Timeline for Checking locally whether a homomorphism is a localization
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2010 at 14:00 | vote | accept | Thomas Nevins | ||
| Jul 5, 2010 at 20:24 | answer | added | Angelo | timeline score: 5 | |
| Jul 5, 2010 at 18:39 | comment | added | Thomas Nevins | Hi Brian, I'm willing to impose noetherian if necessary for understanding. No, it's not idle, I'm curious whether "essentially of finite type" is a flat-local condition. | |
| Jul 5, 2010 at 18:36 | comment | added | BCnrd | Tom, since you mention $\mathbf{C}$-algebras, it sounds like you may be happy to impose some finiteness hypotheses on the situation (otherwise can't imagine what being a $\mathbf{C}$-algebra is meant to do). So are you happy to assume some rings here are noetherian, or anything else? More specifically, is this an idle question or does it come up somewhere? | |
| Jul 5, 2010 at 18:24 | comment | added | BCnrd | Map $f:{\rm{Spec}}(B) \rightarrow {\rm{Spec}}(A)$ is homeo. onto image since true after fpqc base change (EGA IV$_4$, 2.6.2(iii)), so if localize at prime $P$ of $A$ in image then $B$ replaced with local ring at corresponding point $Q$ (since loc. at prime is injlim of rings of fns on base of opens around pt in Spec). Replacing $A'$ by local ring at prime $P'$ above one at which we localized on $A$ gives $B'_ {P'} = A'_ {P'} \otimes_ {A_P} B_ Q$ and this is local. Since $B'$ is loc. of $A'$ at mult. set, their local rings match. Hence, $A_P = B_Q$. Then...??? | |
| Jul 5, 2010 at 16:44 | history | edited | Thomas Nevins | CC BY-SA 2.5 |
deleted 20 characters in body
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| Jul 5, 2010 at 16:35 | history | asked | Thomas Nevins | CC BY-SA 2.5 |