Timeline for When are free modules on sheaves of sets quasicoherent?
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7 events
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| Apr 7, 2017 at 22:36 | comment | added | Ingo Blechschmidt | I agree that the intuition "continuous family of sets" is somewhat vague. Sure, there are skyscraper sheaves, where stalks jump from ${*}$ to a larger set. This kind of jump is acceptable, whereas the jump from $\emptyset$ to a larger set is not. I appreciate the thought you're putting into the question! I like your idea with jumps from 1 to 2. However, the argument "$i_* i^{-1} \mathcal{F} \cong \mathcal{F}$" breaks down, since it's no longer true that the only nontrivial stalk of $\mathcal{F}$ is at $x$. | |
| Apr 7, 2017 at 8:35 | comment | added | Artur Jackson | Are we close to something interesting? Consider then $\mathcal{E} = \mathrm{Sky}_x 1_+$, where I'm using the notation $n_+ = \{*=0,1,\ldots,n\}$. The cardinality of the value set jumps on neighborhoods of $x$ from 1 to 2. Perhaps I'll look at this after some sleep. ^_^ | |
| Apr 7, 2017 at 8:32 | comment | added | Artur Jackson | Dear @Ingo, I'm not sure this reasoning is correct. Certainly there are skyscraper sheaves. The issue here was that there are no maps to the initial object $\emptyset$. | |
| Apr 7, 2017 at 8:31 | comment | added | Artur Jackson | @Dan, thank you for pointing that out. Of course we shouldn't be taking values in the empty set. Sorry! By the way, had I of used the correct notion of skyscraper sheaf (replace $\emptyset$ with {*}) then, as written, this would have been the constant sheaf sending everything to the terminal object in $Ens$. | |
| Apr 6, 2017 at 21:22 | comment | added | Ingo Blechschmidt | I agree with Dan (even though I would have quite liked to have my question resolved with a simple example!). The failure of $\mathcal{E}$ to be a sheaf can be explained geometrically: Recall that there is the familiar saying "a sheaf is a continuous family of sets". This is one of the few times where this saying is strinkingly vivid. The family which is $\{\star\}$ at $x$ and $\emptyset$ at all other points is not continuous in an intuitive sense of the word. | |
| Apr 6, 2017 at 20:33 | comment | added | Dan Petersen | It seems you mean that $\mathcal E$ is the sheaf of sets for which $\mathcal E(S) = \ast$ if $x \in S$ and $\mathcal E(S) = \varnothing$ otherwise. Unfortunately this doesn't even define a presheaf in general since there is no map $\ast \to \varnothing$. So I don't think your example works. (Not also that if the construction worked, it would contradict Ingo's first claim: that the result is always true if $X$ is the spectrum of a local ring.) | |
| Apr 6, 2017 at 0:57 | history | answered | Artur Jackson | CC BY-SA 3.0 |