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Your intuition is confusing the 'fiber over a point' with restriction to a closed subscheme'. In general these can be very different, even if they come from the same place conceptually. Rational functions give a good example of when the fiber is zero but the restriction isn't.

For an example the other way, consider the submodule $O(y)\subset O(-y)\subset O$ consisting of regular functions which vanish at $y$. These restrict to zero at $y$, but the fiber is isomorphic to the zariski cotangent space, which at a smooth point will be a $k$-vector space of the dimension of your scheme.

Restriction of functions is the more intuitive concept, but the fiber construction is more natural from a module theoretic perspective. Consider $x\mathbb{C}[x]\subset \mathbb{C}[x]$ (a case of the previous example). The restrictions of these sets to $x=0$ differ, but they are isomorphic as modules.

There is also a module-theoretic version of 'restriction to a closed subscheme', given by the limit over all open neighborhoods of that subscheme, but this has its own counter-intuitive phenomenon. For example, it can never give different answers for $\mathbb{C}[x]$ and $x\mathbb{C}[x]$. If you take this limit for rational functions, you get all of $\mathcal{K}$, which is at least non-zero, but now it is keeping track of too much information (for example, it is distinguishing between rational functions which differ off of $y$).

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Your intuition is confusing the 'fiber over a point' with restriction to a closed subscheme'. In general these can be very different, even if they come from the same place conceptually. Rational functions give a good example of when the fiber is zero but the restriction isn't.

For an example the other way, consider the submodule $O(y)\subset O$ consisting of regular functions which vanish at $y$. These restrict to zero at $y$, but the fiber is isomorphic to the zariski cotangent space, which at a smooth point will be a $k$-vector space of the dimension of your scheme.

Restriction of functions is the more intuitive concept, but the fiber construction is more natural from a module theoretic perspective. Consider $x\mathbb{C}[x]\subset \mathbb{C}[x]$ (a case of the previous example). The restrictions of these sets to $x=0$ differ, but they are isomorphic as modules.

There is also a module-theoretic version of 'restriction to a closed subscheme', given by the limit over all open neighborhoods of that subscheme, but this has its own counter-intuitive phenomenon. For example, it can never give different answers for $\mathbb{C}[x]$ and $x\mathbb{C}[x]$. If you take this limit for rational functions, you get all of $\mathcal{K}$, which is at least non-zero, but now it is keeping track of too much information (for example, it is distinguishing between rational functions which differ off of $y$).