{"id":573,"date":"2010-06-29T20:35:34","date_gmt":"2010-06-29T20:35:34","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=573"},"modified":"2010-06-29T20:35:34","modified_gmt":"2010-06-29T20:35:34","slug":"the-stack-of-spaces","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=573","title":{"rendered":"The stack of spaces"},"content":{"rendered":"<p>Consider the fibred category p : <em>Spaces<\/em> &#8212;&gt; <em>(Sch)<\/em> where an object of <em>Spaces<\/em> over the scheme U is an algebraic space X over U. A morphism (f, g) : X\/U \\to Y\/V is given by morphisms f : X &#8212;&gt; Y and g : U &#8212;&gt; V fitting into an obvious commutative diagram.<\/p>\n<p>Theorem: This is a stack over (Sch)_{fppf}.<\/p>\n<p>In essence the thing you have to prove here is that any descent data for spaces relative to an fppf covering of a scheme is effective. This follows immediately from the results discussed in <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=511\">this post<\/a>, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04SK\">Lemma Tag 04SK<\/a>. You can find a detailed discussion in the chapter Examples of Stacks of the stacks project (in the stacks project we have only formulated this exact statement for the full subcategory of pairs X\/U whose structure morphism X &#8212;> U is of finite type; this is due to our insistence to be honest about set theoretical issues).<\/p>\n<p>Note how absurdly general this is! There are no assumptions on the morphisms X &#8212;> U at all. Now we can use this to show that suitable full subcategories of <em>Spaces<\/em> form stacks. For example, if we want to construct the stack parametrizing flat families of d-dimensional proper algebraic spaces, all we have to do is show that given an fppf covering {U_i &#8212;> U} of schemes and an algebraic space X &#8212;> U over U such that for each i the base change U_i \\times_U X &#8212;> U_i is flat, proper with d-dimensional fibres, then also the morphism X &#8212;> U is flat, proper and has d-dimensional fibres. This is peanuts (compared to what goes into the theorem above).<\/p>\n<p>Of course, to show that (under additional hypotheses on the families) we sometimes obtain an <em>algebraic<\/em> stack is quite a bit more work! For example you likely will have to add the hypothesis that X &#8212;> U is locally of finite presentation, which I intentionally omitted above, to make this work.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the fibred category p : Spaces &#8212;&gt; (Sch) where an object of Spaces over the scheme U is an algebraic space X over U. A morphism (f, g) : X\/U \\to Y\/V is given by morphisms f : X &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=573\">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-573","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\/573","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=573"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/573\/revisions"}],"predecessor-version":[{"id":777,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/573\/revisions\/777"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=573"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=573"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}