{"id":1046,"date":"2010-12-03T03:54:14","date_gmt":"2010-12-03T03:54:14","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1046"},"modified":"2010-12-03T03:54:14","modified_gmt":"2010-12-03T03:54:14","slug":"completion","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1046","title":{"rendered":"Completion"},"content":{"rendered":"<p>Let R be a ring and I an ideal. For an R-module M we define the <em>completion<\/em> M^* of M to be the limit of the modules M\/I^nM. We say M is <em>complete<\/em> if the natural map M &#8212;> M^* is an isomorphism.<\/p>\n<p>Then you ask yourself: Is the completion M^* complete? The answer is no in general, and I just added an example to the chapter on examples in the stacks project.<\/p>\n<p>But&#8230; it turns out that if I is a finitely generated ideal in R then M^* is always complete. See the section on completion in the algebra chapter. I&#8217;ve found this also on the web in some places&#8230; and apparently it occurs first (?) in a paper by Matlis (1978). Any earlier references anybody?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let R be a ring and I an ideal. For an R-module M we define the completion M^* of M to be the limit of the modules M\/I^nM. We say M is complete if the natural map M &#8212;> M^* &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1046\">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-1046","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\/1046","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=1046"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1046\/revisions"}],"predecessor-version":[{"id":1048,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1046\/revisions\/1048"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1046"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1046"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1046"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}