Timeline for Sums of injective modules, products of projective modules?
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Jan 31, 2017 at 21:55 | answer | added | Leonid Positselski | timeline score: 2 | |
| Dec 24, 2009 at 5:25 | comment | added | Shizhuo Zhang | Yes, it is true. | |
| Dec 23, 2009 at 18:18 | comment | added | Leonid Positselski | @Shizhuo: you mean, the inverse image for quasi-coherent sheaves on a Noetherian scheme, I guess. And it is also true that the class of injectives in a locally Noetherian Grothendieck category (e.g., the category of quasi-coherent sheaves on a Noetherian scheme) is closed under infinite direct sums. | |
| Dec 23, 2009 at 15:12 | comment | added | Shizhuo Zhang | More general statement for VA's question. If C is a locally noetherian category, then any injective in C is direct sum of indecomposable injectives in C. This can be used to prove that inverse image functor of open immersion preserves injectives | |
| Dec 7, 2009 at 18:05 | comment | added | Leonid Positselski | The typo is in Wikipedia. The argument is very simple: over a right coherent ring, any short exact sequence of right modules is a direct limit of short exact sequences of finitely presented right modules; while the tensor product with a finitely presented right module commutes with any direct products of left modules. | |
| Dec 7, 2009 at 17:42 | comment | added | VA. | Good to know! Is there a typo in 1 (left coherent instead of right) or a typo in en.wikipedia.org/wiki/Coherent_ring ? | |
| Dec 7, 2009 at 17:25 | comment | added | Leonid Positselski | 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". | |
| Dec 7, 2009 at 16:32 | comment | added | VA. | 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? | |
| Dec 1, 2009 at 16:38 | comment | added | Leonid Positselski | 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. | |
| Dec 1, 2009 at 15:17 | comment | added | Mark Hovey | 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. | |
| Nov 30, 2009 at 14:47 | history | asked | Leonid Positselski | CC BY-SA 2.5 |