An immediate consequence of the definition of a compact object X of an additive category 𝒜 is the
following lemma.
Compactness Lemma 1.
An object X in an abelian category 𝒜 with arbitrary direct sums (also called coproducts) is compact
if and only if the functor hom𝒜(X,−) commutes with arbitrary direct sums, that is,
if
.
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⊕
I ↠ M, with I being a possibly infinite set. There exists then a section s : M
|A⊕
I. If M
were compact, the image of s would have to lie in a submodule
for some finite subset J ⊆ I. Then p|A⊕
J is still surjective, which proves that M is finitely
generated.
