----- > [!proposition] Proposition. ([[spec functor and global sections functor are adjoint]]) > Let $A$ be a [[ring]], $(X ,\mathcal{O}_{X})$ a [[scheme]]. There is a [[natural transformation|natural]] [[bijection]] $\alpha: \text{Hom}_{\mathsf{Sch}}(X, \text{Spec }A) \to \text{Hom}_{\mathsf{CRing}}\big(A, \Gamma(X)\big)$ > given by associating to a [[morphism of locally ringed spaces|scheme morphism]] $f:X \to \text{Spec }A$ its component map for global sections $f^{\sharp}_{\text{Spec }A}:\Gamma(\text{Spec }A, \mathcal{O}_{\text{Spec }A}) \to \Gamma(X, \mathcal{O}_{X}).$ > Thus, the global sections [[covariant functor|functor]] $\Gamma:\mathsf{Sch}\to \mathsf{CRing}^{\text{op}}$ is [[adjoint functor|left-adjoint]] to $\text{Spec}:\mathsf{CRing}^{\text{op}} \to \mathsf{Sch}$: $\text{Spec}\dashv \Gamma$. > [!note] Remark. > The moral is that it is 'easy' to define morphisms into [[affine scheme|affine schemes]]. It is much harder to construct morphisms [[contravariant functor represented by projective space|into e.g. projective schemes]]. ^note > [!proposition] Corollary. > Putting $A=\mathbb{Z}$ and recalling that $\mathbb{Z}$ [[ring#^properties|is initial in]] $\mathsf{Ring}$, we find that $(\text{Spec } \mathbb{Z}, \mathcal{O}_{\text{Spec } \mathbb{Z}})$ is [[terminal object|final]] in $\mathsf{Sch}$. ^proposition > [!proof]- Proof. ([[spec functor and global sections functor are adjoint]]) > ~ We will define an inverse map $\beta:\text{Hom}_{\mathsf{CRing}}\big(A, \Gamma(X)\big) \to \text{Hom}_{\mathsf{Sch}}(X, \text{Spec }A)$ So fix a [[ring homomorphism]] $A \xrightarrow{\varphi}\Gamma(X)$. Want to induce a [[morphism of locally ringed spaces|scheme morphism]] $X \xrightarrow{\beta(\varphi)} \text{Spec } A$. If $X$ was [[affine scheme|affine]], then we would know how to do this, cf. [[the category of affine schemes is dual to that of rings]]. But we are only given that $X$ is [[cover|covered]] by [[affine scheme|open affines]] $\{ U_{i} \}$, $U_{i}=\text{Spec }B_{i}$. Have composed maps $A \xrightarrow{\varphi}\Gamma(X, \mathcal{O}_{X}) \xrightarrow{ \cdot |_{U_{i}}} \Gamma(U_{i}, \mathcal{O}_{X}) = B_{i}$ inducing morphisms ([[the category of affine schemes is dual to that of rings|cf.]]) $f_{i}:\text{Spec }B_{i} \to \text{Spec }A$. We want to glue the $f_{i}$ together into one morphism $f:X \to \text{Spec } A$. The first thing to do is check agreement of $f_{i},f_{j}$ on overlap $U_{i} \cap U_{j}=:U_{ij}$. - [ ] todo (bring over from ipad, have done of the ground work with things like [[the gluing lemma for sheaf morphisms|the gluing lemma for scheme morphisms]]) ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```