Suppose $f: X \rightarrow Y$ is a morphism of schemes, and $y \in Y$ is closed point.
We know the first projection morphism $p_1: X \ $ x$_{Y} \ k(y) \rightarrow X$ is a homeomorphism onto $f^{-1}(y)$. Is it in fact a closed immersion of schemes?
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It's the base change of a closed immersion, hence it is also a closed immersion.
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$\begingroup$ Thank you, I see it now: the associated morphism Spec $ k(y) \rightarrow Y$ is indeed a closed immersion, and that property is stable under base change. $\endgroup$ Commented Jun 2, 2014 at 15:10
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$\begingroup$ this is of course because $y \in Y$ was a closed point to begin with. $\endgroup$ Commented Jun 3, 2014 at 3:30