PlanetPhysics/Compactness Lemma

An immediate consequence of the definition of a [[../Coproduct/|compact object]] $X$ of an [[../DenseSubcategory/|additive category]] $𝒜$ is the following lemma.

{\mathbf Compactness Lemma 1.}

An \htmladdnormallink{object {http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $X$ in an [[../AbelianCategory2/|abelian category]] $𝒜$ with arbitrary direct sums (also called [[../Coproduct/|coproducts]]) is compact if and only if the [[../TrivialGroupoid/|functor]] $ho{m}_{𝒜}(X,-)$ [[../Commutator/|commutes]] with arbitrary direct sums, that is, if $ho{m}_{𝒜}(X,\underset{\alpha \in S}{⨁}{Y}_{\alpha })=\underset{\alpha \in S}{⨁}ho{m}_{𝒜}(X,Y_{\alpha })$}.

{\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.}

[[../Predicate/|Proposition]] (i) follows immediately from the [[../Generator/|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 $p:A^{⨁I}\twoheadrightarrow M$, with $I$ being a possibly infinite set. There exists then a [[../IsomorphicObjectsUnderAnIsomorphism/|section]] $s:M\hookrightarrow |A^{⨁I}$. If M were compact, the image of $s$ would have to lie in a submodule $A^{⨁J}\subseteq A^{⨁I},$ for some finite subset $J\subseteq I$. Then $p|A^{⨁J}$ is still [[../BCConjecture/|surjective]], which proves that $M$ is finitely generated.