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Tag 01I1

Chapter 25: Schemes > Section 25.6: The category of affine schemes

Lemma 25.6.4. Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. The map $$ \mathop{\rm Mor}\nolimits(X, Y) \longrightarrow \mathop{\rm Hom}\nolimits(\Gamma(Y, \mathcal{O}_Y), \Gamma(X, \mathcal{O}_X)) $$ which maps $f$ to $f^\sharp$ (on global sections) is bijective.

Proof. Since $Y$ is affine we have $(Y, \mathcal{O}_Y) \cong (\mathop{\rm Spec}(R), \mathcal{O}_{\mathop{\rm Spec}(R)})$ for some ring $R$. During the proof we will use facts about $Y$ and its structure sheaf which are direct consequences of things we know about the spectrum of a ring, see e.g. Lemma 25.5.4.

Motivated by the lemmas above we construct the inverse map. Let $\psi_Y : \Gamma(Y, \mathcal{O}_Y) \to \Gamma(X, \mathcal{O}_X)$ be a ring map. First, we define the corresponding map of spaces $$ \Psi : X \longrightarrow Y $$ by the rule of Lemma 25.6.1. In other words, given $x \in X$ we define $\Psi(x)$ to be the point of $Y$ corresponding to the prime in $\Gamma(Y, \mathcal{O}_Y)$ which is the inverse image of $\mathfrak m_x$ under the composition $ \Gamma(Y, \mathcal{O}_Y) \xrightarrow{\psi_Y} \Gamma(X, \mathcal{O}_X) \to \mathcal{O}_{X, x} $.

We claim that the map $\Psi : X \to Y$ is continuous. The standard opens $D(g)$, for $g \in \Gamma(Y, \mathcal{O}_Y)$ are a basis for the topology of $Y$. Thus it suffices to prove that $\Psi^{-1}(D(g))$ is open. By construction of $\Psi$ the inverse image $\Psi^{-1}(D(g))$ is exactly the set $D(\psi_Y(g)) \subset X$ which is open by Lemma 25.6.2. Hence $\Psi$ is continuous.

Next we construct a $\Psi$-map of sheaves from $\mathcal{O}_Y$ to $\mathcal{O}_X$. By Sheaves, Lemma 6.30.14 it suffices to define ring maps $\psi_{D(g)} : \Gamma(D(g), \mathcal{O}_Y) \to \Gamma(\Psi^{-1}(D(g)), \mathcal{O}_X)$ compatible with restriction maps. We have a canonical isomorphism $\Gamma(D(g), \mathcal{O}_Y) = \Gamma(Y, \mathcal{O}_Y)_g$, because $Y$ is an affine scheme. Because $\psi_Y(g)$ is invertible on $D(\psi_Y(g))$ we see that there is a canonical map $$ \Gamma(Y, \mathcal{O}_Y)_g \longrightarrow \Gamma(\Psi^{-1}(D(g)), \mathcal{O}_X) = \Gamma(D(\psi_Y(g)), \mathcal{O}_X) $$ extending the map $\psi_Y$ by the universal property of localization. Note that there is no choice but to take the canonical map here! And we take this, combined with the canonical identification $\Gamma(D(g), \mathcal{O}_Y) = \Gamma(Y, \mathcal{O}_Y)_g$, to be $\psi_{D(g)}$. This is compatible with localization since the restriction mapping on the affine schemes are defined in terms of the universal properties of localization also, see Lemmas 25.5.4 and 25.5.1.

Thus we have defined a morphism of ringed spaces $(\Psi, \psi) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ recovering $\psi_Y$ on global sections. To see that it is a morphism of locally ringed spaces we have to show that the induced maps on local rings $$ \psi_x : \mathcal{O}_{Y, \Psi(x)} \longrightarrow \mathcal{O}_{X, x} $$ are local. This follows immediately from the commutative diagram of the proof of Lemma 25.6.1 and the definition of $\Psi$.

Finally, we have to show that the constructions $(\Psi, \psi) \mapsto \psi_Y$ and the construction $\psi_Y \mapsto (\Psi, \psi)$ are inverse to each other. Clearly, $\psi_Y \mapsto (\Psi, \psi) \mapsto \psi_Y$. Hence the only thing to prove is that given $\psi_Y$ there is at most one pair $(\Psi, \psi)$ giving rise to it. The uniqueness of $\Psi$ was shown in Lemma 25.6.1 and given the uniqueness of $\Psi$ the uniqueness of the map $\psi$ was pointed out during the course of the proof above. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 840–851 (see updates for more information).

    \begin{lemma}
    \label{lemma-morphism-into-affine}
    Let $X$ be a locally ringed space.
    Let $Y$ be an affine scheme.
    The map
    $$
    \Mor(X, Y)
    \longrightarrow
    \Hom(\Gamma(Y, \mathcal{O}_Y), \Gamma(X, \mathcal{O}_X))
    $$
    which maps $f$ to $f^\sharp$ (on global sections) is bijective.
    \end{lemma}
    
    \begin{proof}
    Since $Y$ is affine we have
    $(Y, \mathcal{O}_Y) \cong (\Spec(R), \mathcal{O}_{\Spec(R)})$
    for some ring $R$.
    During the proof we will use facts about $Y$ and
    its structure sheaf which are direct consequences of things
    we know about the spectrum of a ring, see e.g.\ Lemma
    \ref{lemma-spec-sheaves}.
    
    \medskip\noindent
    Motivated by the lemmas above we construct the inverse map.
    Let $\psi_Y : \Gamma(Y, \mathcal{O}_Y) \to \Gamma(X, \mathcal{O}_X)$
    be a ring map. First, we define the corresponding map of
    spaces
    $$
    \Psi : X \longrightarrow Y
    $$
    by the rule of
    Lemma \ref{lemma-morphism-into-affine-where-point-goes}.
    In other words, given $x \in X$ we define $\Psi(x)$
    to be the point of $Y$ corresponding to the prime
    in $\Gamma(Y, \mathcal{O}_Y)$ which is the inverse
    image of $\mathfrak m_x$ under the composition
    $
    \Gamma(Y, \mathcal{O}_Y) \xrightarrow{\psi_Y}
    \Gamma(X, \mathcal{O}_X) \to
    \mathcal{O}_{X, x}
    $.
    
    \medskip\noindent
    We claim that the map $\Psi : X \to Y$ is continuous.
    The standard opens $D(g)$, for $g \in \Gamma(Y, \mathcal{O}_Y)$
    are a basis for the topology of $Y$. Thus it suffices to prove
    that $\Psi^{-1}(D(g))$ is open. By construction of $\Psi$
    the inverse image $\Psi^{-1}(D(g))$ is exactly the set
    $D(\psi_Y(g)) \subset X$ which is open by Lemma \ref{lemma-f-open}.
    Hence $\Psi$ is continuous.
    
    \medskip\noindent
    Next we construct a $\Psi$-map of sheaves from
    $\mathcal{O}_Y$ to $\mathcal{O}_X$. By
    Sheaves, Lemma \ref{sheaves-lemma-f-map-basis-below-structures}
    it suffices to define ring maps
    $\psi_{D(g)} : \Gamma(D(g), \mathcal{O}_Y) \to
    \Gamma(\Psi^{-1}(D(g)), \mathcal{O}_X)$
    compatible with restriction maps.
    We have a canonical isomorphism
    $\Gamma(D(g), \mathcal{O}_Y) = \Gamma(Y, \mathcal{O}_Y)_g$,
    because $Y$ is an affine scheme.
    Because $\psi_Y(g)$ is invertible on $D(\psi_Y(g))$
    we see that there is a canonical map
    $$
    \Gamma(Y, \mathcal{O}_Y)_g
    \longrightarrow
    \Gamma(\Psi^{-1}(D(g)), \mathcal{O}_X)
    =
    \Gamma(D(\psi_Y(g)), \mathcal{O}_X)
    $$
    extending the map $\psi_Y$
    by the universal property of localization.
    Note that there is no choice but to take the canonical map here!
    And we take this, combined
    with the canonical identification
    $\Gamma(D(g), \mathcal{O}_Y) = \Gamma(Y, \mathcal{O}_Y)_g$, to
    be $\psi_{D(g)}$. This is compatible with localization since the
    restriction mapping on the affine schemes are defined in terms
    of the universal properties of localization also, see
    Lemmas \ref{lemma-spec-sheaves} and \ref{lemma-standard-open}.
    
    \medskip\noindent
    Thus we have defined a morphism of ringed spaces
    $(\Psi, \psi) : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$
    recovering $\psi_Y$ on global sections. To see that it is
    a morphism of locally ringed spaces we have to show that
    the induced maps on local rings
    $$
    \psi_x : \mathcal{O}_{Y, \Psi(x)} \longrightarrow \mathcal{O}_{X, x}
    $$
    are local. This follows immediately from the commutative diagram
    of the proof of Lemma \ref{lemma-morphism-into-affine-where-point-goes}
    and the definition of $\Psi$.
    
    \medskip\noindent
    Finally, we have to show that the constructions
    $(\Psi, \psi) \mapsto \psi_Y$ and the construction
    $\psi_Y \mapsto (\Psi, \psi)$ are inverse to each other.
    Clearly, $\psi_Y \mapsto (\Psi, \psi) \mapsto \psi_Y$.
    Hence the only thing to prove is that given $\psi_Y$
    there is at most one pair $(\Psi, \psi)$ giving rise
    to it. The uniqueness of $\Psi$ was shown in Lemma
    \ref{lemma-morphism-into-affine-where-point-goes} and
    given the uniqueness of $\Psi$ the uniqueness of the
    map $\psi$ was pointed out during the course of the proof
    above.
    \end{proof}

    Comments (2)

    Comment #2592 by Rogier Brussee on June 4, 2017 a 7:15 pm UTC

    This lemma basically shows that affine schemes are universal in the following sense.

    For every ring $R$, every locally ringed space $(X, \mathcal{O}_X$ and every ring homomorphism $f: R \to \Gamma(X, \mathcal{O}_X)$, there is a unique map of locally ringed spaces $F: X \to \mathrm{spec}(R)$ such that $f = F^\sharp: \Gamma(\mathrm{spec}(R), \mathcal{O})\to \Gamma(X, \mathcal{O}_X)$ under the identification $\Gamma(\mathrm{spec}(R), \mathcal{O}) = R$.

    This lemma is in SGA 3 somewhere.

    Comment #2622 by Johan (site) on July 7, 2017 a 12:15 pm UTC

    Hi Rogier Brussee! Yes, indeed it does. In general, if you want me to add a reference for this lemma or any other lemma, please point out an exact reference and I will do so.

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