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Given a coherent sheaf F over a noetherian scheme Y, a classical result in algebraic geometry states the upper-semocontinuity of the function sending any point $y \in Y$ to $\mathrm{dim}_{k(y)}(F \otimes k(y))$.

What about similar results for the length function, i.e. $\mathrm{lt}_{O_y}(F_y)$?

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