{"id":1495,"date":"2011-04-22T18:27:12","date_gmt":"2011-04-22T18:27:12","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1495"},"modified":"2011-04-22T18:27:12","modified_gmt":"2011-04-22T18:27:12","slug":"avramovs-theorem","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1495","title":{"rendered":"Avramov’s theorem"},"content":{"rendered":"

Let A —> B be a flat local homomorphism of Noetherian local rings. By the local criterion for flatness this also implies that the map on completions A* —> B* is flat. Hence, in order to prove Avramov’s theorem that I mentioned here<\/a> it suffices to prove it for a flat map of Noetherian complete local rings.<\/p>\n

Let A be a complete local Noetherian ring. Write, using the Cohen structure theorem, A = S\/I where S is a regular complete local ring. The complete intersection defect<\/em> of A is the nonnegative integer<\/p>\n

cid(A) = dim_k(I\/mI) – dim(S) + dim(A)<\/p>\n

where k = S\/m is the residue field of S. Note that dim_k(I\/mI) is the minimal number of generators of the ideal I. Since S is regular we see that A is a complete intersection if and only if cid(A) = 0.<\/p>\n

Next, let A —> B be a flat local homomorphism of complete local Noetherian rings. Avramov proved, among other things, that<\/p>\n

(*) cid(B) = cid(A) + cid(B\/m_AB)<\/p>\n

in this situation. It is clear that this proves the result we mentioned in the previous post.<\/p>\n

What does (*) mean in more elementary terms? For simplicity, let us assume that the residue fields of A and B are identified by the map A —> B. Write A = R\/I. Set S = R[[x_1, …, x_n]] for some large n and choose a surjection S –> B (here we use that the residue fields are equal). Set J \u2282 S equal to the kernel of S —> B so that B = S\/J. Consider the induced map<\/p>\n

(**) I\/m_RI —> J\/m_SJ<\/p>\n

The equality (*) is equivalent to\u00a0 the injectivity of (**). Namely, flatness of A –> B gives dim(B) = dim(A) + dim(B\/m_AB). By construction dim(S) = dim(R) + dim(k[[x_1, …, x_n]]). Finally, the map J\/m_SJ —> (J + m_RS)\/(m_SJ + m_RS) is surjective with kernel equal to the image of (**). (Proof omitted, but see next post.)<\/p>\n

OK, now why is (**) injective? I claim that this is a question about the obstruction space for the deformation theory of the algebra B\/m_AB over k. I will discuss this in the next post.<\/p>\n","protected":false},"excerpt":{"rendered":"

Let A —> B be a flat local homomorphism of Noetherian local rings. By the local criterion for flatness this also implies that the map on completions A* —> B* is flat. Hence, in order to prove Avramov’s theorem that … 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-1495","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\/1495","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=1495"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1495\/revisions"}],"predecessor-version":[{"id":1505,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1495\/revisions\/1505"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1495"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1495"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1495"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}