{"id":1291,"date":"2011-02-28T02:03:18","date_gmt":"2011-02-28T02:03:18","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1291"},"modified":"2011-02-28T02:03:18","modified_gmt":"2011-02-28T02:03:18","slug":"descent-of-locally-free-modules","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1291","title":{"rendered":"Descent of locally free modules"},"content":{"rendered":"

Locally free modules do not satisfy descent for fpqc coverings. I have an example involving a countable “product” of affine curves, which I will upload to the stacks project soon.<\/p>\n

But what about fppf descent? Suppose A —> B is a faithfully flat ring map of finite presentation. Let M be an A-module such that M ⊗_A B is free. Is M a locally free A-module? (By this I mean locally free on the spectrum of A.) It turns out that if A is Noetherian, then the answer is yes. This follows from the results of Bass in his paper on “big” projective modules. But in general I don’t know the answer. If you do know the answer, or have a reference, please email me.<\/p>\n","protected":false},"excerpt":{"rendered":"

Locally free modules do not satisfy descent for fpqc coverings. I have an example involving a countable “product” of affine curves, which I will upload to the stacks project soon. But what about fppf descent? Suppose A —> B is … 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-1291","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\/1291","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=1291"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1291\/revisions"}],"predecessor-version":[{"id":1296,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1291\/revisions\/1296"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1291"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1291"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1291"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}