{"id":3760,"date":"2014-07-09T10:50:09","date_gmt":"2014-07-09T10:50:09","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3760"},"modified":"2014-07-09T10:50:56","modified_gmt":"2014-07-09T10:50:56","slug":"3760","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3760","title":{"rendered":"Lemma of the day"},"content":{"rendered":"<p>Let <em>F<\/em> be a predeformation category which has a versal formal object. Then<\/p>\n<ol>\n<li> <em>F<\/em> has a minimal versal formal object,<\/li>\n<li> minimal versal objects are unique up to isomorphism, and<\/li>\n<li> any versal object is the pushforward of a minimal versal object along a power series ring extension.<\/li>\n<\/ol>\n<p>See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/06T5\" title=\"Tag 06T5\">Lemma Tag 06T5<\/a>.<\/p>\n<p>What is fun about this lemma is that it produces a minimal versal object (as defined in <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/06T4\" title=\"Tag 06T4\">Definition Tag 06T4<\/a>) from a versal one without assuming Schlessinger&#8217;s axioms. If Schlessinger&#8217;s axioms are satisfied and one is in the classical case (see <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/06GC\" title=\"Tag 06GC\">Definition Tag 06GC<\/a>), then a minimal versal formal object is a versal formal object defined over a ring with minimal tangent space. This is discussed in <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/06IL\" title=\"Tag 06IL\">Section Tag 06IL<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let F be a predeformation category which has a versal formal object. Then F has a minimal versal formal object, minimal versal objects are unique up to isomorphism, and any versal object is the pushforward of a minimal versal object &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3760\">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-3760","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\/3760","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=3760"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3760\/revisions"}],"predecessor-version":[{"id":3765,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3760\/revisions\/3765"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3760"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3760"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3760"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}