compactness lemma

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

Compactness Lemma 1.

An object $ 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

$\displaystyle hom_{\mathcal{A}}(X,\bigoplus_{\alpha \in S} Y_{\alpha}) =
\bigoplus_{\alpha \in S} hom_{\mathcal{A}}(X,Y_{\alpha})$

.

Compactness Lemma 2. 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.

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 \vert A^{\bigoplus I}$ . If M were compact, the image of $ s$ would have to lie in a submodule

$\displaystyle A^{\bigoplus J} \subseteq A^{\bigoplus I},$

for some finite subset $ J \subseteq I$ . Then $ p\vert A^{\bigoplus J}$ is still surjective, which proves that $ M$ is finitely generated.



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