P23:Let $f:Y\rightarrow X$ be locally of finite-type.
prove:$\Delta:Y\rightarrow Y\times Y$ is an open immersion,then $f$ is unramified
....we reduce the problem to the case of a morphisim $f:Y\rightarrow spec k$ where $k$ is an algebracally closed field.let $y$ be a closed point of $Y$.because $k$ is an algebracally closed ,there exist a section $g:spec k\rightarrow Y$ whose image is ${y}$.(why ?)......${y}$ is open in $Y$.moreover ,$spec\mathcal{O}_y={y}\rightarrow spec k$ still has the property that $spac\mathcal{O}_y\rightarrow spac\mathcal{O}_y\otimes\mathcal{O}_y$ is an open immersion.($spec\mathcal{O}_y$ is not $V(y)$,why have $spec\mathcal{O}_y={y}$,and I cannot prove the rest statement ) But $\mathcal{O}_y$ is artin ring with residue field $k$(why $k(y)=k$ ?).and $spac\mathcal{O}_y\otimes\mathcal{O}_y$ has only one point.and $\mathcal{O}_y\otimes\mathcal{O}_y\rightarrow\mathcal{O}_y$ must be an isomorphism.(how can we characterize the struction of $\mathcal{O}_y\otimes\mathcal{O}_y$ )