{"id":3104,"date":"2013-02-20T15:41:36","date_gmt":"2013-02-20T15:41:36","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3104"},"modified":"2013-02-20T15:41:36","modified_gmt":"2013-02-20T15:41:36","slug":"a-base-change","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3104","title":{"rendered":"A base change"},"content":{"rendered":"

Let f : X —> B be a morphism of schemes. Suppose that for every open U of X we are given a category C<\/i>U<\/sub> whose opposite is a subcategory of the category of surjections A —> OU<\/sub> of sheaves of f-1<\/sup>OB<\/sub>-algebras. Moreover, assume that these categories fit together to give a stack C<\/i> over XZar<\/sub> with the usual notion of restriction of sheaves.<\/p>\n

Since the purpose of this discussion is to study deformation theory, it make sense to assume the stalks of A are local rings, which means exactly that the localization of A as in the previous blog post doesn’t do anything. I will assume this from now on.<\/p>\n

Example:<\/strong> Assume f locally of finite type. Given U let C<\/i>U<\/sub> be the full subcategory of surjections i-1<\/sup>OT<\/sub> —> OU<\/sub> where i : U —> T is a closed immersion of U into a scheme T smooth over B. As maps we can take those maps that come from morphisms between smooth schemes over B. This does not form a stack over XZar<\/sub> but we can stackify.<\/p>\n

Now suppose we have a third scheme S and a morphism of schemes g : S —> B. Then I claim there is a natural stack C<\/i>S<\/sub> over (XS<\/sub>)Zar<\/sub> which can be called the base change of C<\/i>. I will construct this by saying what the objects and morphisms look like locally on XS<\/sub> and you’ll have to stackify to get the real thing.<\/p>\n

OK, suppose that V is an open of XS<\/sub> which maps into the open U of X. Denote p : V —> S and q : V —> U the projections and h : V —> B the structure morphism. Let A —> OU<\/sub> be an object of C<\/i>U<\/sub>. Then the map<\/p>\n

p-1<\/sup>OS<\/sub> ⊗h-1<\/sup>OB<\/sub><\/sub> q-1<\/sup>A —-> OV<\/sub><\/p><\/blockquote>\n

is a surjection (see previous post) and we can consider its localization A’ —> OV<\/sub> (as in previous post). This will be what our objects look like locally. Moreover, morphisms are maps which are locally the pullback of maps in C<\/i>.<\/p>\n

Here is how we can use this: The stack C<\/i>S<\/sub> is naturally a ringed site with topology inherited from the Zariski topology on XS<\/sub>. Moreover, in the example above the rings A are all “smooth” over B thus the rings A’ in C<\/i>S<\/sub> are all “smooth” over S. I think we can think of XS<\/sub> as a closed subspace of C<\/i>S<\/sub> and use this to compute obstruction groups for deformations of modules, etc. I’ll come back to this later (and if not then it didn’t work).<\/p>\n","protected":false},"excerpt":{"rendered":"

Let f : X —> B be a morphism of schemes. Suppose that for every open U of X we are given a category CU whose opposite is a subcategory of the category of surjections A —> OU of sheaves … 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-3104","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\/3104","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=3104"}],"version-history":[{"count":14,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3104\/revisions"}],"predecessor-version":[{"id":3118,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3104\/revisions\/3118"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3104"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3104"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}