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/XSch/X of schemes over a fixed ground scheme XX. 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:XYf:X\to Y satisfies a property PP iff there is a Zariski cover {U α} α\{U_\alpha\}_\alpha of YY by open subschemes such that for all α\alpha the preimage f 1(U α)f^{-1}(U_\alpha) is a scheme satisfying property PP.

Last revised on March 6, 2013 at 19:27:22. See the history of this page for a list of all contributions to it.