1. Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?

  2. 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?

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    $\begingroup$ Just wanted to say that this question is interesting to me, but you have covered all the cases that occur to me. One comment: I think your question is equivalent to asking when finite injective dimension modules are closed under (countable) direct sums, and the obvious dual thing for finite projective dimension. $\endgroup$ – Mark Hovey Dec 1 '09 at 15:17
  • $\begingroup$ I think the following assertions (and their obvious duals) are true: if a countable sum of injective R-modules always has a finite injective dimension, then this dimension is bounded by a constant d depending on R only. Moreover, a countable sum of R-modules of injective dimensions not exceeding n then never exceeds n+d. However, I do not see why what you are saying is true. E.g., if R is a Noetherian ring for which there are modules of arbitrarily high finite injective dimension, then a countable sum of such modules would have an infinite injective dimension, providing a counterexample. $\endgroup$ – Leonid Positselski Dec 1 '09 at 16:38
  • $\begingroup$ How about a much easier question: when are sums of injective themselves injective; similarly for projectives? Over a field or the ring of dual numbers $k[a]/(a^2)$, injectives=projectives, so the answer is yes. Over a PID, sums of divisible modules are divisible, so yes for injectives (and no for projectives e.g. over $R=\mathbb Z$). Are there any other interesting cases? $\endgroup$ – VA. Dec 7 '09 at 16:32
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    $\begingroup$ The answer to this easier question is well-known. Sums of injective left R-modules are injective if and only if R is left Noetherian. Products of projective left R-modules are projective if and only if two conditions hold: 1. products of flat left R-modules are flat (which is equivalent to R being right coherent) and 2. all flat left R-modules are projective (in which case R is called left perfect). Concerning the latter assertion, see Chase, "Direct products of modules", and Bass, "Finitistic dimension and a homological generalization of semi-primary rings". $\endgroup$ – Leonid Positselski Dec 7 '09 at 17:25
  • $\begingroup$ Good to know! Is there a typo in 1 (left coherent instead of right) or a typo in en.wikipedia.org/wiki/Coherent_ring ? $\endgroup$ – VA. Dec 7 '09 at 17:42

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 .


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