# Sums of injective modules, products of projective modules?

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?

• 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. Commented Dec 1, 2009 at 15:17
• 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. Commented Dec 1, 2009 at 16:38
• 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?
– VA.
Commented Dec 7, 2009 at 16:32
• 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". Commented Dec 7, 2009 at 17:25
• Good to know! Is there a typo in 1 (left coherent instead of right) or a typo in en.wikipedia.org/wiki/Coherent_ring ?
– VA.
Commented Dec 7, 2009 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 .