Affine morphisms

Idea and definition

An affine morphism of schemes is a relative version of an affine scheme: given a scheme $X$, the canonical morphism $X \to Spec \mathbb{Z}$ is affine iff $X$ is an affine scheme. By the basics of spectra, every morphism of affine schemes $Spec S \to Spec R$ corresponds to a morphism $f^\circ\colon R \to S$ of rings. The affine morphisms of general schemes are defined as the ones which are locally of that form:

• a morphism $f\colon X\to Y$ of (general) schemes is affine if there is a cover of $Y$ (as a ringed space) by affines $U_\alpha$ such that $f^{-1} U_\alpha$ is an affine subscheme of $X$.

A seemingly stronger, but in fact equivalent, characterization follows: $f\colon X\to Y$ is affine iff for every affine $U \subset Y$, the inverse image $f^{-1}(U)$ is affine.

Relative spectra and affine schemes

Grothendieck constructed a spectrum of a (commutative unital) algebra in the category of quasicoherent $\mathcal{O}X$-modules. The result is a scheme over $X$; relative schemes of that form are called relative affine schemes.

Functorial point of view

Now notice that a map of (associative) rings, possibly noncommutative (and possibly nonunital), induces an adjoint triple of functors $f^*\dashv f_*\dashv f^!$ among the categories of (say left) modules where $f^*$ is the extension of scalars, $f_*$ the restriction of scalars and $f^!\colon M \mapsto Hom_R(S,M)$ where the latter is an $R$-module via $(r x) (s) = x (s r)$. In particular, $f_*$ is exact.

In fact, if $f\colon X\to Y$ is a quasicompact morphism of schemes and $X$ is separated, then $f$ is affine iff it is cohomologically affine, that is, the direct image $f_*$ is exact (Serre's criterion of affineness, cf. EGA II 5.2.2, EGA IV 1.7.17).

An affine localization is a localization functor among categories of quasicoherent $\mathcal{O}$-modules which is also the inverse image functor of an affine morphism; or an abstraction of this situation.