Affine morphisms in different settings coincide? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:52:40Z http://mathoverflow.net/feeds/question/15291 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15291/affine-morphisms-in-different-settings-coincide Affine morphisms in different settings coincide? Peter Lee 2010-02-14T23:03:27Z 2011-03-15T08:21:00Z <p>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? </p> <p>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 ? </p> <p>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)?</p> <p>If such definitions of affine morphisms exists, are they equivalent or not?</p> http://mathoverflow.net/questions/15291/affine-morphisms-in-different-settings-coincide/15313#15313 Answer by Emerton for Affine morphisms in different settings coincide? Emerton 2010-02-15T04:38:22Z 2010-02-15T04:38:22Z <p>The following answer relates contexts 1 and 3.</p> <p>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. </p> http://mathoverflow.net/questions/15291/affine-morphisms-in-different-settings-coincide/58486#58486 Answer by Zoran Škoda for Affine morphisms in different settings coincide? Zoran Škoda 2011-03-14T22:31:21Z 2011-03-14T22:36:34Z <p>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 $f^{!}$. </p> <p>If $f$ is quasi-compact and $X$ separated, then $f$ is affine iff it is <em>cohomologically affine</em>, that is, $f_*$ is exact (Serre's criterium of affiness, cf. EGA II 5.2.2, EGA IV 1.7.17). </p> http://mathoverflow.net/questions/15291/affine-morphisms-in-different-settings-coincide/58510#58510 Answer by Anton Geraschenko for Affine morphisms in different settings coincide? Anton Geraschenko 2011-03-15T08:21:00Z 2011-03-15T08:21:00Z <p>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.</p> <p>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 <em>relative Spec</em>. 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 &sect;1 for this development of affine morphisms.</p> <p>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 <em>canonical</em> factorization through an affine morphism $X\to Spec_Y(f_*\O_X)\to Y$, called the <em>Stein factorization</em> (the first morphism is <em>Stein</em>, 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.</p>