nLab
affine morphism

Affine morphisms

Idea and definition

An affine morphism of schemes is a relative version of an affine scheme: given a scheme XX, the canonical morphism XSpecX \to Spec \mathbb{Z} is affine iff XX is an affine scheme. By the basics of spectra, every morphism of affine schemes SpecSSpecRSpec S \to Spec R corresponds to a morphism f :RSf^\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:XYf\colon X\to Y of (general) schemes is affine if there is a cover of YY (as a ringed space) by affines U αU_\alpha such that f 1U αf^{-1} U_\alpha is an affine subscheme of XX.

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

Relative spectra and affine schemes

Grothendieck constructed a spectrum of a (commutative unital) algebra in the category of quasicoherent 𝒪X\mathcal{O}X-modules. The result is a scheme over XX; 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 !f^*\dashv f_*\dashv f^! among the categories of (say left) modules where f *f^* is the extension of scalars, f *f_* the restriction of scalars and f !:MHom R(S,M)f^!\colon M \mapsto Hom_R(S,M) where the latter is an RR-module via (rx)(s)=x(sr)(r x) (s) = x (s r). In particular, f *f_* is exact.

In fact, if f:XYf\colon X\to Y is a quasicompact morphism of schemes and XX is separated, then ff is affine iff it is cohomologically affine, that is, the direct image f *f_* is exact (Serre's criterion 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\mathcal{O}-modules which is also the inverse image functor of an affine morphism; or an abstraction of this situation.

See also monad in algebraic geometry.

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 February 5, 2014 04:37:20 by Zoran Škoda (94.250.151.219)