Exercise 111.40.5. Simple examples.

Let $k$ be a field. Let $A = k[x]$. Show that $X = \mathop{\mathrm{Spec}}(A)$ has only trivial invertible ${\mathcal O}_ X$-modules. In other words, show that every invertible $A$-module is free of rank 1.

Let $A$ be the ring

\[ A = \{ f\in k[x] \mid f(0) = f(1) \} . \]Show there exists a nontrivial invertible $A$-module, unless $k = {\mathbf F}_2$. (Hint: Think about $\mathop{\mathrm{Spec}}(A)$ as identifying $0$ and $1$ in ${\mathbf A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$.)

Same question as in (2) for the ring $A = k[x^2, x^3] \subset k[x]$ (except now $k = {\mathbf F}_2$ works as well).

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