{"id":3345,"date":"2013-07-09T15:32:51","date_gmt":"2013-07-09T15:32:51","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3345"},"modified":"2013-07-09T15:32:51","modified_gmt":"2013-07-09T15:32:51","slug":"lemma-of-the-day-8","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3345","title":{"rendered":"Lemma of the day"},"content":{"rendered":"

Let A —> B be a ring map. Assume<\/p>\n

    \n
  1. A ⊂ B is an extension of domains,<\/li>\n
  2. A is Noetherian,<\/li>\n
  3. A —> B is of finite type, and<\/li>\n
  4. the extension f.f.(A) ⊂ f.f.(B) is finite.<\/li>\n<\/ol>\n

    Let p ⊂ A be a prime such that dim(Ap<\/sub>) = 1. Then there are at most finitely many primes of B lying over p. See Tag 02MA<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Let A —> B be a ring map. Assume A ⊂ B is an extension of domains, A is Noetherian, A —> B is of finite type, and the extension f.f.(A) ⊂ f.f.(B) is finite. Let p ⊂ A be … Continue reading →<\/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-3345","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\/3345","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=3345"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3345\/revisions"}],"predecessor-version":[{"id":3347,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3345\/revisions\/3347"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3345"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3345"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3345"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}