{"id":2078,"date":"2011-12-05T01:24:54","date_gmt":"2011-12-05T01:24:54","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2078"},"modified":"2011-12-05T01:24:54","modified_gmt":"2011-12-05T01:24:54","slug":"rf_-not-preserving-quasi-coherence","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2078","title":{"rendered":"Rf_* not preserving quasi-coherence"},"content":{"rendered":"

Here is a simple example that shows that in order to obtain a derived functor Rf_* on unbounded<\/em> complexes with quasi-coherent cohomology sheaves we need some additional hypothesis beyond just requiring f to be quasi-compact and quasi-separated.<\/p>\n

Let k be a field of characteristic p > 0. Let G = Z\/pZ be the cyclic group of order p. Set S = Spec(k[x]) and let X = [S\/G] be the stacky quotient where G acts trivially on S. Consider the morphism f : X —> S. Then Rf_*O_X is a complex with cohomology sheaves isomorphic to O_S for all p >= 0. In fact Rf_*O_X is quasi-isomorphic to \u2295 O_S[-n] where n runs over nonnegative integers.<\/p>\n

Now consider the complex K = \u2295 O_X[m] where m runs over the nonnegative integers. This is an object of D_{QCoh}(X) but it isn’t bounded below. So we have to pay attention if we want to compute Rf_*K. Namely, in D(O_X) the complex K is also K = \u220f O_X[m]. Since cohomology commutes with products, we see that<\/p>\n

Rf_*K = \u220f Rf_*O_X[m] = \u220f (\u2295 O_S[m – n]).<\/p><\/blockquote>\n

In degree 0 we get an infinite product of copies of O_S which isn’t quasi-coherent.<\/p>\n

Conclusion: Rf_* does not map D_{QCoh}(X) into D_{QCoh}(S).<\/p>\n

Of course if f is a quasi-compact and quasi-separated morphism between algebraic spaces, then this kind of thing doesn’t happen.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here is a simple example that shows that in order to obtain a derived functor Rf_* on unbounded complexes with quasi-coherent cohomology sheaves we need some additional hypothesis beyond just requiring f to be quasi-compact and quasi-separated. Let k be … 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-2078","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\/2078","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=2078"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2078\/revisions"}],"predecessor-version":[{"id":2088,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2078\/revisions\/2088"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2078"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2078"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2078"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}