{"id":1622,"date":"2011-06-07T13:09:08","date_gmt":"2011-06-07T13:09:08","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1622"},"modified":"2011-06-07T13:32:33","modified_gmt":"2011-06-07T13:32:33","slug":"slicing-presentations","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1622","title":{"rendered":"Slicing presentations"},"content":{"rendered":"

In the stacks project a Deligne-Mumford stack<\/em> is an algebraic stack X such that there exists a scheme U and a surjective etale morphism U —> X. An algebraic stack X is said to be DM<\/em> if the diagonal \u0394 : X —> X x X is unramified. In fact Theorem Tag 06N3<\/a> says:<\/p>\n

X is DM if and only if X is Deligne-Mumford.<\/p><\/blockquote>\n

An algebraic stack X is said to be quasi-DM<\/em> if the diagonal \u0394 : X —> X x X is locally quasi-finite. The analogue of the theorem above is Theorem Tag 06MF<\/a> which says:<\/p>\n

X is quasi-DM if and only if there exists a scheme U and a surjective, flat, locally finitely presented, and locally quasi-finite morphism U —> X.<\/p><\/blockquote>\n

The proofs of these theorems are completely parallel. Assume X is DM (resp. quasi-DM). We try to construct etale (resp. loc fp + flat + loc quasi-finite) maps from schemes toward X. In both cases the strategy is the following:<\/p>\n

    \n
  1. Pick a smooth morphism U —> X,<\/li>\n
  2. choose a suitable point x of X,<\/li>\n
  3. let F be the fibre of U over x, and<\/li>\n
  4. “slice” U, i.e., find a complete intersection V(f_1, …, f_d) ⊂ U such that f_1, …, f_d form a regular system of parameters (resp. regular sequence) at some point of F<\/li>\n<\/ol>\n

    In both cases the proof shows that, after possibly shrinking U, the morphism V(f_1, …, f_d) —> X is flat, locally finitely presented, and unramified (resp. locally quasi-finite). A bit of care is needed in choosing the point x on X. I decided to use “finite type points”; in both cases one then has to do a bit of work to show that the “fibre F” has desirable properties: in the DM case one need to produce x such that F —> U is unramified and in the quasi-DM case such that F —> U is locally quasi-finite.<\/p>\n

    The reasoning above is completely standard. However, there is a way to deduce the first theorem from the second. I decided against arguing like this in the stacks project as it is perhaps a little nonstandard. Here is the argument. Let X be DM. By the second theorem we can find U —> X which is surjective, flat, locally of finite presentation, and locally quasi-finite. Let H_{d, lci}(U\/X) be the LCI locus in the relative degree d Hilbert stack of U over X (see Section Tag 06CJ<\/a>). Then H_{d, lci}(U\/X) —> X is smooth (this is explained in the proof of Theorem Tag 06DC<\/a>). But of course it is clear that H_{d, lci}(U\/X) —> X has relative dimension 0, hence it is etale. This doesn’t quite finish the proof because H_{d, lci}(U\/X) is (as defined in the stacks project) an algebraic stack and not an algebraic space; but a straightforward argument shows (because X is DM) that the disjoint union for varying d of the open substacks of H_{d, lci}(U\/X) having trivial inertia surjects onto X.<\/p>\n","protected":false},"excerpt":{"rendered":"

    In the stacks project a Deligne-Mumford stack is an algebraic stack X such that there exists a scheme U and a surjective etale morphism U —> X. An algebraic stack X is said to be DM if the diagonal \u0394 … 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-1622","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\/1622","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=1622"}],"version-history":[{"count":14,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1622\/revisions"}],"predecessor-version":[{"id":1636,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1622\/revisions\/1636"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1622"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1622"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1622"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}