# 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.

## 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 4, 2015 21:17:16 by Beren Sanders (87.72.55.143)