{"id":806,"date":"2010-09-05T19:35:27","date_gmt":"2010-09-05T19:35:27","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=806"},"modified":"2010-09-05T19:35:27","modified_gmt":"2010-09-05T19:35:27","slug":"raynaud-gruson","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=806","title":{"rendered":"Raynaud-Gruson"},"content":{"rendered":"<p>As I mentioned earlier I have started to look into Hilbert and Quot schemes\/spaces\/stacks. This led me to think about the existence of a flattening stratification and then I started thinking about some of the results obtained in the paper [RG] by Raynaud and Gruson. For example, as I first mentioned <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=697#comment-1774\">here<\/a>, I think that using Theorem 4.1.2 of their paper would be a good way to prove the existence of flattening stratifications (as formulated in <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=697\">this post<\/a>) in the &#8220;correct&#8221; level of generality.<\/p>\n<p>Thus it now seems to me that the basic &#8220;devissage&#8221; result of [RG] would be a worthwhile addition to the stacks project. Consequently, I have started adding material on geometric fibres in families which is needed for one of the key geometric lemmas in [RG].<\/p>\n<p>I think there is a lot more, besides flattening stratifications, to gain from adding this to the stacks project. For example, there are several places in [RG] with directions for the reader to rewrite several parts of EGA using their methods, and I feel that others will show up as we go along.<\/p>\n<p>I&#8217;d love to hear about any interesting foundational applications of [RG] you know about!<\/p>\n<p>[RG] Raynaud, Michel; Gruson, Laurent<br \/>\nCrit\u00e8res de platitude et de projectivit\u00e9. Techniques de &#8220;platification&#8221; d&#8217;un module. (French)<br \/>\nInvent. Math. 13 (1971), 1&#8211;89.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>As I mentioned earlier I have started to look into Hilbert and Quot schemes\/spaces\/stacks. This led me to think about the existence of a flattening stratification and then I started thinking about some of the results obtained in the paper &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=806\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-806","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/806","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=806"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/806\/revisions"}],"predecessor-version":[{"id":811,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/806\/revisions\/811"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=806"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=806"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=806"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}