{"id":365,"date":"2010-05-05T17:54:27","date_gmt":"2010-05-05T17:54:27","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=365"},"modified":"2010-05-05T17:54:27","modified_gmt":"2010-05-05T17:54:27","slug":"affines-over-algebraic-spaces","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=365","title":{"rendered":"Affines over algebraic spaces"},"content":{"rendered":"

Suppose that f : Y —> X is a morphism of schemes with f locally of finite type and Y affine. Then there exists an immersion Y —> A^n_X of Y into affine n-space over X. See the slightly more general Lemma Tag 04II<\/a>.<\/p>\n

Now suppose that f : Y —> X is a morphism of algebraic spaces with f locally of finite type and Y an affine scheme. Then it is not true in general that we can find an immersion of Y into affine n-space over X.<\/p>\n

A first (nasty) counter example is Y = Spec(k) and X = [A^1_k\/Z] where k is a field of caracteristic zero and Z acts on A^1_k by translation (n, t) —> t + n. Namely, for any morphism Y —> A^n_X over X we can pullback to the covering A^1_k of X and we get an infinite disjoint union of A^1_k’s mapping into A^{n + 1}_k which is not an immersion.<\/p>\n

A second counter example is Y = A^1_k —> X = A^1_k\/R with R = {(t, t)} \\coprod {(t, -t), t not 0}. Namely, in this case the morphism Y —> A^n_X would be given by some regular functions f_1, …, f_n on Y and hence the fibre product of Y with the covering A^{n + 1}_k —> A^n_X would be the scheme<\/p>\n

{(f_1(t), …, f_n(t), t)} \\coprod {(f_1(t), …, f_n(t), -t), t not 0}<\/p>\n

with obvious morphism to A^{n + 1} which is not an immersion. Note that this gives a counter example with X quasi-separated.<\/p>\n

I think the statement does hold if X is locally separated, but I haven’t written out the details. Maybe it is somehow equivalent to X being locally separated?<\/p>\n

Perhaps the correct weakening of the lemma that holds in general is that given Y —> X with Y affine and f locally of finite type, there exists a morphism Y —> A^n_X which is “etale locally on X and then Zariski locally on Y” an immersion? (This does not seem to be a very useful statement however… although you never know.)<\/p>\n","protected":false},"excerpt":{"rendered":"

Suppose that f : Y —> X is a morphism of schemes with f locally of finite type and Y affine. Then there exists an immersion Y —> A^n_X of Y into affine n-space over X. See the slightly more … 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-365","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\/365","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=365"}],"version-history":[{"count":7,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/365\/revisions"}],"predecessor-version":[{"id":372,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/365\/revisions\/372"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=365"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=365"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}