{"id":1490,"date":"2011-04-22T14:14:49","date_gmt":"2011-04-22T14:14:49","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1490"},"modified":"2011-04-22T14:14:49","modified_gmt":"2011-04-22T14:14:49","slug":"complete-intersections","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1490","title":{"rendered":"Complete intersections"},"content":{"rendered":"

Let us say that a Noetherian local ring is a strong complete intersection<\/em> if it is of the form S\/(f_1, …, f_r) where S is a regular local ring and f_1, …, f_r is a regular sequence. It turns out that if R = S\/I = S’\/I’ where S, S’ are regular local rings, then I is generated by a regular sequence if and only if I’ is (this is not a triviality!). But, as there exist Noetherian local rings which are not the quotient of any regular local rings, this definition of a strong complete intersection does not make a whole lot of sense. Note that it is clear that if R is a strong complete intersection, then so is R_p for any prime ideal p of R.<\/p>\n

The Cohen structure theorem tells us we can write the completion of any Noetherian local ring as the quotient of a regular local ring. Thus we say a Noetherian local ring is a complete intersection<\/em> if its completion is a strong complete intersection. By the Cohen structure theorem we can write the completion as a quotient of a regular local ring, so this definition makes sense.<\/p>\n

The problem: Why is the localization of a complete intersection at a prime a complete intersection?<\/p>\n

The solution to this conundrum comes from a theorem of Avramov: If R —> R’ is a flat local homomorphism of Noetherian local rings, and if R’ is a complete intersection, then R is a complete intersection. How do we use this? Suppose that R is a complete intersection and p a prime ideal of R. Let R’ be the completion of R. As R —> R’ is faithfully flat, we can find a prime p’ of R’ lying over p. By assumption R’ is a strong complete intersection, hence R’_{p’} is a strong complete intersection. Since the local ring map R_p —> R’_{p’} is flat, Avramov’s theorem kicks in and we see that R_p is a complete intersection!<\/p>\n

Cool, no?<\/p>\n","protected":false},"excerpt":{"rendered":"

Let us say that a Noetherian local ring is a strong complete intersection if it is of the form S\/(f_1, …, f_r) where S is a regular local ring and f_1, …, f_r is a regular sequence. It turns out … 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-1490","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\/1490","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=1490"}],"version-history":[{"count":4,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1490\/revisions"}],"predecessor-version":[{"id":1494,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1490\/revisions\/1494"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1490"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1490"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1490"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}