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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