Sums of injective modules, products of projective modules? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:19:47Z http://mathoverflow.net/feeds/question/7279 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7279/sums-of-injective-modules-products-of-projective-modules Sums of injective modules, products of projective modules? Leonid Positselski 2009-11-30T14:47:52Z 2011-12-15T14:39:29Z <ol> <li><p>Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?</p></li> <li><p>Analogously, under what assumptions on R does a countable product of projective left R-modules necessarily have a finite projective dimension?</p></li> </ol> <p>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. </p> <p>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.</p> <p>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.</p> <p>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.</p> <p>Any other sufficient conditions?</p>