{"id":4346,"date":"2019-02-14T02:11:58","date_gmt":"2019-02-14T02:11:58","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4346"},"modified":"2019-02-18T14:18:29","modified_gmt":"2019-02-18T14:18:29","slug":"coh-proper-henselian-schemes","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4346","title":{"rendered":"Coh proper henselian schemes"},"content":{"rendered":"<p>In this blog post we (partially?) answer one of the questions posed in this <a href=\"https:\/\/mathoverflow.net\/q\/323048\">mathoverflow post<\/a>. Namely, let A be a complete discrete valuation ring with uniformizer t and fraction field K. Let X = P^1_A be the projective line over A. Let (X^h, O^h) be the henselian scheme you get by henselizing along t = 0.<\/p>\n<p><b>Lemma.<\/b> If the characteristic of K is zero, then H^1(X^h, O^h) is not zero.<\/p>\n<p><b>Proof.<\/b> Recall that the underlying topological space of X^h is the projective line over the residue field of A. Consider the standard open covering of this projective line. Then the first Cech cohomology for O^h with respect to this covering is the cokernel of the map A[x]^h x A[1\/x]^h \u2192 A[x, 1\/x]^h. Here the henselizations are taken with respect to the ideal generated by t. Since for H^1 Cech cohomology always injects into cohomology, it suffices to show that this cokernel is nonzero.<\/p>\n<p>What does this mean? Well, by Artin approximation the ring A[x, 1\/x]^h is the set of algebraic elements of the t-adic completion of A[x, 1\/x] as defined in <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4332\">the previous blog post<\/a>. A similar statement holds for A[x]^h and A[1\/x]^h. See <a href=\"https:\/\/stacks.math.columbia.edu\/tag\/0A1W\">Tag 0A1W<\/a> for an explanation. Thus we see immediately that the result of the previous blog post exactly gives the nonvanishing of the cokernel. QED.<\/p>\n<p>What is mildly interesting is that this counter example doesn&#8217;t work if the characteristic of K is p &gt; 0. Moreover, in <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4323\">this blog post<\/a> we proved that one does have theorem B for henselian affine schemes in characteristic p. It could still be true that there is some good theory of henselian schemes and quasi-coherent modules on them in positive characteristic. Let me know if you have one. Thanks!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this blog post we (partially?) answer one of the questions posed in this mathoverflow post. Namely, let A be a complete discrete valuation ring with uniformizer t and fraction field K. Let X = P^1_A be the projective line &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4346\">Continue reading <span class=\"meta-nav\">&rarr;<\/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-4346","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\/4346","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=4346"}],"version-history":[{"count":6,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4346\/revisions"}],"predecessor-version":[{"id":4352,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4346\/revisions\/4352"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4346"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4346"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}