L} \otimes [n - 1]^*\mathcal{L} \cong [2]^*\mathcal{L} \otimes [n]^*\mathcal{L} \otimes [n]^*\mathcal{L}
and the result follows by elementary arithmetic.
\square
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