{"id":2206,"date":"2012-03-03T14:03:46","date_gmt":"2012-03-03T14:03:46","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2206"},"modified":"2012-03-03T14:03:46","modified_gmt":"2012-03-03T14:03:46","slug":"rlim","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2206","title":{"rendered":"Rlim"},"content":{"rendered":"

Let N<\/strong> be the natural numbers. Think of N<\/strong> as a category with a unique morphisms n —> m whenever m \u2265 n and endow it with the chaotic topology to get a site. Then a sheaf of abelian groups on N<\/strong> is an inverse system (M_e) and H^0 corresponds to the limit lim M_e of the system. The higher cohomology groups H^i correspond to the right dervided functors R^i lim. A good exercise everybody should do is prove directly that R^i lim is zero when i > 1.<\/p>\n

Consider D(Ab N). <\/strong>This is the derived category of the abelian category of inverse systems. An object of D(Ab N<\/strong>) is a complex of inverse systems (and not an inverse system of complexes — we will get back to this). The functor R\u0393(-) corresponds to a functor Rlim. Given a bounded below complex of inverse systems where all the transition maps are surjective, then Rlim is computed by simply taking the lim in each degree.<\/p>\n

What if we have an inverse system with values in D(Ab)? In other words, we have objects K_e in D(Ab) and transition maps K_{e + 1} —> K_e in D(Ab). By choosing suitable complexes K^*_e representing each K_e we can assume that there are actual maps of complexes K^*_{e + 1} —> K^*_e representing the transition maps in D(Ab). Thus we obtain an object K of D(Ab N<\/strong>) lifting<\/em> the inverse system (K_e). Having made this choice, we can compute Rlim K.<\/p>\n

During the lecture on crystalline cohomology yesterday morning I asked the following question: Is Rlim K independent of choices? The reason for this question is that there are a priori many isomorphism classes of objects K in D(Ab N<\/strong>) which give rise to the inverse system (K_e) in D(Ab). It turns out that Rlim K is somewhat independent of choices, as Bhargav explained to me after the lecture. Namely, you can identify Rlim K with the homotopy limit, i.e., Rlim K sits in a distinguished triangle<\/p>\n

Rlim K —–> \u03a0 K_e —–> \u03a0 K_e ——> Rlim K[1]<\/p>\n

in D(Ab) where the second map is given by 1 – transition maps. And this homotopy limit depends, up to non-unique isomorphism, only on the inverse system in D(Ab).<\/p>\n

One of the things I enjoy about derived categories is how things don’t work, but how in the end it sort of works anyway. The above is a nice illustration of this phenomenon.<\/p>\n","protected":false},"excerpt":{"rendered":"

Let N be the natural numbers. Think of N as a category with a unique morphisms n —> m whenever m \u2265 n and endow it with the chaotic topology to get a site. Then a sheaf of abelian groups … 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-2206","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\/2206","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=2206"}],"version-history":[{"count":8,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2206\/revisions"}],"predecessor-version":[{"id":2214,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2206\/revisions\/2214"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2206"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2206"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}