Are there examples of rings $R$ with nonzero global dimension over which inverse limits of projectives are projective?
If not, are there any $R$'s over which products of projectives are projective (equivalently, $R^{\Bbb N}$ projective)?
2026-03-26 17:52:11.1774547531
When limits preserve projectivity
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A theorem of Chase states that projective (right) $R$-modules are closed under arbitrary products if and only if $R$ is right perfect and left coherent.
If (and only if) $R$ also has global dimension at most two, then projective modules are also closed under kernels, and hence under arbitrary limits.
If by inverse limits you mean only directed limits, there may be more examples, but I don’t know.