nLab relativization in algebraic geometry

Grothendieck insisted that all of the theory of schemes should in foundational works be developed in a relative setup, that is working in the slice category $Sch/X$ of schemes over a fixed ground scheme $X$. Moreover the properties should not be studied for schemes only but for morphisms of schemes instead, where the properties are seen often at a more fundamental level.

For local properties of schemes, there is a standard way to extend them to morphisms: a morphism $f:X\to Y$ satisfies a property $P$ iff there is a Zariski cover $\{U_\alpha\}_\alpha$ of $Y$ by open subschemes such that for all $\alpha$ the preimage $f^{-1}(U_\alpha)$ is a scheme satisfying property $P$.

category: algebraic geometry