# PlanetPhysics/Compactness Lemma

An immediate consequence of the definition of a compact object ${\displaystyle X}$ of an additive category ${\displaystyle {\mathcal {A}}}$ is the following lemma.

{\mathbf Compactness Lemma 1.}

An \htmladdnormallink{object {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} ${\displaystyle X}$ in an abelian category ${\displaystyle {\mathcal {A}}}$ with arbitrary direct sums (also called coproducts) is compact if and only if the functor ${\displaystyle hom_{\mathcal {A}}(X,-)}$ commutes with arbitrary direct sums, that is, if ${\displaystyle hom_{\mathcal {A}}(X,\bigoplus _{\alpha \in S}Y_{\alpha })=\bigoplus _{\alpha \in S}hom_{\mathcal {A}}(X,Y_{\alpha })}$}.

{\mathbf Compactness Lemma 2.} {\em Let ${\displaystyle A}$ be a ring and ${\displaystyle M}$ an ${\displaystyle A}$-module. (i) If ${\displaystyle M}$ is a finitely generated ${\displaystyle A}$-module, then (M) is a compact object of ${\displaystyle A}$-mod. (ii) If ${\displaystyle M}$ is projective and is a compact object of ${\displaystyle A}$-mod, then ${\displaystyle M}$ 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 ${\displaystyle M}$ is projective, and then also choose any surjection ${\displaystyle p:A^{\bigoplus I}\twoheadrightarrow M}$, with ${\displaystyle I}$ being a possibly infinite set. There exists then a section ${\displaystyle s:M\hookrightarrow |A^{\bigoplus I}}$. If M were compact, the image of ${\displaystyle s}$ would have to lie in a submodule ${\displaystyle A^{\bigoplus J}\subseteq A^{\bigoplus I},}$ for some finite subset ${\displaystyle J\subseteq I}$. Then ${\displaystyle p|A^{\bigoplus J}}$ is still surjective, which proves that ${\displaystyle M}$ is finitely generated.