{"id":1367,"date":"2011-04-01T23:55:10","date_gmt":"2011-04-01T23:55:10","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1367"},"modified":"2011-04-01T23:55:10","modified_gmt":"2011-04-01T23:55:10","slug":"differentials","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1367","title":{"rendered":"Differentials"},"content":{"rendered":"

The sheaf of differentials \u03a9_{X\/S} of one scheme X over another scheme S is the target of the universal O_S-derivation d_{X\/S} : O_X —> \u03a9_{X\/S}. I remember being surprised to learn that people habitually define this sheaf using the conormal sheaf C_{X\/Xx_SX} of the diagonal morphism of X over S[1]<\/sup><\/a>.<\/p>\n

Why is it not the “right thing” to do? The reason is that both the conormal sheaf and the sheaf of differentials have a natural functoriality, and that the identification of C_{X\/Xx_SX} with Ω_{X\/S} is not compatible with this! Namely, consider the morphism that flips the factors on Xx_SX. This should clearly act by -1 on the conormal sheaf C_{X\/Xx_SX} and by +1 on Ω_{X\/S}. So there you go!<\/p>\n

When X —> S is a morphism of algebraic spaces, then the diagonal morphism isn’t an immersion in general so the conormal sheaf is harder to define. In this case defining \u03a9_{X\/S} as the target of the universal O_S-derivation d_{X\/S} : O_X —> \u03a9_{X\/S} on the small etale site of X works fine, see Tag 04CR<\/a>.<\/p>\n

Finally, suppose that X —> S is a morphism of algebraic stacks. We have yet to choose (in the stacks project) which site to use to define quasi-coherent sheaves on X. But in order to study differentials the only reasonable choice seems to be the lisse-etale site X_{lisse, etale}. Again there is a universal O_{S_{lisse, etale}}-derivation d : O_{X_{lisse, etale}} —> Ω. Now, (I think) Ω is not a quasi-coherent O_{X_{lisse, etale}}-module, and it is not what authors on algebraic stacks define as Ω_{X\/S}, but for some purposes it might be the right thing to look at (e.g., deformation theory?).<\/p>\n

Footnote 1:<\/a> Yes, currently the stacks project also introduces sheaves of differentials for morphisms of schemes using this method. The first result is then that d_{X\/S} is a universal derivation, see Lemma Tag 01UR<\/a>. Having proven this, maps involving Ω_{X\/S} are defined using the universal property.<\/p>\n","protected":false},"excerpt":{"rendered":"

The sheaf of differentials \u03a9_{X\/S} of one scheme X over another scheme S is the target of the universal O_S-derivation d_{X\/S} : O_X —> \u03a9_{X\/S}. I remember being surprised to learn that people habitually define this sheaf using the conormal … 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-1367","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\/1367","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=1367"}],"version-history":[{"count":20,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1367\/revisions"}],"predecessor-version":[{"id":1387,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1367\/revisions\/1387"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}