example of inverse limit and direct limit

Question

Does a direct limit of projective need to be projective? And is the inverse limit of injectives injective? I guess they need not, but I can't find an example.

Can you help please?

Answer 1

In his paper "Finitistic dimension...", Bass proved that a ring $R$ has the property that every direct limit of projective modules is projective $\Leftrightarrow R$ is perfect (meaning that every module has a projective cover) $\Leftrightarrow R$ satisfies the descending chain condition on principal ideals $\Leftrightarrow R/J(R)$ is semisimple and $J(R)$ is T-nilpotent (meaning that for any sequence $\{x_n\}_{n=1}^\infty \subseteq J(R)$ there is some $n$ with $x_1\cdots x_n = 0$). To be more concrete (although I am not sure how useful this part is), he implicitly shows that $R$ is perfect $\Leftrightarrow$ certain direct limits of countably generated free modules are projective, namely ones of the form $\lim_{n \to \infty} F/G_n$, where $F$ is the free module spanned by $\{e_n\}_{n=1}^\infty$, $\{x_n\}_{n=1}^\infty \subseteq R$, and $G_n = \{e_i - x_ie_{i+1}\}_{i=1}^n$.

Answer 2

Neither.

The $\mathbb{Z}$-module $\mathbb{Q}$ is the direct limit of its finitely generated submodules. A finitely generated submodule of $\mathbb{Q}$ is actually infinite cyclic and so free. However $\mathbb{Q}$ is not projective.

So the claim is false even for filtered direct limits.

For inverse limits of injective modules, see this paper by Bergman on arXiv where it is proved that every module is the inverse limit of injective modules.