{"id":1766,"date":"2011-08-17T20:45:51","date_gmt":"2011-08-17T20:45:51","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1766"},"modified":"2011-08-17T20:45:51","modified_gmt":"2011-08-17T20:45:51","slug":"countable-rings","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1766","title":{"rendered":"Countable rings"},"content":{"rendered":"

Some types of questions in algebra immediately reduce to the “countable” case. Simple example: Let A be a ring and let φ : A —> A be an automorphism. Then for every finite subset E of A there exists a countable subring A’ ⊂ A containing E such that φ induces an automorphism of A’. The proof is to let A’ be the subring of A generated by φ^n(e) for all e in E and all n ∈ Z<\/strong>.<\/p>\n

Another example of this phenomenon is that any projective module is a direct sum of countably generated (projective) modules (Kaplansky’s theorem).<\/p>\n

The technique also applies to the following problem: Let A ⊂ B be an integral extension of rings with A Noetherian. Let M be an A-module such that M ⊗_A B is flat over B. Problem: Show M is flat over A. (This is equivalent to the direct summand conjecture by a 1 page paper of Ohi.) A key case is to show M ⊗_A B = 0 implies M = 0. Picking suitable families of elements this reduces to the case where B is countably generated over A and M is a countably generated A-module.<\/p>\n

Here are two examples involving algebraic stacks: (1) Suppose X is a quasi-compact algebraic stack with affine diagonal. I claim you can write X as a filtered limit of stacks X’ of the form X’ = [U’\/R’] with U’ and R’ spectra of countable rings. I haven’t written out the details but it seems to me one can do this by just “adding elements” as above. (2) A quasi-coherent module F on a quasi-separated and quasi-compact algebraic stack is a filtered colimit of countably generated quasi-coherent modules.<\/p>\n

I wonder if this type of argument can ever be used to bootstrap? Do some general arguments become easier if you assume all rings\/modules in question are countable? Are there some _useful_ properties that hold for countable rings? Things like “the topology on Spec has a countable basis” aren’t really useful, or are they?<\/p>\n

Before you say “No!” let me just point out that Kaplansky’s theorem is used in the proof of faithfully flat descent for projectivity of modules, so sometimes…<\/p>\n","protected":false},"excerpt":{"rendered":"

Some types of questions in algebra immediately reduce to the “countable” case. Simple example: Let A be a ring and let φ : A —> A be an automorphism. Then for every finite subset E of A there exists a … 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-1766","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\/1766","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=1766"}],"version-history":[{"count":19,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1766\/revisions"}],"predecessor-version":[{"id":1785,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1766\/revisions\/1785"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1766"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1766"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1766"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}