Editing Direct limit (section)

== Properties ==
Direct limits are linked to [[inverse limit]]s via

:<math>\mathrm{Hom} (\varinjlim X_i, Y) = \varprojlim \mathrm{Hom} (X_i, Y).</math>

An important property is that taking direct limits in the category of [[module (mathematics)|modules]] is an [[exact functor]]. This means that for any directed system of [[short exact sequence]]s <math>0 \to A_i \to B_i \to C_i \to 0</math>, the sequence <math>0 \to \varinjlim A_i \to \varinjlim B_i \to \varinjlim C_i \to 0</math> of direct limits is also exact.