Affine morphisms in different settings coincide? - MathOverflow

Affine morphisms in different settings coincide?

2010-02-14T23:03:27Z

1.If we identify two schemes $X$ and $Y$ as two presheaves of set on category of affine schemes.($Aff:=\text{CRing}^{op}$) If there is a morphism as natural transformations $f:X\to Y$, then, how to give the definition of the affine morphism?

2.If we identify two schemes $X$ and $Y$ as two category of quasi coherent sheaves $QCoh_{X}$ and $QCoh_{Y}$. Then morphism between this two schemes is a functor $f: Qcoh_{X}\to Qcoh_{Y}$, how to give the definition of affine morpshim ?

3.If we just consider the classical case. X and Y are two schemes. suppose $f:X\to Y$ is an affine morphism. Does this definition of affineness coincide with the other two definitions(if there exists)?

If such definitions of affine morphisms exists, are they equivalent or not?

Answer by Emerton for Affine morphisms in different settings coincide?

2010-02-15T04:38:22Z

The following answer relates contexts 1 and 3.

Suppose that we have a morphism of functors $X \to Y$ in the functor--of--points setting of 1. Then to see if this morphism is affine, we consider any open immersion $U \to Y$ where $U$ is a representable functor (i.e. an affine scheme, thought of in the functor--of--points way), form the fibre product $U \times_Y X$, and see whether this is always representable (i.e. whether it is again an affine scheme). This is just a rephrasing of the usual definition in context 3 in the language of context 1.

Answer by Zoran Škoda for Affine morphisms in different settings coincide?

2011-03-14T22:31:21Z

For case 2) $f: X\to Y$ is affine if its direct image functor $f_*:Qcoh_X\to Qcoh_Y$ is faithful and admits not only a left adjoint (inverse image) $f^{*}$, but also a right adjoint, say $f^{!}$.

If $f$ is quasi-compact and $X$ separated, then $f$ is affine iff it is cohomologically affine, that is, $f_*$ is exact (Serre's criterium of affiness, cf. EGA II 5.2.2, EGA IV 1.7.17).

Answer by Anton Geraschenko for Affine morphisms in different settings coincide?

2011-03-15T08:21:00Z

Emerton and Zoran's answers completely answer the question as stated, but there's another way to think about affine morphisms that is worth mentioning.

Given any quasi-compact and quasi-separated morphism of schemes $f:X\to Y$, $\newcommand{\O}{\mathcal O}f_*\O_X$ is a quasi-coherent sheaf of $\O_Y$-algebras. This functor has an adjoint, called relative $Spec_Y$, or relative Spec. Given a quasi-coherent sheaf of $\O_Y$-algebras $\newcommand{\A}{\mathcal A}\A$, we get a scheme over $Y$, $\phi^\A:Spec_Y \A\to Y$, with the property that $\phi^\A_*(\O_{Spec_Y \A})=\A$ and $Hom_Y(X,Spec_Y \A)\cong Hom_{\O_Y\text{-alg}}(\A,f_*\O_X)$ for any $f:X\to Y$. A morphism $f:X\to Y$ is affine if and only if $X\cong Spec_Y(\A)$ (as a $Y$-scheme) for some $\A$ (which must be $f_*\O_X$). See EGA II §1 for this development of affine morphisms.

I find this way of thinking about affine morphisms is useful for two reasons. First, if I'm working with a bunch of schemes affine over $Y$, it's often easier for me to think about a bunch of $\O_Y$-algebras with algebra morphisms between them. Second, the adjunction in the previous paragraph tells you that any (quasi-compact quasi-separated) morphism $f:X\to Y$ has a canonical factorization through an affine morphism $X\to Spec_Y(f_*\O_X)\to Y$, called the Stein factorization (the first morphism is Stein, meaning that the structure sheaf pushes forward to the structure sheaf). This factorization is often extremely handy; for example, if a morphism is quasi-affine, the Stein factorization is a witness of its quasi-affineness.