Timeline for Colimits of schemes
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 10, 2010 at 4:17 | comment | added | Emerton | Dear Martin, My reconstruction effort has failed, at least for the moment, and so I have added a further edit, scaling back the claim of the previous one. | |
| Dec 10, 2010 at 4:17 | history | edited | Emerton | CC BY-SA 2.5 |
added 409 characters in body
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| May 9, 2010 at 4:20 | comment | added | Emerton | Dear Martin, I will try to reconstruct the proof, and then post it. | |
| May 8, 2010 at 19:42 | comment | added | Martin Brandenburg | I still have not understood the claim in the EDIT if $I$ is not maximal. | |
| Dec 29, 2009 at 10:42 | comment | added | Martin Brandenburg | in the general case, let $U \subseteq X$ be an open affine. since the transition maps $Spec A/I^n \to Spec A/i^{n+1}$ are homeomorphisms, the images of the $f_n$ are all equal. consider the preimage of $U$ in $Spec A/I$. let $f \in A$ such $D(\overline{f})$ is a basic open subset of this preimage. then all the $f_n$ restrict to compatible $Spec A_f / (I_f)^n \to U$. the affine case yields $Spec \hat{A_f} \to U$. now we want to glue these morphisms to $Spec \hat{A} \to X$. so it would be nice that $Spec \hat{A_f}$ is an open cover of $Spec \hat{A}$, but this seems to be unlikely ... | |
| Dec 29, 2009 at 10:41 | comment | added | Martin Brandenburg | in general, the colimit of local affine schemes and local transition maps exists and is a local affine scheme. this is because you can describe morphisms on local schemes via points and local homomorphisms on the stalks. in particular, the colimit of the $Spec A/I^n$ is $Spec \hat{A}$, when $I$ is a maximal ideal. I try to prove the general case: if $f_n : Spec A/I^n \to X$ are compatible morphisms, we want to glue them to $Spec \hat{A} \to X$. if $X$ is affine, this is trivial. | |
| Dec 29, 2009 at 8:32 | history | edited | Emerton | CC BY-SA 2.5 |
added 284 characters in body
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| Dec 29, 2009 at 3:56 | history | answered | Emerton | CC BY-SA 2.5 |