PlanetPhysics/Compactness Lemma

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

{\mathbf Compactness Lemma 1.}

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

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