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?



Closed immersion of closed fiber?
It's the base change of a closed immersion, hence it is also a closed immersion.


