The Stacks project

Exercise 111.40.8. Let $R$ be a ring.

  1. Show that if $R$ is a Noetherian normal domain, then $\mathop{\mathrm{Pic}}\nolimits (R) = \mathop{\mathrm{Pic}}\nolimits (R[t])$. [Hint: There is a map $R[t] \to R$, $t \mapsto 0$ which is a left inverse to the map $R \to R[t]$. Hence it suffices to show that any invertible $R[t]$-module $M$ such that $M/tM \cong R$ is free of rank $1$. Let $K$ be the fraction field of $R$. Pick a trivialization $K[t] \to M \otimes _{R[t]} K[t]$ which is possible by Exercise 111.40.5 (1). Adjust it so it agrees with the trivialization of $M/tM$ above. Show that it is in fact a trivialization of $M$ over $R[t]$ (this is where normality comes in).]

  2. Let $k$ be a field. Show that $\mathop{\mathrm{Pic}}\nolimits (k[x^2, x^3, t]) \not= \mathop{\mathrm{Pic}}\nolimits (k[x^2, x^3])$.

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