{"id":1134,"date":"2017-02-23T18:52:33","date_gmt":"2017-02-23T17:52:33","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1134"},"modified":"2022-11-19T15:22:18","modified_gmt":"2022-11-19T14:22:18","slug":"ordinal-heights-of-noetherian-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1134","title":{"rendered":"Ordinal heights of Noetherian spaces"},"content":{"rendered":"

Update.<\/strong>\u00a0 These problems (and a few others) were solved, by Bastien Laboureix and me: see this arXiv paper<\/a>. \u00a0Bastien is a gifted young man, whom I have had the pleasure to have had as a student in various courses at ENS Paris-Saclay, and who spent his M2 (masters, second year) research internship in the Spring of 2021 on those questions with me. \u00a0Bastien is also a coauthor of an award-winning paper<\/a> on a new, intriguing model of computation, with Yoan G\u00e9ran, Corto Mascle, and Valentin Richard; they discovered it after spilling a cup of coffee on a computer keyboard, and asking themselves what kinds of languages one could produce with a partially dysfunctional keyboard. \u00a0As of late 2021, Bastien is a PhD student with Isabelle Debled-Rennesson and Eric Domenjoud at LORIA, in Nancy, France, and works in a third area of research: discrete geometry. \u00a0He had already coauthored a paper<\/a> with Eric Domenjoud and Laurent Vuillon, in that area, while an L3 (license, 3rd year) research intern in Nancy.<\/p>\n

The only remaining problems is to determine the stature (and dimension) of finite trees, and a few other Noetherian spaces of interest. \u00a0See the followup<\/a>\u00a0to this post<\/p>\n

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<\/p>\n

The ordinal height (or length, or rank) of a well-founded poset Z<\/em> is the supremum of the ordinal lengths of its chains.\u00a0 This can be defined explicitly by first defining the rank of an element z<\/em> in Z<\/em> by well-founded induction by rk z<\/em> = sup {rk y<\/em>+1 | y<\/em><z<\/em>} (a suprema of a family of ordinals), and defining the rank of the whole space Z<\/em> by rk Z<\/em> = sup {rk z<\/em>+1 | z<\/em> in Z<\/em>}.<\/p>\n

A Noetherian space X<\/em> is the same thing as a topological space whose lattice of closed subsets\u00a0H<\/strong>X<\/em> is well-founded.\u00a0 Hence rk\u00a0H<\/strong>X<\/em> makes sense.<\/p>\n

Also, the sobrification S<\/strong>X<\/em> of X<\/em> is well-founded, so rk S<\/strong>X<\/em> makes sense.\u00a0 Indeed S<\/strong>X<\/em> is a sublattice of H<\/strong>X<\/em>, consisting of irreducible closed subsets.<\/p>\n

There is a list of standard constructions of Noetherian spaces in the book<\/a> [1]: finite products, finite coproducts, words with the subword topology, terms with the tree topology, etc.\u00a0 The questions I would like to solve is:<\/p>\n

Given a construction X<\/em>=C(X1<\/sub><\/em>,…,Xn<\/sub><\/em>) of a Noetherian space from Noetherian spaces X1<\/sub><\/em>,…,Xn<\/sub><\/em>, can you express rk X<\/em> as a function of rk X1<\/sub>,…,rk Xn<\/sub><\/em>?<\/p>\n

Can you express rk S<\/strong>X<\/em> as a function of rk S<\/strong>X1<\/sub>,…,rk S<\/strong>Xn<\/sub><\/em>?<\/p>\n

Can you express rk H<\/strong>X<\/em> as a function of rk H<\/strong>X1<\/sub>,…,rk H<\/strong>Xn<\/sub><\/em>?<\/p><\/blockquote>\n

The purposes of those questions are the following:<\/p>\n