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Example 10.82.2. Examples of universally exact sequences.

  1. A split short exact sequence is universally exact since tensoring commutes with taking direct sums.

  2. The colimit of a directed system of universally exact sequences is universally exact. This follows from the fact that taking directed colimits is exact and that tensoring commutes with taking colimits. In particular the colimit of a directed system of split exact sequences is universally exact. We will see below that, conversely, any universally exact sequence arises in this way.

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