## Pushouts in the Category of Schemes

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When does it make sense to glue schemes together along subschemes?

In particular: is there a way to glue two schemes together along a closed point (say we're working over a field)? Can you glue two closed points of the same scheme together? Is it easier to glue in the category of algebraic spaces?

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## 4 Answers

Given schemes $X,Y$ and $Z$ such that $Z$ is a closed subscheme of both $X$ and $Y$ the pushout exists in the category of schemes. So in particular one can glue schemes along a closed point. A reference for this (carried out via the category of locally ringed spaces) is given in this paper of Schwede (Corollary 3.9).

In general though the pushout in the category of locally ringed spaces need not be a scheme even if one pushes out along a subscheme - see for instance Example 3.3 in Schwede's paper.

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Wow.. interesting work there by Schwede! :) – Jose Capco Nov 12 2009 at 7:29

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Schwede's Example 3.3 doesn't really show that the coproduct doesn't exist in the scheme category; just that the coproduct in the ringed space category isn't a scheme. – Andrew Critch Nov 12 2009 at 7:40
Fixed this. Do you know if the pushout in the category of sheaves is representable in this example? I can think of plenty of non-representable pushouts but none I am sure about along a subscheme. – Greg Stevenson Nov 12 2009 at 7:45

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I'm sure I'm missing something, but: how does your first sentence imply that "in particular one can ... glue a scheme to itself along a pair of closed points"? Example I have in mind that gives me pause without being a counterexample to your statement: if you try to glue an elliptic curves times the $\mathbb{P}^1$ (over $\mathbb{C}$) to itself, by gluing one fiber over $\mathbb{P}^1$ to another, not by the identity but by a nontorsion element, you just get an algebraic space that is not a scheme. – Ravi Vakil Apr 8 2010 at 5:19

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Ravi, you are absolutely right. I assume the author meant just choose two closed points, and glue them (but not glue other things), which is true say for quasi-projective schemes I think. You should always be able to do this assuming both points can be represented as points in the same affine patch (if not more generally). Hm, if you have two points which can't live on the same affine patch due to failure of separatedness, I'm not sure what happens if you try to glue them. – Karl Schwede Apr 16 2010 at 16:33
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Consider a commutative local ring R, say a valuation domain, with maximal ideal M. Consider the fiber product $R \times_M R$ (I wrote M instead of R/M), coming from the pullback in commutative rings $R\rightarrow R/M$. Then the corresponding prime spectra of this fibered product (in rings) is actually a form of gluing of the same same (affine) scheme Spec R along the closed point M. So this is the case where this happens.

So I think you can do such things for affine Schemes. For affine schemes, you can at least reverse the topology (they are sometimes called inverse spectrum) and you can form a sheaf over this topology similar to the canonical structure sheaf, but the closed points becomes the generic points in this topology. I cannot recall correctly, but I think the stalks of this sheaves become integral domains (so it is some form of dual to the affine schemes, local becomes integral and so on)

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is there a way to glue two schemes together along a closed point (say we're working over a field)? Is it easier to glue in the category of algebraic spaces?

For this particular pushout, the geometric intuition is quite simple: given two algebraic varieties, one of which lives in $\mathbb A^m$, another in $\mathbb A^n$, combine them in two complementary hyperplanes in $\mathbb A^{m+n}$. Algebraically, this easily generalizes to an affine scheme $\mathrm{Spec}\, R_1\times R_2/{\mathrm{relationship}}$ and then you glue everything together.

As correctly said above, general pushouts of schemes may not be schemes themselves.

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In the affine case, let $X=Spec A$, $Y=Spec B$, and $Z=Spec R$. If you have morphisms $f:Z\rightarrow X$ coming from $\phi:A\rightarrow R$ and $g:Z\rightarrow Y$ coming from $\psi: B\rightarrow R$ (because $Aff$ is anti-equivalent to
$CRing$), then the pushout $X \coprod_{Z} Y$, gluing $X$ and $Y$ along $Z$ is given by $Spec D$, where

$D=A\times_{ R} B:=$ { $(a,b) \in A\times B | \phi (a)= \psi (b)$ } .

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The coproduct in CRing is given by the tensor product over $\mathbb{Z}$, and the pushout over $R$ is given by the tensor product over $R$. I suspect you meant the pullback of A and B over R, written $A\times_R B$.  Also, the "coproduct" that you're referring to is called the pushout, the gluing, or the "fibered coproduct", although this last one is nonstandard.  The coproduct of affine schemes is specifically the disjoint union. – Harry Gindi Apr 17 2010 at 5:52
I think I meant something like the "fibered coproduct of schemes". That is, the opposite notion to the actual fibered product in CRings (and the latter corresponds, if I'm not mistaken, to my definition of $D$). – Qfwfq Apr 17 2010 at 6:36
I fixed it for you. – Harry Gindi Apr 17 2010 at 7:16
@Qfwfg: Your pushout is really only the pushout in the category of affine schemes. For non-affine schemes, in general, your pushout does noes have the desired universal property. One has to require, for example, that $\phi$ or $\psi$ is surjective (see the paper by Karl Schwede). – Martin Brandenburg Nov 12 2011 at 11:22
@Martin: yes, my answer was only meant to be a (very) partial answer. – Qfwfq Nov 12 2011 at 13:59
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