Timeline for Gluing two affine schemes along a different intersection
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 7, 2020 at 23:20 | comment | added | user127776 | I see. But here our gluings are along open affines that seems pretty easy by itself, I mean I can't see how Karl's article can make it more clear. I'm just not sure about the existence of the two maps from $X$ to the new gluing. Or the construction of $\text{Spec}(A)^f$ with aforementioned properties is also very helpful. | |
| Dec 7, 2020 at 23:16 | comment | added | pinaki | The general construction given in Section 2 of Schwede's article is very general - I think it works with any collection of "ringed subspaces" of ringed spaces. | |
| Dec 7, 2020 at 22:40 | history | edited | user127776 | CC BY-SA 4.0 |
added 661 characters in body
|
| Dec 7, 2020 at 20:32 | comment | added | user127776 | Isn't Karl Schwede's article good when gluing along closed subschemes? It might be helpful to give meaning to $\text{Spec}(A)^f$. Let the zeros of $f$ extend in $\text{Spec}(A)$, it intersects the complement of $\text{Spec}(C)$ in a closed subscheme. Then we can glue the complement of the zero set in $\text{Spec}(A)$ and the complement of $\text{Spec}(C)$ to get an affine scheme. This is a candidate for $\text{Spec}(A)^f$. But I'm not sure whether gluing it to $\text{Spec}(B)$ along $\text{Spec}(C_f)$ gives back $X$. | |
| Dec 7, 2020 at 19:37 | comment | added | LSpice | Title of Schwede's article referenced by @auniket: Gluing schemes and a scheme without closed points. | |
| Dec 7, 2020 at 18:54 | comment | added | pinaki | Karl Schwede's article (math.stanford.edu/~vakil/files/schwede03.pdf) has probably been for sometimes a canonical reference on matters related to gluing schemes - what you want is a special case of the construction in Section 2 | |
| Dec 7, 2020 at 16:59 | history | edited | user127776 | CC BY-SA 4.0 |
Added an extra question
|
| Dec 7, 2020 at 2:01 | history | asked | user127776 | CC BY-SA 4.0 |