As it is shown in "Goertz,Whedorn - Algebraic geometry I" there is an equivalence of categories between the category of prevarieties over $k$ (field algebraically closed) and the category of integral schemes of finite type over $k$. Now the fibered product of schemes, coincides with the classical product of prevarieties, so I ask if the following proposition is true:
A prevariety $X$ over $k$ is a variety in the classical setting (i.e. the diagonal is closed in $X\times X$) if and only if $X$ is separated as scheme over $k$.
Moreover I ask if there is an equivalence of categories between:
- varieties over $k$ as separated prevarieties in the "classical setting" (closedness of the diagonal in $X\times X$)
- separated,integral schemes of finite type over $k$