An immediate consequence of the definition of a compact object
of an additive category
is the following lemma.
Compactness Lemma 1.
An object
in an abelian category
with arbitrary direct sums (also called coproducts) is compact if and only if the functor
commutes
with arbitrary direct
sums, that is, if
.
Compactness Lemma 2.
Let
be a ring and
an
-module.
(i) If
is a finitely generated
-module, then (M) is a compact object of
-mod.
(ii) If
is projective and is a compact object of
-mod, then
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
is projective, and then also choose any surjection
, with
being a possibly infinite set. There exists then a section
. If M were compact, the image of
would have to lie in a submodule
for some finite subset
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