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.