Affine morphisms

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

Last revised on March 4, 2015 at 21:17:16. See the history of this page for a list of all contributions to it.