Lemma 110.14.1. There exists a ring $R$ and a nonsplit sequence of modules which becomes split Zariski locally.
110.14 Nonsplit locally split sequence
Consider the short exact sequence
\[ 0 \to M \to \bigoplus \nolimits _{p\text{ prime}} \mathbf{Z}_{(p)} \to \mathbf{Q} \to 0 \]
where $M$ is the kernel of the map on the right which uses the inclusion $\mathbf{Z}_{(p)} \to \mathbf{Q}$ on each summand. This sequence of $\mathbf{Z}$-modules is nonsplit because there are no nonzero homomorphisms $\mathbf{Q} \to \mathbf{Z}_{(p)}$. On the other hand, if we localize at any prime $p$, then the sequence becomes split.
Proof. See discussion above. $\square$
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