{"id":67,"date":"2010-02-09T17:05:18","date_gmt":"2010-02-09T17:05:18","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=67"},"modified":"2010-02-09T17:05:18","modified_gmt":"2010-02-09T17:05:18","slug":"geometric-quotients-for-finite-groupoids","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=67","title":{"rendered":"Geometric quotients for finite groupoids"},"content":{"rendered":"

In this comment<\/a> David Rydh formulates the conjecture that for a finite affine groupoid (U, R, s, t, c) the spectrum of the ring of invariants may be a geometric quotient for the stack [U\/R]. In fact, the same question came up in a recent conversation with Jarod Alper here in the department.<\/p>\n

I have an idea for generating invariant functions, which sounds so familiar to me that I am sure it is in the literature (let me know if you have a reference), or maybe I have already tried using it in the past. First, recall that if s,t are finite locally free flat so B is finite locally free over A then for any element x of A gives rise to an invariant element y by taking y = Nm_s(t(x)). In words y is the norm of t(x) with respect to the finite locally free ring map s : A —> B. Thus, in the general case where s, t are finite we try to find an element y in A which behaves like the norm of t(x) with respect to s. Maybe a falsifiable version of the conjecture above would be to conjecture the existence of a y in A such that for every prime p of A the value of y in k(p) is a power of the Nm of t(x) restricted to B \\otimes_{s, A} k(p)?<\/p>\n

My idea is to try to do the following. Take a finite free extension phi : A —> B’ and a surjection pi : B —> B’ such that pi o phi = s. (It may be convenient for later arguments to allow only certain types of ring maps A —> B, such as my personal favorite: finite flat relative complete intersections.) Now for any element x of A we can let y = Nm_phi(x’) where x’ in B’ is any element with pi(x’) = t(x). It is clear that y will NOT be R-invariant in general, simply because we have put too little restrictions on B’. But on the other hand, I am pretty confident that the ideal generated by all y of the form Nm_phi(x’) will be R-invariant. Namely, it should just cut out the set of points which are R-equivalent to a zero of the function x.<\/p>\n

However, if A is an Artinian ring, then we can choose B’ so that B’ and B have the same maximal ideals. In this case if A has positive residue characteristics then it is quite easy to show that y^{p^n} for large n is independent of the choice of x’ and presumably is an invariant element of A (I haven’t checked this completely). This could then be the start of a kind of induction argument in the Noetherian case. But in characteristic zero I do not even know how to produce enough invariant functions in the Artinian case.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this comment David Rydh formulates the conjecture that for a finite affine groupoid (U, R, s, t, c) the spectrum of the ring of invariants may be a geometric quotient for the stack [U\/R]. In fact, the same question … 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-67","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\/67","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=67"}],"version-history":[{"count":3,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/67\/revisions"}],"predecessor-version":[{"id":70,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/67\/revisions\/70"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=67"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=67"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=67"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}