Timeline for Is there a formal local criterion of finiteness?
Current License: CC BY-SA 3.0
9 events
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| Dec 22, 2016 at 14:10 | history | edited | Emre | CC BY-SA 3.0 |
Original question demonstrated to be the wrong thing to ask. Modified it accordingly.
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| Dec 22, 2016 at 14:04 | comment | added | Emre | @FrancescoPolizzi Thanks for your edits. Though indeed I was asking about the deformation of points, e.g. the complete local rings at those points. | |
| Dec 22, 2016 at 14:01 | comment | added | Emre | @PiotrAchinger Ah! Indeed a good point about finiteness being local on the target. Can you think of some global standard hypothesis (e.g. properness etc.) which combined with this source local criteria implies finiteness? | |
| Dec 22, 2016 at 12:42 | comment | added | Piotr Achinger | In any case, the answer to the question seems to be negative already for schemes in the case of an open immersion. Indeed, the diagonal of an open immersion $f:X\to Y$ is an isomorphism, so finite and unramified. However, if $x$ is a point over $y$, then the morphism on completed local rings is an isomorphism. The problem is that being finite is local on the target rather than on the source. | |
| Dec 22, 2016 at 12:38 | comment | added | Piotr Achinger | @FrancescoPolizzi I'm not sure this is what the question was asking. It seems that you have confused deforming $X$ with deforming a point $x\in X$. If $y\in Y$ is the point you blow up and $x\in X$ is a point over it, then $\widehat{\mathcal{O}}_{Y, y} \to \widehat{\mathcal{O}}_{X,x}$ is not finite. Or am I missing the point of your comment? | |
| Dec 22, 2016 at 6:59 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
added 8 characters in body
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| Dec 21, 2016 at 23:33 | history | edited | Francesco Polizzi | CC BY-SA 3.0 |
added 26 characters in body; edited tags; edited title
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| Dec 21, 2016 at 23:25 | comment | added | Francesco Polizzi | If $Y$ is an abelian variety and $X$ is the blow-up of $Y$ at one point, then the blow-up morphism $\pi \colon X \to Y$ is not finite, but it induces an isomorphism of deformation functors $$\mathrm{Def}(X) \simeq \mathrm{Def}(Y),$$ see mathoverflow.net/questions/198049/deformations-of-a-blowup | |
| Dec 21, 2016 at 21:01 | history | asked | Emre | CC BY-SA 3.0 |