User henning krause - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:51:07Z http://mathoverflow.net/feeds/user/15394 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66321/a-compactness-property-of-posets A compactness property of posets Henning Krause 2011-05-28T22:40:14Z 2011-05-29T00:19:02Z <p>Consider a poset \$P\$ and suppose that every finite subset admits a supremum. Call an ideal \$I\$ of \$P\$ <em>minimal infinite</em> if it is infinite and every ideal properly contained in \$I\$ is finite. I am interested in the following "compactness property": Every infinite ideal of \$P\$ contains a minimal infinite ideal.</p> <p>Question: Has this property studied before? Are there posets arising in "mathematical nature" having this property?</p> http://mathoverflow.net/questions/14992/why-do-people-forget-verdier-abelianization-functorlooking-for-application/66110#66110 Answer by Henning Krause for Why do people "forget" Verdier abelianization functor?(Looking for application) Henning Krause 2011-05-26T21:16:42Z 2011-05-26T21:16:42Z <p>There is a proof of Brown representability using the abelianisation. One can identify the abelianisation with the category of coherent functors (in the sense of M. Auslander). It is an amusing fact that Auslander and Brown were both colleagues at Brandeis in the early 1960s, and most likely not aware about the close relationship of their work. </p> http://mathoverflow.net/questions/66321/a-compactness-property-of-posets/66326#66326 Comment by Henning Krause Henning Krause 2011-05-29T22:02:32Z 2011-05-29T22:02:32Z Many thanks for this answer. I like in particular the 3rd example, because it is in spirit related to the example I have in mind: A recent result of C. M. Ringel [Minimal infinite submodule-closed subcategories, available from <a href="http://www.math.uni-bielefeld.de/sfb701/preprints/]" rel="nofollow">math.uni-bielefeld.de/sfb701/preprints/]</a> says that a specific poset attached to the representations of a finite dimensional algebra has this property. http://mathoverflow.net/questions/14992/why-do-people-forget-verdier-abelianization-functorlooking-for-application/66110#66110 Comment by Henning Krause Henning Krause 2011-05-26T22:06:04Z 2011-05-26T22:06:04Z The reference is: H. Krause, A Brown representability theorem via coherent functors, Topology 41 (2002), 853–861.