Timeline for Colimits of schemes
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Mar 3, 2021 at 18:06 | comment | added | user127776 | Never mind! I found it. It is a variant of Grothendieck's algebraization theorem. See here: stacks.math.columbia.edu/tag/09ZT | |
| Mar 3, 2021 at 6:25 | comment | added | user127776 | @B.B. This has been a very long time but I still do not quite get the formal GAGA argument! What version of formal GAGA are you using? The version that I have seen is about equivalence of category of coherent sheaves. I am not sure how that applies to the morphism of schemes. | |
| Dec 29, 2009 at 18:00 | comment | added | David Zureick-Brown | Ah, you are right. | |
| Dec 29, 2009 at 8:58 | comment | added | Bhargav | Actually, if X is proper over a field k, then X-hat = X x k[[t]] is a colimit of X_n = X x k[t]/t^{n+1} in Sch/k by formal GAGA (Proof: Given noetherian finite type k-scheme Y and k-maps X_n -> Y, we get k[t]/t^{n+1}-maps X_n -> Y_n. By formal GAGA over k[[t]] (which only needs the source proper), there is a unique map X-hat -> Y inducing the given maps on the truncations.) | |
| Dec 29, 2009 at 8:02 | comment | added | David Zureick-Brown | For affine schemes the colimit does exist, it is just the completion. The point behind the P^1 example is that you can't glue the `completed A^1's' together to get a completed P^1. I'll see if I can think of a succinct way to see this. | |
| Dec 29, 2009 at 0:54 | comment | added | Martin Brandenburg | for example, we could try that A^1 -> A^2 -> ... (the canonical closed immersions) has no colimit in the categors of schemes. assume X is a colimit. the global sections are morphisms to A^1, thus we get that the global sections of X are k[[x_1,x_2,...]]. I don't know how to go on. | |
| Dec 28, 2009 at 20:19 | comment | added | Martin Brandenburg | thank you. how can we prove that the colimit in your example (infinitesemal neighborhood in P^1) does not exist? so we have to conclude a contradiction, if X = colim(Z^n) exists. I have no idea how this can be done. I know nothing about algebraic spaces. sorry that I repeat my question again and again, but as I said the problem is not really in "finding" examples, but checking them. | |
| Dec 28, 2009 at 19:17 | comment | added | David Zureick-Brown | Ah. You are right, too hasty on my part. I edited my answer. | |
| Dec 28, 2009 at 19:15 | history | edited | David Zureick-Brown | CC BY-SA 2.5 |
deleted 33 characters in body; deleted 262 characters in body
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| Dec 28, 2009 at 19:06 | comment | added | Martin Brandenburg | 1) seems that something in your answer is not visible. anyway, is there are a easy prove that the colimit does not exist in this example? 2) as I already said, this is no prove that the pushout does not exist. | |
| Dec 28, 2009 at 18:53 | history | answered | David Zureick-Brown | CC BY-SA 2.5 |