# PlanetPhysics/Compactness Lemma

An immediate consequence of the definition of a compact object of an additive category is the following lemma.

{\mathbf Compactness Lemma 1.}

*An \htmladdnormallink{object* {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in an abelian category with arbitrary direct sums (also called coproducts) is compact if and only if the functor commutes with arbitrary direct
sums, that is, if
}.

{\mathbf Compactness Lemma 2.} {\em Let be a ring and an -module. (i) If is a finitely generated -module, then (M) is a compact object of -mod. (ii) If is projective and is a compact object of -mod, then is finitely generated.}

{\mathbf Proof.}

Proposition *(i)* follows immediately from the generator definition for the case of an Abelian category.

To prove statement (ii), let us assume that is projective, and then also choose any surjection , with being a possibly infinite set. There exists then a section . If M were compact, the image of would have to lie in a submodule for some finite subset . Then is still surjective, which proves that is finitely generated.