is the preorder of locally closed immersions complete? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:17:54Z http://mathoverflow.net/feeds/question/20915 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/20915/is-the-preorder-of-locally-closed-immersions-complete is the preorder of locally closed immersions complete? Martin Brandenburg 2010-04-10T11:44:41Z 2010-04-15T12:25:11Z <p>Let \$X\$ be a scheme. Consider the preorder of locally closed immersions into \$X\$ (considered, say, as a full subcategory of \$Sch/X\$). Is it complete or cocomplete? That is, are there infima or suprema? </p> <p>Ok it's easy to see that every nontrivial locally closed subscheme of \$\mathbb{A}^1_\mathbb{C}\$ contains only finitely or cofinitely many rational points of \$\mathbb{A}^1_{C}\$. Therefore we should restrict to <strong>finite</strong> infima or suprema.</p> <p>I guess everything works fine for infima if we restrict to seperated schemes. Take the ideal sheaves in the open subschemes which correspond to the closed immersions, restrict them to their intersection, take the smallest quasi coherent ideal which contains them and consider the corresponding closed subscheme of the intersection.</p> http://mathoverflow.net/questions/20915/is-the-preorder-of-locally-closed-immersions-complete/21448#21448 Answer by Angelo for is the preorder of locally closed immersions complete? Angelo 2010-04-15T12:09:11Z 2010-04-15T12:09:11Z <p>It does not have suprema. For example, consider the affine plane \$\mathbb A^2_{\mathbb C}\$, and let \$\mathbb A^1_{\mathbb C}\$ by the line \$y=0\$. Consider the locally closed subschemes \$U := \mathbb A^2_{\mathbb C} \smallsetminus \mathbb A^1_{\mathbb C}\$ and \$V := {(0,0)}\$ with the reduced scheme structure. The locally closed subschemes that contain \$U\$ and \$V\$ are the complements of finite set of points in \$\mathbb A^2_{\mathbb C} \smallsetminus {(0,0)}\$; so there is no supremum.</p> http://mathoverflow.net/questions/20915/is-the-preorder-of-locally-closed-immersions-complete/21450#21450 Answer by Joël Riou for is the preorder of locally closed immersions complete? Joël Riou 2010-04-15T12:25:11Z 2010-04-15T12:25:11Z <p>The maximum may not exist in general. Take X=Spec k[T,U] the affine plane, A the complement of the vertical line L passing through the origin (A=Spec k[T,U,1/T]) and B the origin (Spec k[T,U]/(T,U)). Then, the maximum C of A and B in the ordered set of subschemes of X does not exist. If it was the case, then C should be some subscheme of X. Such are usually described as closed subschemes in some open subset of X, but can also be described as open subsets in some closed subschemes Z of X (I cannot find the reference in general, but it is certainly true when the schemes are noetherian). As U is (schematically)-dense in X, this Z can be nothing else as X itself, then C should be an open subset of X containing A and B, that is the open complement of a finite set of closed points of the line L except the origin, in which case it is clear that we could find an open subset of X containing A and B and strictly contained in C, which would give a contradiction.</p>