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Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?
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Analogously, under what assumptions on R does a countable product of projective left R-modules necessarily have a finite projective dimension?
These questions arise in the study of the coderived and contraderived categories of (CDG-)modules, or, if one wishes, the homotopy categories of unbounded complexes of injective or projective modules.
There are some obvious sufficient conditions and some less-so-obvious ones. For both #1 and #2, it clearly suffices that R have a finite left homological dimension.
More interestingly, in both cases it suffices that R be left Gorenstein, i.e., such that the classes of left R-modules of finite projective dimension and left R-modules of finite injective dimension coincide.
For #1, it also suffices that R be left Noetherian. For #2, it suffices that R be right coherent and such that any flat left module has a finite projective dimension.
Any other sufficient conditions?
Best Answer
For #1, it suffices that $R$ be left coherent and such that any fp-injective left $R$-module has finite injective dimension. In particular, these conditions hold when $R$ is left coherent and every left ideal in $R$ has a set of generators of the cardinality not exceeding $\aleph_n$ for some nonnegative integer $n$ (e.g., a countable set of generators). See Section 2 in https://arxiv.org/abs/1504.00700 .