{"id":3772,"date":"2014-07-11T11:05:42","date_gmt":"2014-07-11T11:05:42","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3772"},"modified":"2014-07-11T11:05:42","modified_gmt":"2014-07-11T11:05:42","slug":"lemma-of-the-day-31","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3772","title":{"rendered":"Lemma of the day"},"content":{"rendered":"

Let A be a ring and let I be a finitely generated ideal. Let M and N be I-power torsion modules.<\/p>\n

    \n
  1. Hom_{D(A)}(M, N) = Hom_{D(I^\u221e-torsion)}(M, N),<\/li>\n
  2. Ext^1_{D(A)}(M, N) = Ext^1_{D(I^\u221e-torsion)}(M, N),<\/li>\n
  3. Ext^2_{D(I^\u221e-torsion)}(M, N) \u2192 Ext^2_{D(A)}(M, N) is not surjective in general,<\/li>\n
  4. (0A6N<\/a>) is not an equivalence in general.<\/li>\n<\/ol>\n

    See Lemma Tag 0592<\/a>.<\/p>\n

    Discussion: Let A be a ring and let I be an ideal. The derived category of complexes of A-modules with I-power torsion cohomology modules is not the same as the derived category of the category of I-power torsion modules in general, even if I is finitely generated. However, if the ring is Noetherian then it is true, see Lemma Tag 0955<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Let A be a ring and let I be a finitely generated ideal. Let M and N be I-power torsion modules. Hom_{D(A)}(M, N) = Hom_{D(I^\u221e-torsion)}(M, N), Ext^1_{D(A)}(M, N) = Ext^1_{D(I^\u221e-torsion)}(M, N), Ext^2_{D(I^\u221e-torsion)}(M, N) \u2192 Ext^2_{D(A)}(M, N) is not surjective in … 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-3772","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\/3772","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=3772"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3772\/revisions"}],"predecessor-version":[{"id":3774,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3772\/revisions\/3774"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3772"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3772"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3772"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}