{"id":3775,"date":"2014-07-12T21:00:05","date_gmt":"2014-07-12T21:00:05","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3775"},"modified":"2014-07-12T21:00:05","modified_gmt":"2014-07-12T21:00:05","slug":"lemma-of-the-day-32","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3775","title":{"rendered":"Lemma of the day"},"content":{"rendered":"<p>Let (A,I) be a henselian pair with A Noetherian. Let A^* be the I-adic completion of A. Assume at least one of the following conditions holds<\/p>\n<ol>\n<li> A \u2192 A^* is a regular ring map,<\/li>\n<li> A is a Noetherian G-ring, or<\/li>\n<li> (A,I) is the henselization (More on Algebra, Lemma 15.7.10) of a pair (B,J) where B is a Noetherian G-ring.<\/li>\n<\/ol>\n<p>Given f_1, &#8230;, f_m \u2208 A[x_1, &#8230;, x_n] and a_1, &#8230;, a_n \u2208 A^* such that f_j(a_1, &#8230;, a_n) = 0 for j = 1, &#8230;, m, for every N \u2265 1 there exist b_1, &#8230;, b_n \u2208 A such that a_i \u2212 b_i \u2208 I^N and such that f_j(b_1, &#8230;, b_n) = 0 for j = 1, &#8230;, m. See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/0AH5\">Lemma Tag 0AH5<\/a>.<\/p>\n<p>Slogan: Approximation for henselian pairs.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let (A,I) be a henselian pair with A Noetherian. Let A^* be the I-adic completion of A. Assume at least one of the following conditions holds A \u2192 A^* is a regular ring map, A is a Noetherian G-ring, or &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3775\">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-3775","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\/3775","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=3775"}],"version-history":[{"count":1,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3775\/revisions"}],"predecessor-version":[{"id":3776,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3775\/revisions\/3776"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3775"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3775"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3775"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}