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

• a morphism $f:X\to Y$ 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: $f:X\to Y$ is affine iff for every affine $U\subset Y$, the inverse image ${f}^{-1}\left(U\right)$ is affine.

## Relative spectra and affine schemes

Grothendieck constructed a spectrum of a (commutative unital) algebra in the category of quasicoherent $𝒪X$-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 ${f}^{*}⊣{f}_{*}⊣{f}^{!}$ among the categories of (say left) modules where ${f}^{*}$ is the extension of scalars, ${f}_{*}$ the restriction of scalars and ${f}^{!}:M↦{\mathrm{Hom}}_{R}\left(S,M\right)$ where the latter is an $R$-module via $\left(rx\right)\left(s\right)=x\left(sr\right)$. In particular, ${f}_{*}$ is exact.

In fact, if $f:X\to Y$ 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 criterium of affiness, cf. EGA II 5.2.2, EGA IV 1.7.17).

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

## Extensions

One can extend the notion of an affine morphism to algebraic spaces, the noncommutative schemes of Rosenberg, Durov’s generalized schemes, algebraic stacks and so on. The affinity is a local property so for algebraic stacks and the like one looks at the pullback to affine charts and checks if the resulting morphism is affine; for Durov’s and Rosenberg’s schemes one is basically generalizing the functorial criterium by definition. (more on this later)

## Literature

Some of the material is extracted from MathOverflow http://mathoverflow.net/questions/15291/affine-morphisms-in-different-settings-coincide/58486.

• R. Hartshorne, Algebraic geometry, exercise II.5.17

• A. L. Rosenberg, Noncommutative schemes, Compositio Math. 112 (1998) 93–125, MR99h:14002, doi

Revised on March 6, 2013 19:30:15 by Zoran Škoda (161.53.130.104)