The Stacks project

Lemma 15.121.3. Let $R$ be a valuation ring with fraction field $K$ and residue field $\kappa $. Let $R \to A$ be a homomorphism of rings such that

  1. $A$ is local and $R \to A$ is local,

  2. $A$ is flat and essentially of finite type over $R$,

  3. $A \otimes _ R \kappa $ regular.

Then $\mathop{\mathrm{Pic}}\nolimits (A \otimes _ R K) = 0$.

Proof. Let $L$ be an invertible $A \otimes _ R K$-module. In particular $L$ is a finite module. There exists a finite $A$-module $M$ such that $M \otimes _ R K \cong L$, see Algebra, Lemma 10.126.3. We may assume $M$ is torsion free as an $R$-module. Thus $M$ is flat as an $R$-module (Lemma 15.22.10). From Lemma 15.25.6 we deduce that $M$ is of finite presentation as an $A$-module and $A$ is essentially of finite presentation as an $R$-algebra. By Lemma 15.83.4 we see that $M$ is perfect relative to $R$, in particular $M$ is pseudo-coherent as an $A$-module. By Lemma 15.77.6 we see that $M$ is perfect, hence $M$ has a finite free resolution $F_\bullet $ over $A$. It follows that $L$ is quasi-isomorphic to a finite complex of free $A \otimes _ R K$-modules. Hence by Lemma 15.119.1 we see that $[L] = n[A \otimes _ R K]$ in $K_0(A \otimes _ R K)$ for some $n \in \mathbf{Z}$. Applying the map of Lemma 15.118.7 we see that $L$ is trivial. $\square$

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