{"id":3750,"date":"2014-07-07T13:56:47","date_gmt":"2014-07-07T13:56:47","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3750"},"modified":"2014-07-07T13:56:47","modified_gmt":"2014-07-07T13:56:47","slug":"lemma-of-the-day-30","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3750","title":{"rendered":"Lemma of the day"},"content":{"rendered":"<p> Let (A,m) be a Noetherian local ring. Let I \u2282 J \u2282 A be proper ideals. Assume<\/p>\n<ol>\n<li> A\/J has finite tor dimension over A\/I, and<\/li>\n<li> J is generated by a regular sequence.<\/li>\n<\/ol>\n<p>Then I is generated by a regular sequence and J\/I is generated by a regular sequence. See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/09PW\" title=\"Tag 09PW\">Lemma Tag 09PW<\/a>.<\/p>\n<p>Here is the graph of this lemma<br \/>\n<a href=\"http:\/\/stacks.math.columbia.edu\/tag\/09PW\/graph\/force\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2014\/07\/09PW.png\" alt=\"09PW\" width=\"642\" height=\"554\" class=\"aligncenter size-full wp-image-3751\" srcset=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2014\/07\/09PW.png 642w, https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2014\/07\/09PW-300x258.png 300w, https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2014\/07\/09PW-347x300.png 347w\" sizes=\"auto, (max-width: 642px) 100vw, 642px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let (A,m) be a Noetherian local ring. Let I \u2282 J \u2282 A be proper ideals. Assume A\/J has finite tor dimension over A\/I, and J is generated by a regular sequence. Then I is generated by a regular sequence &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3750\">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-3750","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\/3750","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=3750"}],"version-history":[{"count":3,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3750\/revisions"}],"predecessor-version":[{"id":3754,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3750\/revisions\/3754"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3750"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3750"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3750"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}