{"id":1507,"date":"2011-04-22T18:52:54","date_gmt":"2011-04-22T18:52:54","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1507"},"modified":"2011-04-22T18:52:54","modified_gmt":"2011-04-22T18:52:54","slug":"obstructions","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1507","title":{"rendered":"Obstructions"},"content":{"rendered":"

We continue the discussion<\/a>. Let R be a regular complete local ring with residue field k. Let S = R[[x_1, …, x_n]]. Let I ⊂ R and J ⊂ S be ideals such that IS &sub J and such that A = R\/I —> B = S\/J is a flat ring homomorphism. Consider the map<\/p>\n

(**) I\/m_RI —> J\/m_SJ<\/p>\n

In the previous post we claimed that the cokernel of this map is (J + m_RS)\/(m_SJ + m_RS). To see this choose f’_1, …, f’_r ∈ J whose images f_1, …, f_r in S\/m_RS form a minimal system of generators for the ideal (J + m_RS)\/m_RS which is the ideal cutting out B\/m_AB in S\/m_RS. Think of B as a flat deformation of B\/m_AB over A. Then by the discussion in this post<\/a>, the flatness assumption implies that any relation between f_1, …, f_r in S\/m_RS lifts to a relation in S\/IS. Hence any element h of J ∩ m_RS is an element of IS + m_RJ as desired.<\/p>\n

But more is true. Namely, recall from this post<\/a> that with these choices we obtain an obstruction map<\/p>\n

Ob : Rel\/TrivRel —> S\/(JS + m_RS) \\otimes_k I\/m_RI<\/p>\n

(unfortunately the notation between these two posts isn’t compatible) where Rel is the module of relations between the f_1, …, f_r in S\/m_RS. Now because J ⊂ m_S we can compose Ob with the canonical map S\/(JS + m_R S) —> k to get a reduced map<\/p>\n

Ob_reduced : Rel\/TrivRel —> I\/m_RI.<\/p>\n

At this point an argument along the lines of the argument in the first paragraph shows that (**) is injective if this reduced obstruction map is zero (in fact I think it is equivalent, but I didn’t check this).<\/p>\n

Having arrived at this point we see that it suffices to prove the following (changing back to the notation in the post on deformation theory):<\/p>\n

(***) Given a proper ideal I in R = k[[x_1, …, x_n]], a minimal set of generators f_1, …, f_r for I with module of relations Rel and submodule of trivial relations TrivRel ⊂ Rel. Then any R-module homomorphism Rel\/TrivRel —> R\/I has image contained in m_R\/I.<\/p>\n

It turns out that this result is contained in a paper<\/a> by Vasconcelos and with a little bit more detail on the proof it is Lemma 2 in this paper<\/a> by Rodicio. The key appears to be the technique of Tate<\/a> to find divided power dga resolutions of R\/I combined with a technique for constructing derivations on dgas which is due to Gulliksen. We will return to this in a future post.<\/p>\n","protected":false},"excerpt":{"rendered":"

We continue the discussion. Let R be a regular complete local ring with residue field k. Let S = R[[x_1, …, x_n]]. Let I ⊂ R and J ⊂ S be ideals such that IS &sub J and such 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-1507","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\/1507","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=1507"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1507\/revisions"}],"predecessor-version":[{"id":1512,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1507\/revisions\/1512"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1507"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1507"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1507"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}