Question

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 1

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 2

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 3

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.