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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?

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 finite infima or suprema.

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.

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  • $\begingroup$ I would find this question easier to read if you stated explicitly which way the order relation goes. There may be a standard convention in Sch/X, but not everyone who might find the question interesting is necessarily familiar with the convention. $\endgroup$
    – Charles Staats
    Commented Apr 10, 2010 at 15:37
  • $\begingroup$ As I said, the preorder is a full subcategory of $Sch/X$. Thus for locally closed immersions $i : U \to X, j : V \to X$, we have $i \leq j$ iff there is a $X$-morphism $U \to V$ (which is unique since $j$ is mono). $\endgroup$
    – Martin Brandenburg
    Commented Apr 10, 2010 at 16:36
  • $\begingroup$ Also it can be proven without much difficulty that this category is "essentially-small" in the sense that there is a small skeleton. Thus for many purposes it makes sense to regard this category as a small preorder. $\endgroup$
    – Martin Brandenburg
    Commented Apr 10, 2010 at 17:22

2 Answers 2

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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.

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  • $\begingroup$ alright :). can someone give a reference for the desccription of locally closed immersions? so far, I know only the other direction. $\endgroup$
    – Martin Brandenburg
    Commented Apr 15, 2010 at 14:50
  • $\begingroup$ Well, it follows from EGA I 9.5.10 : Let Y be a subscheme of a scheme X, such that the canonical injection i:Y->X be a quasi-compact morphism. Then, there exists a smallest closed subscheme Z of X containing Y ; as a set, Z is the closure of Y ; and Y identifies to an open subscheme of Z. This precisely gives the factorisation Y -> Z -> X of Y -> X as an open immersion followed by a closed immersion. $\endgroup$
    – Joël Riou
    Commented Apr 15, 2010 at 15:21
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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.

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