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How can we prove true that:

If $f$ is proper, then the set $T_d=\{s\in S\mid \dim f^{-1}(s) \geq d\}$ is closed?

Is this true if we do not require $f$ to be proper?

(I think the semicontinuity in the dimension of fibres look like Hartshorne Thm 12.8., but I don't know how to express dimension as the cohomology of certain sheaf if it works)

Are there any references?

Michael Hardy
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  • Note \mid and \dim, as in my edit to your question. \dim is already an operator name in LaTeX and MathJax, and \mid provides proper spacing without a need to put in spacing manually. – Michael Hardy Mar 26 '14 at 03:09
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    Consider an open immersion and $d=0$. There is no need to use cohomology. A good reference is EGA IV, §13.1. – Cantlog Mar 26 '14 at 17:18

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