Timeline for morphisms of affine schemes question
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 28, 2011 at 6:00 | comment | added | Will Chen | nevermind. He says maps to the stalk, not "is the stalk". sorry. | |
| Jul 28, 2011 at 5:31 | vote | accept | Will Chen | ||
| Jul 28, 2011 at 5:28 | comment | added | Will Chen | $\newcommand{\fF}{\mathcal{F}}$ Sorry to come back to this, but after rereading Hartshorne's definition of a morphism of locally ringed spaces, that even though as $V$ ranges over all open nbhd's of $f(P)$, $f^{−1}(V)$ ranges over a subset of the nbhd's of $P$, he still claims that $\lim_V \oO_X(f^{−1}(V)) = \oO_{X,P}$. (In your addendum to your original response, you said in general this limit, which you wrote as $(f_∗\fF)_{f(x)}$ is not ismorphic to $\fF_x$) | |
| Jun 7, 2011 at 17:18 | vote | accept | Will Chen | ||
| Jul 28, 2011 at 5:11 | |||||
| Jun 7, 2011 at 17:13 | vote | accept | Will Chen | ||
| Jun 7, 2011 at 17:13 | |||||
| Jun 5, 2011 at 20:32 | history | edited | Emerton | CC BY-SA 3.0 |
added 1148 characters in body
|
| Jun 5, 2011 at 20:24 | comment | added | Emerton | Dear oxeimon, Yes, your computations of the stalks in your first comment are correct. As for "the induced map on stalks" remark of Hartshorne, the point is that if $\mathfrak p \mapsto \mathfrak q,$ and if $V$ is a n.h. of $\mathfrak p$, then $f^{-1}(V)$ will be a n.h. of $\mathfrak q$, so there is an induced map on stalks $\mathcal O_{\mathrm{Spec} B,\mathfrak p} \to \mathcal O_{\mathrm{Spec} A,\mathfrak q},$ which is the map Hartshorne is discussing. (See my updated answer for slightly more detail, although it may be superfluous at this point.) Regards, Matthew | |
| Jun 5, 2011 at 1:32 | comment | added | Will Chen | so I guess these stalks in general are just the direct limit of localizations, where the maps are just inclusions. (at least for integral domains) | |
| Jun 5, 2011 at 1:24 | comment | added | Will Chen | Also, I thought that $(f_*\oO_{\Spec B})_\mf{p} = (\oO_{\Spec B})_\mf{q}$ because of the way he defined the morphism of sheaves $f^\sharp$ and the local homomorphisms $\varphi_\mf{p} : A_{\varphi^{-1}}(\mf{p})\rightarrow B_\mf{p}$. Ie, he said "The induced maps $f^\sharp$ on the stalks are just the local homomorphisms $\varphi_\mf{p}$", but these local homomorphisms are only defined for $\mf{p}\in\Spec B$, whereas they should be defined for all $\mf{q}\in\Spec A$, so it seemed like he was saying that as $\mf{p}$ ranges over $\Spec B$, $f(\mf{p})$ ranges over all of $\Spec A$, which is false.. | |
| Jun 5, 2011 at 0:50 | comment | added | Will Chen | Alright, so in (1), I guess the direct limit in the definition of the stalk is just the direct limit of a single group, which is just $O_B(B) = k\times k$. And in (2), since the primes of B are just the primes of A except for $(t)$, the direct limit just limits over all nonempty open sets of Spec B, so if we only consider the open sets $D(f)$ for $f\in B$, for which $O_B(D(f)) = B_f$, so it seems like the direct limit (ie the stalk) is just $k(t)$? | |
| Jun 2, 2011 at 17:19 | history | answered | Emerton | CC BY-SA 3.0 |