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# colimitsColimits of schemes

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I've found many remarks that the category of schemes is not cocomplete. the The category of locally ringed spaces is cocomplete, and in some special cases this turns out to be the colimit of schemes, but in other cases not (which is, of course, no evidence that the colimit does not exist). howeverHowever, I want to understand in detail a counterexample where the colimit does not exist, but I hardly found one. in In FGA explained I've found the reference, that example Example 3.4.1 in hartshorneHartshorne, appendix Appendix B is a smooth proper scheme over $\mathbb{C}$ with a free $\mathbb{Z}/2$-action, but the quotient does not exist (without proof). to To be honest, this is too complicated to me. are Are there easy examples? you You won't help me just giving the example, because there are lots of them, but the hard part is to prove that the colimit really does not exist.

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this is related to another question.

I've found many remarks that the category of schemes is not cocomplete. the category of locally ringed spaces is cocomplete, and in some special cases this turns out to be the colimit of schemes, but in other cases not (which is, of course, no evidence that the colimit does not exist). however, I want to understand in detail a counterexample where the colimit does not exist, but I hardly found one. in FGA explained I've found the reference, that example 3.4.1 in hartshorne, appendix B is a smooth proper scheme over $\mathbb{C}$ with a free $\mathbb{Z}/2$-action, but the quotient does not exist (without proof). to be honest, this is too complicated to me. are there easy examples? you won't help me just giving the example, because there are lots of them, but the hard part is to prove that the colimit really does not exist.

edit: up to now, no proof is given here

3 added 4 characters in body

this is related to another question.

I've found many remarks that the category of schemes is not cocomplete. the category of locally ringed spaces is cocomplete, and in some special cases this turns out to be the colimit of schemes, but in other cases not (which is, of course, no evidence that the colimit does not exist). however, I want to understand in detail a counterexample where the colimit does not exist, but I hardly found one. in FGA explained I've found the reference, that example 3.4.1 in hartshorne, appendix B is a smooth proper scheme over $\mathbb{C}$ with a free $\mathbb{Z}/2$-action, but the quotient does not exist (without proof). to be honest, this is too complicated to me. are there easy examples? you won't help me just giving the example, because there are lots of them, but the hard part is to prove that the colimit really does not exist.

edit: up to now, no proof is given here

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