Lemma 52.26.1. Let (A, \mathfrak m) be a Noetherian local ring. Let f \in \mathfrak m be a nonzerodivisor and assume that \text{depth}(A/fA) \geq 2, or equivalently \text{depth}(A) \geq 3. Let U, resp. U_0 be the punctured spectrum of A, resp. A/fA. The map
is injective on torsion.
Proof. Let \mathcal{L} be an invertible \mathcal{O}_ U-module. Observe that \mathcal{L} maps to 0 in \mathop{\mathrm{Pic}}\nolimits (U_0) if and only if we can extend \mathcal{L} to an invertible coherent triple (\mathcal{L}, \mathcal{L}_0, \lambda ) as in Section 52.25. By Proposition 52.25.4 the function
is a polynomial. By Lemma 52.25.5 the value of this polynomial is zero if and only if \mathcal{L}^{\otimes n} is trivial. Thus if \mathcal{L} is torsion, then this polynomial has infinitely many zeros, hence is identically zero, hence \mathcal{L} is trivial. \square
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