Timeline for Is this essentially of finite type algebra actually of finite type?
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| Dec 31, 2014 at 1:33 | history | edited | Question Mark |
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| Nov 6, 2014 at 4:38 | comment | added | user27920 | @QuestionMark: Right, one should use the smoothening process. (That was something paramount in my mind back when I read this stuff.) Anyway, my insert for p. 92 in too long to type as an answer, so here is the key idea. You "spread out" the essentially finite type guy to something affine of finite type over the henselian local base $S$ and use openness of the quasi-finite locus (EGA IV$_3$, 13.1.4) to get an open quasi-finite neighborhood upstairs with one point in the closed fiber, and use Zariski's Main Theorem to split off something $S$-finite from that. (Maybe just read Artin?) | |
| Nov 6, 2014 at 0:55 | comment | added | Question Mark | Thanks, this resolves my confusion regarding Prop. 7 (even though I've gone through the proof of 3.5/6, I didn't realize that it gives more than claimed, namely, that the $X_i'$ cover $X$). I don't see though why $X$ is smooth if so are the $X_i$, but that doesn't matter for 3.6/7 because one can do a further smoothening facilitated by 3.1/3. In conclusion, Prop. 7 is correct as stated, except that it possibly lacks the assumption that $I \neq \emptyset$. | |
| Nov 5, 2014 at 22:58 | comment | added | user27920 | @QuestionMark: Concerning Prop. 7, 3.5/6 has a separatedness hypothesis and its conclusion should include the property that the $X'_i$ constitute an open cover of $X$ (clear when one reads the proof). So that takes care of smoothness (since smoothness can be checked on the constituents of a Zariski-open cover) and your issue #2 (since $R'$ is local) for that Prop. 7. As for page 92, I'll have to look in the old filing cabinet later to remind myself what is involved in fixing that glitch. I've never read Artin's paper, but it has no more or less reason to be glitch-free than this book. | |
| Nov 5, 2014 at 21:31 | comment | added | Question Mark | @user52824: would you mind copying your margin/insert comments here (e.g., as an answer)? Other than myself, I think it would be useful for whoever else may be reading this proof in the future. I am not motivated to possibly spend a lot of time trying to fix the proof (presumably Artin's original proof is correct anyway...), but I would gladly proofread your argument. | |
| Nov 5, 2014 at 21:25 | comment | added | Question Mark | Regarding, Prop. 7, I don't see how separatedness is of relevance for the issues I've raised. A "weak Neron model" is required to be smooth by definition, but 3.5/6 says nothing about smoothness of the glueing output it constructs (issue #1), and, even if the "output" $X$ were smooth, it is then not clear why $X(R')$ is contained in $\cup X_i(R')$ (issue #2), as it should be if $(X_i \otimes_R R')$ were to be a weak Neron model of $X_{K'}$ as is claimed there. | |
| Nov 5, 2014 at 20:42 | comment | added | user27920 | @QuestionMark: Also, since I was only interested in Neron models of groups, in which case $X_K(K)$ is non-empty, it is automatic that $I$ is non-empty in such cases. | |
| Nov 5, 2014 at 20:36 | comment | added | user27920 | @QuestionMark: Concerning Prop. 7, the definition of "weak Neron model" requires that $X_K$ is separated. So separatedness of $X_K$ is implicit in the hypotheses of Prop. 7. Hence, reduction to $\#I=1$ goes easily via 3.5/6 as claimed in the book. I wrote in the margin of my copy at the "quasi-finite" claim on p. 92 that this is not quite true, as $S_0$ is only essentially finite type over $A_0$, and that the margin was too small to contain the extra argument to fix this, so I wrote it up in a separate insert. Have fun working it out as I did for myself when I was a student. :) | |
| Nov 5, 2014 at 16:51 | comment | added | Question Mark | This is slightly off-topic, but I think there is one further inaccuracy in section 3.6: in Proposition 7 one should assume that $\#I = 1$. Surely, they must've intended $I$ to be at least nonempty (why no rat'l points over $K^{sh}$ implies the same over $K^{'sh}$?), but also the first sentence of the proof doesn't justify how to "easily reduce to the case $\# I = 1$," since there is no reason why separated glueing be smooth, or (if one also does the smoothening afterwards) why all of its $R'$-points factor through the open immersions of the blowups of the $X_i$'s. | |
| Nov 5, 2014 at 16:44 | comment | added | Question Mark | OK, I agree that this question is a bit contrived. This is a step in the proof of 3.6/16 in "Neron models." Namely, this concerns paragraph 3 on p. 92 and its sentence "it is easily seen that $Spec(A_0) \rightarrow Spec(S_0)$ is quasi-finite at..." My question is why that map is of finite type to begin with (as is part of the definition of quasi-finite). The reason why I sometimes avoid giving context is because I don't like answers of the sort "oh, but that doesn't matter, since in the main case of interest so-and-so this can be seen directly..." (the case of interest depends on taste...). | |
| Nov 5, 2014 at 16:12 | comment | added | user27920 | @JasonStarr: Sure, but I have responded to many of QM's questions and often only find out afterwards what the motivation was. It would be nice to know beforehand (e.g., "I am reading book X and on page Y the following thing happens which is unclear..."). In the end if I or you are doing the actual "correcting" then it can help in making a more informed response if we know where the question is coming from. For example, maybe the question is about an "error" which you or I can see does not affect the intended application but QM doesn't realize that. | |
| Nov 5, 2014 at 13:32 | comment | added | Jason Starr | @user52824: Maybe Question Mark is just trying to figure out what is correct and what is incorrect in those sources. There are mistakes, and it is good for students to try to find them and correct them. | |
| Nov 5, 2014 at 7:47 | comment | added | user27920 | Please give more motivation for your questions, so people can recognize the context before spending time to try to answer them. For example, is your question related to some step in the proof of Artin approximation as presented in the book "Neron Models"? | |
| Nov 5, 2014 at 5:51 | history | edited | Question Mark | CC BY-SA 3.0 |
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| Nov 5, 2014 at 4:49 | history | asked | Question Mark | CC BY-SA 3.0 |