Timeline for Local blowup versus global blowup
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2013 at 16:08 | vote | accept | DavidWayne | ||
| Jul 12, 2013 at 6:43 | comment | added | Karl Schwede | Great! I guess the other way to do this is to deal with some sort of multicative system sheaf, form that product, and then take sheafy Spec. | |
| Jul 12, 2013 at 3:59 | comment | added | DavidWayne | I agree that this description of the localization makes the most sense geometrically, but I was having a hard time with the algebra in my argument. After reading more, I used what you said about localization commuting with the Rees algebra formation. Let $S=R-\mathfrak{m}$. Then $S$ is included into $R[\mathfrak{m}t]$ as $St^0$, and $R_{\mathfrak{m}}[\mathfrak{m}_{\mathfrak{m}}t] =R_S[\mathfrak{m}_St]=R[\mathfrak{m}t]_{St^0}$. Using this, it is more obvious to me exactly how blowups commute with localization, and I have an idea of how the details of the original question should work out. | |
| Jul 11, 2013 at 22:11 | comment | added | Karl Schwede | Probably the best way to do it is as follows. Form the Cartesian product: $$Bl_m(X) \times_X \text{Spec} R_{\mathfrak m}.$$ It's not exactly localization in the usual sense. | |
| Jul 11, 2013 at 21:50 | comment | added | DavidWayne | This is sort of what I was thinking, but I wasn't totally convinced by my train of thought. To clarify, when we say that the blowup commutes with localization, we mean something like $Bl_{\mathfrak{m}_{\mathfrak{m}}}(R_{\mathfrak{m}})=Bl_{\mathfrak{m}}(X)_{f^{-1}(\mathfrak{m})}$ which is the localization of the blowup at the exceptional fiber? Or is it the localization at the fiber over $\mathfrak{m}$ in the strict transform? | |
| Jul 11, 2013 at 9:57 | history | answered | Karl Schwede | CC BY-SA 3.0 |