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compactness lemma (Theorem)

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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See Also: generator, compact object

Keywords:  compactness lemma

Cross-references: surjective, section, generator, proposition, commutes, functor, coproducts, Abelian category, object, additive category, compact object

This is version 6 of compactness lemma, born on 2009-06-15, modified 2009-06-15.
Object id is 803, canonical name is CompactnessLemma.
Accessed 491 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
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