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Consider the following property of an abelian group $G$:

S: $G$ is torsionfree and a directed limit of finitely generated (hence free) subgroups $\{F_i\}_i$, such that for all $i \leq j$, $F_i$ is a direct summand of $F_j$.

Clearly, every free abelian group satisfies S. Does also the converse hold? If not, is S satisfied by $A^\mathbb{N} / A^{(\mathbb{N})}$, where $A=\mathbb{Q}_+=\mathbb{Z}^{(\mathbb{P})}$?

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Edit: The answer below has been modified to reflect the comments.

My guess is that $G$ is forced to be projective, hence free, in this situation. To show this, we need to verify that $\mathrm{Hom}(G,-)$ is an exact functor. As we have an identification of functors $\mathrm{Hom}(G,-) \simeq \lim_i \mathrm{Hom}(F_i,-)$, applying $\mathrm{Hom}(G,-)$ to an exact sequence

$0 \to A \to B \to C \to 0$

of abelian groups, we get an induced sequence

$0 \to \lim_i \mathrm{Hom}(F_i,A) \to \lim_i \mathrm{Hom}(F_i,B) \to \lim_i \mathrm{Hom}(F_i,C) \to R^1 \lim_i \mathrm{Hom}(F_i,A) \to ...$

So it suffices to show that $R^1 \lim_i \mathrm{Hom}(F_i,A)$ vanishes for any abelian group $A$. Is this true?

Here's a not-so-well-thought-out idea: if I chased elements correctly, an affirmative answer to the question above follows from the bijectivity of the natural map $\mathrm{Hom}(F_j,A) \to \lim_{i < j} \mathrm{Hom}(F_i,A)$, for j sufficiently big. After making the harmless assumption that the system $(F_i)$ consists of all finitely generated saturated subgroups of $G$, the preceding bijectivity question translates to: given a free abelian group F of finite rank, when is the natural map $\mathrm{colim}_i H_i \to F$ an isomorphism, where the indexing set I is the poset of all proper saturated subgroups $H_i \subset F$. I think the answer to this question is yes when the rank of $F$ is at least $3$ (which is enough for the application at hand), but I'm not sure.

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  • $\begingroup$ Is the surjectivity condition sufficient to guarantee vanishing of $\lim^1$ for a directed limit which is not necessarily a sequential limit? $\endgroup$
    – Reid Barton
    Commented Jan 30, 2010 at 21:51
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    $\begingroup$ Probably not. If I is a directed poset with at most $\aleph_d$ morphisms, then $lim^n$ over $I$ vanishes for $n \geq d+2$ (Mitchell, Rings with several objects). I don't think that bhargav's proofs works this easily. $\endgroup$
    – Martin Brandenburg
    Commented Jan 31, 2010 at 2:19
  • $\begingroup$ You're both right. I think I implicitly assumed the system was indexed by the natural numbers, which is definitely not the case for this question. $\endgroup$
    – Bhargav
    Commented Jan 31, 2010 at 7:55
  • $\begingroup$ Taking the direct limit of modules over a directed index set is an exact functor, non? $\endgroup$
    – Joel Dodge
    Commented Jan 31, 2010 at 9:13
  • $\begingroup$ In any case, I think direct limits are pretty clearly right exact, no matter what index set you look at. The left exactness is what is less clear and one can prove exactness just using the fact that your index set is directed. $\endgroup$
    – Joel Dodge
    Commented Jan 31, 2010 at 9:16

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