Grothendieck emphasised the relative point of view: the main foundational effort should be devoted to the study of properties of morphisms rather than spaces. Thus one can consider an affine morphism instead of an affine scheme; affine morphism can be considered as a relative affine scheme over , intuitively a bundle of affine schemes. If is itself an affine scheme, corresponding to a ring and is affine then is the opposite of a morphism where .
By the Gabriel-Rosenberg theorem a scheme can be reconstructed from the category of quasicoherent sheaves over a scheme. Thus one is tempted to replace the properties of morphisms by properties of the “geometric functors” among the corresponding categories of quasicoherent sheaves. The properties of affine morphisms were first elucidated in Serre’s criterium of affiness. It appears that the direct image functor of an affine morphism is faithful and having its own right adjoint . The adjunction induces a monad making monadic over . If were just then this is clear as then and .