{"id":2896,"date":"2012-10-26T18:33:53","date_gmt":"2012-10-26T18:33:53","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2896"},"modified":"2012-10-26T21:48:51","modified_gmt":"2012-10-26T21:48:51","slug":"openness-of-versality","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2896","title":{"rendered":"Openness of versality"},"content":{"rendered":"

This post is a followup on this<\/a> series<\/a> of posts<\/a>. Basically, I in some sense forgot the punchline of the whole story and just now it came back to me.<\/p>\n

Let’s consider the case of the Quot functor for example. First let’s consider it in a reasonable level of generality: assume we have a proper morphism X —> S with S Noetherian and a coherent O_X-module F flat over S. Then given a Noetherian S-algebra A and a short exact sequence<\/p>\n

0 —> E —> F_A —> G —> 0<\/p>\n

on X_A = X \u00d7_S Spec(A) with G flat over A the obstructions to deforming this to a thickening A’ of A lie in Ext^1_{O_X}(E, G \u2297 I) and if the obstruction vanishes the set of deformations is principal homogeneous under Ext^0_{O_X}(E, G \u2297 I).<\/p>\n

In order to apply Artin’s method to get openness of versality exactly as in Section 07YV<\/a> we need to prove there exists a perfect complex of A-modules K such that we have<\/p>\n

Ext^1_{O_X}(E, G \u2297 M) = H^2(K \u2297_A M)<\/p>\n

and<\/p>\n

Ext^0_{O_X}(E, G \u2297 M) = H^1(K \u2297_A M)<\/p>\n

functorially in the A-module M and agreeing with boundary maps. What is important here is that you only need to have this for low cohomology groups.<\/p>\n

Guess:<\/strong> There exists a bounded below complex of finite projective A-modules P and functorial isomorphisms Ext^i_{O_X}(E, G \u2297 M) = H^{i + 1}(P \u2297_A M).<\/p>\n

Firstly, if the guess is true, then we get what we want by just taking K to be a stupid<\/em> truncation of P!<\/p>\n

Secondly, if X —> S is projective, then the guess is correct. Namely, in this case you can find a resolution<\/p>\n

…—> E_1 —> E_0 —> E —> 0<\/p>\n

of E where each E_i is a finite direct sum O_X(-n) where n \u226b 0. Then the complex<\/p>\n

P = (f_*Hom(E_0, G) —> f_*Hom(E_1, G) —> … )<\/p>\n

works because each of the flat sheaves Hom(E_i, G) will have only cohomology in degree 0 and hence f_*Hom(E_i, G) is finite locally free on Spec(A). Moreover, the complex will compute the correct cohomology after tensoring with M. (Observation: you do not need E to be flat over A for this argument to work; the key is that G is flat over A.)<\/p>\n

Thirdly, in general, if we can find a perfect complex E’ on X and a map E’ –> E which is a quasi-isomorphism in degrees > -5 (for example), then in order to compute Ext^i(E, G \u2297 M) for i < 4 (or something) we can replace E by E’. This leads us to something like Rf_*Hom(E’, G) which is a perfect complex on Spec(A) by standard arguments. Grothendieck uses this argument to study Ext^0 in EGA III, Cor 7.7.8. Anyhow, this should allow us to handle the case where you have an ample family of invertible sheaves on X (that’s not much better than the projective case of course). [Edit:<\/strong> Jack just emailed that the existence of E’ (in the case of schemes) is in a paper by Lipman and Neeman entitled “QUASI-PERFECT SCHEME-MAPS AND BOUNDEDNESS OF THE TWISTED INVERSE IMAGE FUNCTOR”.]<\/p>\n

Fourthly, it may well be that the guess follows from Jack Hall’s paper “Coherence results for algebraic stacks”. It is obviously very close and it may just be a translation, but I haven’t tried to think about it. I’d like to know if this is so. [Edit:<\/strong> Jack just explained to me that this is true for example if the base is of finite type over Z which we can reduce to I think.]<\/p>\n

Fifthly, if F is not assumed flat over S, then we should probably do something like look at distinguished triangles<\/p>\n

E —> F \u2297^L<\/strong> A —> G —> E[1]<\/p>\n

where now E is an object of D(X_A) but G is a usual O_{X_A}-module (placed in degree 0) flat over A and hopefully the obstructions lie in Ext^1(E, G \u2297 I), etc, and we can try imitating the above. This is just pure speculation; I’ll have to check with my friends to see what would be the correct thing to do. Just leave a comment if you know what to do.<\/p>\n","protected":false},"excerpt":{"rendered":"

This post is a followup on this series of posts. Basically, I in some sense forgot the punchline of the whole story and just now it came back to me. Let’s consider the case of the Quot functor for example. … 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-2896","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\/2896","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=2896"}],"version-history":[{"count":21,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2896\/revisions"}],"predecessor-version":[{"id":2915,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2896\/revisions\/2915"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2896"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2896"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2896"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}