스킴
Schemes
Definition of a scheme as a locally affine locally ringed space
This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.
Definition of a Scheme
Now we can define what a scheme is.
Definition 1 A locally ringed space \((X, \mathcal{O}_X)\) is called a scheme if for every \(x\in X\), there exists an open neighborhood \(U\) of \(x\) such that \((U, \mathcal{O}_X\vert_U)\) is an affine scheme. The notions of morphisms and isomorphisms between schemes are defined as those of locally ringed spaces.
Given any point \(x\in X\) of a scheme \(X\), choose an affine open neighborhood \(U\) of \(x\). Then by definition there exists a ring \(A\) such that \((U, \mathcal{O}_X\vert_U)\cong (\Spec A, \mathcal{O}_{\Spec A})\) as locally ringed spaces. In particular, writing \(\mathfrak{p}_x\) for the point of \(\Spec A\) corresponding to \(x\in U\) via the above isomorphism, we know that
\[\mathcal{O}_{X,x}=\varinjlim_{V\ni x} \mathcal{O}_X(V)=\varinjlim_{U\supseteq V\ni x} \mathcal{O}_X(V)=\mathcal{O}_{U, x}\cong \mathcal{O}_{\Spec A, \mathfrak{p}_x}\tag{$\ast$}\]In general, for an affine scheme \((\Spec A, \mathcal{O}_{\Spec A})\), the restriction \((U, \mathcal{O}_{\Spec A}\vert_U)\) to an arbitrary open subset is not always an affine scheme. (Example 8) However, the principal open subset \(D(f)\) of an affine scheme \(\Spec A\) is always an affine scheme via the isomorphism
\[(D(f), \mathcal{O}_{\Spec A}\vert_{D(f)})\cong (\Spec A_f, \mathcal{O}_{\Spec A_f})\tag{$\ast\ast$}\]Lemma 2 For an affine scheme \(\Spec A\) and \(f\in A\), the subset \(D(f)\) is always an affine scheme via the isomorphism (\(\ast\ast\)) above.
Proof
From the third result of §The Spectrum, ⁋Proposition 9, we know that the ring homomorphism \(\epsilon: A \rightarrow A_f\) gives an injective continuous map \(\Spec\epsilon: \Spec A_f \rightarrow \Spec A\) that is a homeomorphism \(D(f)\cong \Spec A_f\), but to justify the above isomorphism we must justify this as a scheme morphism.
First, the ring homomorphism \(\epsilon: A \rightarrow A_f\) induces a scheme morphism
\[(\Spec \epsilon, (\Spec\epsilon)^\sharp): (\Spec A_f, \mathcal{O}_{\Spec A_f}) \rightarrow (\Spec A, \mathcal{O}_{\Spec A})\]and in this case, the homeomorphism between \(D(f)\) and \(\Spec A_f\) comes from the following decomposition, considering the canonical inclusion \(\iota: D(f)\hookrightarrow \Spec A\):
Here, we write \(\Spec\epsilon\vert^{D(f)}\) for the homeomorphism \(\Spec A_f\cong D(f)\) obtained by restricting the codomain of \(\Spec\epsilon\) to its image \(D(f)\). In general, we will write \(g\vert^W\) for the function obtained by restricting the codomain of a function \(g\) to a subset \(W\) containing its image. Now, for the above isomorphism (\(\ast\ast\)) to be an isomorphism of schemes, we must define the corresponding morphism \((\Spec\epsilon\vert^{D(f)})^\sharp\) in \(\Sh(D(f); \cRing)\). Then by [Topology] §Sheaves, ⁋Example 12, defining
\[(\Spec\epsilon\vert^{D(f)})^\sharp: \mathcal{O}_{D(f)} \rightarrow (\Spec\epsilon\vert^{D(f)})_\ast \mathcal{O}_{\Spec A_f}\]is exactly the same as defining
\[(\Spec\epsilon\vert^{D(f)})^{\sharp}: \iota^{-1}\mathcal{O}_{\Spec A} \rightarrow (\Spec\epsilon\vert^{D(f)})_\ast \mathcal{O}_{\Spec A_f}\]and again by [Topology] §Sheaves, ⁋Lemma 11,
\[\begin{aligned}\Hom_{\Sh(D(f); \cRing)}(\iota^{-1}\mathcal{O}_{\Spec A}, (\Spec\epsilon\vert^{D(f)})_\ast\mathcal{O}_{\Spec A_f})&\cong \Hom_{\Sh(\Spec A; \cRing)}(\mathcal{O}_{\Spec A}, \iota_\ast(\Spec\epsilon\vert^{D(f)})_\ast \mathcal{O}_{\Spec A_f})\\&\cong \Hom_{\Sh(\Spec A; \cRing)}(\mathcal{O}_{\Spec A}, (\iota\circ\Spec\epsilon\vert^{D(f)})_\ast \mathcal{O}_{\Spec A_f})\\&=\Hom_{\Sh(\Spec A; \cRing)}(\mathcal{O}_{\Spec A}, (\Spec\epsilon)_\ast \mathcal{O}_{\Spec A_f})\end{aligned}\]so we can define \((\Spec\epsilon\vert^{D(f)})^\sharp\) via \((\Spec\epsilon)^\sharp:\mathcal{O}_{\Spec A} \rightarrow (\Spec \epsilon)_\ast \mathcal{O}_{\Spec A_f}\).
Now we show that this is an isomorphism. For any \(g\in A\) we have \(D(fg)\subseteq D(f)\), and since any open subset of \(D(f)\) is a union of such \(D(fg)\), the sets \(D(fg)\) form a base for \(D(f)\). On the other hand, since \(\epsilon(f)\) is a unit in \(A_f\),
\[(\Spec\epsilon)^{-1}(D(fg))=D(\epsilon(fg))=D(\epsilon(g))\qquad\text{in $\Spec A_f$}\]and therefore the map \((\Spec\epsilon\vert^{D(f)})^\sharp\) on \(D(fg)\) is, by the construction in §Affine Scheme, ⁋Proposition 9, the canonical homomorphism
\[A_{fg} \rightarrow (A_f)_{\epsilon(g)}\]extending \(A \rightarrow A_f\). But since the images of both \(f\) and \(g\) are units in both \(A_{fg}\) and \((A_f)_{\epsilon(g)}\), applying universality from [Commutative Algebra] §Localization, ⁋Proposition 6 to both sides shows that \(A\rightarrow A_{fg}\) factors uniquely through \((A_f)_{\epsilon(g)}\), and the two composites of the map thus obtained with the above canonical homomorphism each extend \(A\rightarrow A_{fg}\) and \(A \rightarrow (A_f)_{\epsilon(g)}\), so by the same uniqueness they are identity maps. That is, \((\Spec\epsilon\vert^{D(f)})^\sharp\) is an isomorphism on every basic open set, and since stalks are computed as direct limits over basic open neighborhoods, it is an isomorphism at every stalk. Therefore by [Topology] §Sheaves, ⁋Proposition 4, \((\Spec\epsilon\vert^{D(f)})^\sharp\) is a sheaf isomorphism.
In particular, by definition a scheme can be covered by affine schemes, and since these have a base consisting of principal open sets, any scheme has a base of affine open subsets.
The following lemma shows, by a similar argument, that any open subset of a scheme is always a scheme.
Lemma 3 Let \(U\) be an arbitrary open subset of a scheme \((X, \mathcal{O}_X)\). Then \((U, \mathcal{O}_X\vert_U)\) is also a scheme.
Proof
Let \(x\) be an arbitrary point of the open subset \(U\). Since \(X\) is a scheme, there exist a suitable open neighborhood \(V\) of \(x\) (in \(X\)) and a ring \(A\) such that \((V, \mathcal{O}_X\vert_V)\cong(\Spec A, \mathcal{O}_{\Spec A})\). Also, since \(U\) is open, \(U\cap V\) is an open subset of \(V\). Now by §The Spectrum, ⁋Lemma 11, there exist \(f_i\in A\) satisfying
\[U\cap V=\bigcup_{i\in I} D(f_i)\qquad\text{in $\Spec A$}\]so \(x\in D(f_i)\) for some \(i\). Now \((D(f_i), \mathcal{O}_X\vert_{D(f_i)})\) is an affine scheme and \(x\in D(f_i)\subseteq V\), giving the desired result.
Definition 4 For any open subset \(U\) of a scheme \((X, \mathcal{O}_X)\), we call the scheme \((U, \mathcal{O}_X\vert_U)\) an open subscheme of \(X\).
Algebraic Functions on a Scheme
In §Affine Scheme, §§Algebraic functions on \(\Spec A\), we decided to think of the value of a function \(f\in A\) at an arbitrary point \(\mathfrak{p}\) as the image of \(f\) in \(A/\mathfrak{p}\). Applying this directly to a general scheme \((X, \mathcal{O}_X)\) is somewhat tricky, not only because this process is unclear in \((X, \mathcal{O}_X)\), but also because even if we repeat the above discussion using an affine open neighborhood of \(x\in X\), it is not obvious that this is independent of the choice of affine open neighborhood.
To resolve this, we use a little ingenuity and make the following definition.
Definition 5 Consider an arbitrary point \(x\in X\) of a scheme \((X, \mathcal{O}_X)\) and the local ring \((\mathcal{O}_{X, x}, \mathfrak{m}_x)\). Then the residue field of \(X\) at \(x\) is defined as \(\mathcal{O}_{X,x}/\mathfrak{m}_x\), and we denote this by \(\kappa(x)\). For a function \(f\in \mathcal{O}_X(U)\) defined on an open subset \(U\) containing \(x\), the value of \(f\) at \(x\) is defined as the image of \(f\) in \(\kappa(x)\).
Choose an arbitrary point \(x\in X\) of a scheme \(X\) and an affine open neighborhood \((U, \mathcal{O}_X\vert_U)\cong(\Spec A, \mathcal{O}_{\Spec A})\) of \(x\). Then, writing \(\mathfrak{p}_x\) for the point of \(\Spec A\) corresponding to \(x\in U\), we verified from the isomorphism (\(\ast\)) above that \(\mathcal{O}_{X,x}\cong \mathcal{O}_{\Spec A, \mathfrak{p}_x}\), and then from §Affine Scheme, ⁋Lemma 8 we obtain the isomorphism
\[\mathcal{O}_{X,x}\cong \mathcal{O}_{\Spec A, \mathfrak{p}_x}\cong A_{\mathfrak{p}_x}\]and in particular this isomorphism corresponds the maximal ideal \(\mathfrak{p}_xA_{\mathfrak{p}_x}\) of the local ring \(A_{\mathfrak{p}_x}\) with \(\mathfrak{m}_x\). Therefore, recalling the definition in [Commutative Algebra] §Localization, ⁋Definition 10, we obtain
\[\kappa(x)=\mathcal{O}_{X,x}/\mathfrak{m}_x\cong A_{\mathfrak{p}_x}/\mathfrak{p}_xA_{\mathfrak{p}_x}=\kappa(\mathfrak{p}_x)\]On the other hand, since localization and quotient commute ([Commutative Algebra] §Properties of Localization, ⁋Proposition 2), we obtain
\[\kappa(x)\cong A_{\mathfrak{p}_x}/\mathfrak{p}_xA_{\mathfrak{p}_x}\cong \Frac(A/\mathfrak{p}_x)\]and therefore we see that Definition 5 generalizes the notion of function values on an affine scheme well. Then from this expression, for any \(f\in \mathcal{O}_X(X)\) define
\[X_f=\{x\in X\mid\text{$f_x\neq 0$ in $\kappa(x)$}\}=\{x\in X\mid f_x\not\in \mathfrak{m}_x\}\]Then for any \(x\in X\), choose an affine open neighborhood \(U\cong\Spec A\) containing \(x\) and let \(a\in A\) be the element corresponding to \(f\vert_U\) under this isomorphism. Then from the above discussion, the value of \(f\) at each \(\mathfrak{p}\in \Spec A\) is the image of \(a\) in \(\kappa(\mathfrak{p})\), so \(U\cap X_f=D(a)\), and therefore by §The Spectrum, ⁋Lemma 11 we know that \(X_f\) is an open subset of \(X\).
On the other hand, at a point \(x\in X\) a function \(f\) can be examined not only by its value but also by using its germ. Let us make the following definition.
Definition 6 For a scheme \((X, \mathcal{O}_X)\) and \(f\in \mathcal{O}_X(X)\), the support of \(f\) is given by the following formula:
\[\supp(f)=\{x\in X\mid f_x\neq 0\text{ in $\mathcal{O}_{X,x}$}\}\]Here \(f_x\) denotes the stalk of \(f\) at \(x\).
Then if the stalk of \(f\) at \(x\in X\) is zero, we can choose a suitable open neighborhood \(U=\Spec A\) of \(x\) so that \(f\) is identically zero on \(U\), and therefore we know that \(\supp(f)\) is a closed subset of \(X\). On the other hand, if \(f_x=0\) in \(\mathcal{O}_{X,x}\) then it is trivial that \(f_x\) is zero in \(\kappa(x)\), so the following inclusion
\[X\setminus \supp(f)\subseteq X\setminus X_f\iff X_f\subseteq \supp(f)\]holds.
Examples of Schemes
For convenience, from now on we will omit the structure sheaf of a scheme and write simply \(X\) instead of \((X, \mathcal{O}_X)\), and write simply \(\varphi\) instead of \((\varphi, \varphi^\sharp)\) for a scheme morphism.
Example 7 In [Algebraic Varieties] §Affine Varieties, ⁋Definition 1 we defined the (classical) affine \(n\)-space as
\[\mathbb{A}_{\mathbb{K},\mathrm{classical}}^n=\{(x_1,\ldots, x_n)\mid x_i\in \mathbb{K}\}=\MaxSpec \mathbb{K}[\x_1,\ldots, \x_n]\]Generalizing this, from now on we define the affine \(n\)-space over \(\mathbb{K}\) as
\[\mathbb{A}_\mathbb{K}^n=\Spec \mathbb{K}[\x_1,\ldots, \x_n]\]More generally, replacing the field \(\mathbb{K}\) by an arbitrary ring \(A\), we obtain the affine \(n\)-space over \(A\):
\[\mathbb{A}_A^n=\Spec A[\x_1,\ldots, \x_n]\]Passing from the maximal spectrum \(\MaxSpec A\) to \(\Spec A\) not only makes algebra easier, but also has geometric meaning. That is, now the points of \(\mathbb{A}^n\) represent not only the points of the \(n\)-space \(\mathbb{A}_{\mathbb{K},\mathrm{classical}}^n\) in the traditional sense, but each irreducible closed subset of this space is also represented by a single point.
Earlier we mentioned that an open subset of an affine scheme need not be an affine scheme; the following example illustrates this.
Example 8 Consider the affine plane \(\mathbb{A}_\mathbb{K}^2=\Spec \mathbb{K}[\x_1,\x_2]\). Then
\[\{(0,0)\}=Z(\x_1)\cap Z(\x_2)\]so \(\{(0,0)\}\) is a closed set, and therefore
\[U=\mathbb{A}_\mathbb{K}^2\setminus\{0\}=D(\x_1)\cup D(\x_2)\]is an open set. On the other hand, functions defined on \(D(\x_1)\) and \(D(\x_2)\) are of the form
\[\mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(D(\x_1))\cong \mathbb{K}[\x_1,\x_2]_{\x_1}=\mathbb{K}[\x_1,\x_2, \x_1^{-1}],\qquad \mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(D(\x_2))\cong \mathbb{K}[\x_1,\x_2]_{\x_2}=\mathbb{K}[\x_1,\x_2, \x_2^{-1}]\]Now functions on \(D(\x_1)\cup D(\x_2)\) are those obtained by gluing these, and such functions must agree when restricted to the intersection \(D(\x_1)\cap D(\x_2)=D(\x_1\x_2)\). Now comparing elements of \(\mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(D(\x_1))\) and \(\mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(D(\x_2))\) in
\[\mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(D(\x_1\x_2))=\mathbb{K}[\x_1,\x_2]_{\x_1\x_2}\]we see that such functions are only polynomial functions. That is,
\[\mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(U)=\mathbb{K}[\x_1,\x_2]\]If \((U, \mathcal{O}_{\mathbb{A}_\mathbb{K}^2}\vert_U)\) were affine, then \(U\) would have to be isomorphic to \(\Spec \mathbb{K}[\x_1,\x_2]\) obtained by taking the global sections, but considering the point of \(U\) corresponding to the prime ideal \((\x_1,\x_2)\),
\[Z(\x_1,\x_2)=Z(\x_1)\cap Z(\x_2)=\emptyset\qquad\text{ in $U$}\]so this is impossible.
By definition a scheme is made by gluing affine schemes, and in the above example we also understood the sections on \(U\) by thinking of it as glued from \(D(\x_1)\) and \(D(\x_2)\). Before looking at more examples, it will be helpful to first examine the following lemma, which will be useful in handling various examples.
Lemma 9 Fix an index set \(I\), and suppose the following data are given.
- Schemes \(X_i\),
- Open subschemes \(X_{ij}\) of \(X_i\),
- For each \(i,j\in I\), isomorphisms \(\varphi_{ij}:X_{ij} \rightarrow X_{ji}\)
Here we define \(X_{ii}=X_i\) and \(\varphi_{ii}=\id_{X_i}\). Then if these data satisfy the cocycle condition
\[\varphi_{ik}\vert_{X_{ij}\cap X_{ik}}=\varphi_{jk}\vert_{X_{ji}\cap X_{jk}}\circ \varphi_{ij}\vert_{X_{ij}\cap X_{ik}}\qquad\text{for all $i,j,k\in I$}\]there exist a scheme \(X\), open subschemes \(U_i\) covering \(X\), and isomorphisms \(\sigma_i: X_i \rightarrow U_i\) such that for each \(i,j\),
\[\sigma_i(X_{ij})=U_i\cap U_j,\qquad \sigma_j\circ\varphi_{ij}=\sigma_i\quad\text{on $X_{ij}$}\]Moreover, such \((X, (\sigma_i)_{i\in I})\) is unique. That is, if \((X', (\sigma_i')_{i\in I})\) also satisfies the same conditions, then there exists a unique isomorphism \(\chi: X\rightarrow X'\) such that \(\chi\circ \sigma_i=\sigma_i'\) for all \(i\).
Proof
First, for the composition on the right-hand side of the cocycle condition to be defined, we must have \(\varphi_{ij}(X_{ij}\cap X_{ik})\subseteq X_{ji}\cap X_{jk}\), which we understand as part of the given data. Now the cocycle condition in the case \(k=i\), together with \(X_{ii}=X_i\) and \(\varphi_{ii}=\id_{X_i}\), gives
\[\id_{X_{ij}}=\varphi_{ji}\circ\varphi_{ij}\]so \(\varphi_{ji}=\varphi_{ij}^{-1}\), and therefore applying the above inclusion to \((i,j,k)\) and \((j,i,k)\) respectively yields
\[\varphi_{ij}(X_{ij}\cap X_{ik})=X_{ji}\cap X_{jk}\]Now on the set \(\coprod_{i\in I} X_i\), define a relation \(\sim\) for \(x\in X_i\) and \(y\in X_j\) by
\[x\sim y\iff \text{$x\in X_{ij}$ and $\varphi_{ij}(x)=y$}\]Then from \(X_{ii}=X_i\) and \(\varphi_{ii}=\id_{X_i}\), the relation \(\sim\) is reflexive, and from \(\varphi_{ji}=\varphi_{ij}^{-1}\) it is symmetric. To show transitivity, suppose \(x\in X_i\), \(y=\varphi_{ij}(x)\), and \(z=\varphi_{jk}(y)\) are given. Then \(y\in X_{ji}\cap X_{jk}\), so by the above equality \(x\in X_{ij}\cap X_{ik}\), and by the cocycle condition
\[z=\varphi_{jk}(\varphi_{ij}(x))=\varphi_{ik}(x)\]so \(x\sim z\). That is, \(\sim\) is an equivalence relation.
Now consider the set \(X=\coprod_{i\in I}X_i\big/{\sim}\) and the canonical map \(\sigma_i: X_i \rightarrow X\), and give \(X\) the final topology with respect to the \(\sigma_i\). ([Topology] §Initial and Final Topology, ⁋Definition 4) Then by [Topology] §Initial and Final Topology, ⁋Proposition 5, \(U\subseteq X\) is open if and only if \(\sigma_i^{-1}(U)\) is open in \(X_i\) for all \(i\), which is exactly the quotient space topology on \(\coprod_{i\in I} X_i\). ([Topology] §Quotient Spaces, ⁋Definition 3)
First, each \(\sigma_i\) is injective, because for \(x,x'\in X_i\), \(x\sim x'\) is equivalent to \(\varphi_{ii}(x)=x'\), i.e., \(x=x'\). Now let \(U_i=\sigma_i(X_i)\). Then for any \(y\in X_j\), \(\sigma_j(y)\in U_i\) is equivalent to the existence of an element of \(X_i\) equivalent to \(y\), i.e., \(y\in X_{ji}\), so
\[\sigma_j^{-1}(U_i)=X_{ji}\]and therefore \(U_i\) is an open subset of \(X\). By the same calculation, for any open subset \(V\) of \(X_i\),
\[\sigma_j^{-1}(\sigma_i(V))=\varphi_{ij}(V\cap X_{ij})\]which is an open subset of \(X_{ji}\), hence of \(X_j\), so \(\sigma_i\) is an open map. From the above, \(\sigma_i\) is a homeomorphism from \(X_i\) onto the open subset \(U_i\), and from the above calculation
\[\sigma_i(X_{ij})=U_i\cap U_j,\qquad \sigma_j\circ\varphi_{ij}=\sigma_i\quad\text{on $X_{ij}$}\]also holds.
Now we define the structure sheaf on \(X\). Since each \(\sigma_i\) is a homeomorphism, the pushforward \((\sigma_i)_\ast\mathcal{O}_{X_i}\) is a sheaf on \(U_i\). ([Topology] §Sheaves, ⁋Example 9) On the other hand, since \(\varphi_{ij}\) is an isomorphism of schemes, for any open subset \(W\subseteq U_i\cap U_j\), \(\varphi_{ij}^\sharp\) gives an isomorphism
\[\mathcal{O}_{X_j}(\sigma_j^{-1}(W)) \rightarrow \mathcal{O}_{X_i}(\sigma_i^{-1}(W))\]and writing its inverse as \(\theta_{ij}(W)\), we obtain a sheaf isomorphism on \(U_i\cap U_j\)
\[\theta_{ij}: ((\sigma_i)_\ast\mathcal{O}_{X_i})\vert_{U_i\cap U_j} \rightarrow ((\sigma_j)_\ast\mathcal{O}_{X_j})\vert_{U_i\cap U_j}\]Then the cocycle condition is an equality of scheme morphisms including the sheaf components, so this means exactly that \(\theta_{jk}\circ\theta_{ij}=\theta_{ik}\) holds on \(U_i\cap U_j\cap U_k\), and in particular \(\theta_{ii}=\id\) and \(\theta_{ji}=\theta_{ij}^{-1}\). Now for any open subset \(U\subseteq X\) define
\[\mathcal{O}_X(U)=\left\{(s_i)_{i\in I}\in \prod_{i\in I}\mathcal{O}_{X_i}(\sigma_i^{-1}(U))\middle\vert \theta_{ij}\left(s_i\vert_{\sigma_i^{-1}(U\cap U_j)}\right)=s_j\vert_{\sigma_j^{-1}(U\cap U_i)}\quad\text{for all $i,j$}\right\}\]and define restriction maps componentwise. Then \(\mathcal{O}_X\) becomes a sheaf, which we verify by checking the two conditions of [Topology] §Sheaves, ⁋Definition 1 componentwise. That is, the identity axiom is trivial since each \((\sigma_i)_\ast \mathcal{O}_{X_i}\) is a sheaf, and for the gluability axiom, the fact that the resulting \((s_i)\) obtained by gluing the given sections componentwise satisfies the above condition follows from the fact that this condition holds on each element of the given open covering and that \(\theta_{ij}\) commutes with restriction, again by the identity axiom.
If \(U\subseteq U_i\), then for any \(k\) we have \(\sigma_k^{-1}(U)\subseteq \sigma_k^{-1}(U_i)=X_{ki}\), so the above condition can be rewritten as \(s_k=\theta_{ik}(s_i\vert_{\sigma_i^{-1}(U\cap U_k)})\). That is, \((s_k)_{k\in I}\) is completely determined by \(s_i\), and conversely for any \(s_i\in \mathcal{O}_{X_i}(\sigma_i^{-1}(U))\), the fact that \((s_k)_{k\in I}\) defined by this formula satisfies the above condition follows from \(\theta_{kl}\circ\theta_{ik}=\theta_{il}\). Therefore the projection \((s_k)_{k\in I}\mapsto s_i\) defines a sheaf isomorphism
\[\mathcal{O}_X\vert_{U_i}\cong (\sigma_i)_\ast\mathcal{O}_{X_i}\]and thinking of this together with the homeomorphism \(\sigma_i\), we have \((X_i,\mathcal{O}_{X_i})\cong (U_i, \mathcal{O}_X\vert_{U_i})\) as locally ringed spaces. In particular, for any \(x\in X_i\), \(\mathcal{O}_{X, \sigma_i(x)}\cong \mathcal{O}_{X_i,x}\) is a local ring, so \((X,\mathcal{O}_X)\) is a locally ringed space, and since \(X_i\) is a scheme, choosing an affine open neighborhood \(V\subseteq X_i\) of \(x\) makes \(\sigma_i(V)\) an affine open neighborhood of \(\sigma_i(x)\). Since the \(U_i\) cover \(X\), it follows that \((X,\mathcal{O}_X)\) is a scheme in the sense of Definition 1. Also, \(U_i\) is an open subscheme of \(X\) (Lemma 3), and restricting the above isomorphism to \(U_i\cap U_j=\sigma_i(X_{ij})\) yields \(U_i\cap U_j\cong X_{ij}\).
Finally we show uniqueness. Let \(X'\) be a scheme, \(U_i'\) open subschemes covering \(X'\), and \(\psi_i: X_i \rightarrow U_i'\) isomorphisms satisfying the two conditions of the claim, i.e., \(\psi_i(X_{ij})=U_i'\cap U_j'\) and \(\psi_j\circ\varphi_{ij}=\psi_i\) on \(X_{ij}\). Then we must show that there exists a unique isomorphism \(\chi:X \rightarrow X'\) such that \(\chi\circ \sigma_i=\psi_i\) for all \(i\).
First, as a function between sets, \(\chi\) is defined by \(\chi(\sigma_i(x))=\psi_i(x)\), and this is well-defined because whenever \(x\in X_{ij}\), we have \(\psi_j(\varphi_{ij}(x))=\psi_i(x)\). Since each restriction \(\chi\vert_{U_i}=\psi_i\circ \sigma_i^{-1}\) is continuous and the \(U_i\) form an open covering of \(X\), by [Topology] §Presheaves, ⁋Lemma 1 \(\chi\) is a continuous function. Also, since \(X'=\bigcup U_i'\), \(\chi\) is surjective, and if \(\chi(\sigma_i(x))=\chi(\sigma_j(y))\) then
\[\psi_i(x)=\psi_j(y)\in U_i'\cap U_j'=\psi_i(X_{ij})=\psi_j(X_{ji})\]so \(x\in X_{ij}\) and \(y\in X_{ji}\), and now from \(\psi_j(\varphi_{ij}(x))=\psi_i(x)=\psi_j(y)\) and the injectivity of \(\psi_j\) we obtain \(\varphi_{ij}(x)=y\), i.e., \(\sigma_i(x)=\sigma_j(y)\). Therefore \(\chi\) is bijective, and since each \(\chi\vert_{U_i}\) is a homeomorphism onto the open subset \(U_i'\), \(\chi\) is a homeomorphism.
On the other hand, the sheaf components \(\psi_i^\sharp\) of the \(\psi_i\) give isomorphisms \(\mathcal{O}_{X'}\vert_{U_i'} \rightarrow (\chi\vert_{U_i})_\ast (\mathcal{O}_X\vert_{U_i})\), and since the condition \(\psi_j\circ\varphi_{ij}=\psi_i\) holds as scheme morphisms, these agree with each other on \(U_i'\cap U_j'\). Therefore by the two conditions of [Topology] §Sheaves, ⁋Definition 1 they glue to a unique morphism \(\chi^\sharp: \mathcal{O}_{X'} \rightarrow \chi_\ast\mathcal{O}_X\), which is a sheaf isomorphism since it is an isomorphism on all stalks. ([Topology] §Sheaves, ⁋Proposition 4) That is, \(\chi=(\chi,\chi^\sharp)\) is an isomorphism of schemes. Finally, the condition \(\chi\circ \sigma_i=\psi_i\) determines \(\chi\) as a function between topological spaces and as a sheaf morphism on the restriction to \(U_i\), and since the \(U_i\) cover \(X\), such \(\chi\) is unique by the identity axiom.
That is, we can glue schemes along open subsets to make a new scheme. Through this, let us see the following example.
Example 10 Let two affine lines \(X_0=\mathbb{A}_\mathbb{K}^1=\Spec \mathbb{K}[\x_0]\), \(X_1=\mathbb{A}_\mathbb{K}^1=\Spec \mathbb{K}[\x_1]\), and their open subsets
\[U_0=X_0\setminus \{(\x_0)\}=D(\x_0),\quad U_1=X_1\setminus \{(\x_1)\}=D(\x_1)\]be given. Then by definition
\[\mathcal{O}_{X_0}(U_0)\cong \mathbb{K}[\x_0]_{\x_0}=\mathbb{K}[\x_0,1/\x_0]\]and similarly \(\mathcal{O}_{X_1}(U_1)\cong \mathbb{K}[\x_1,1/\x_1]\). In this example we concretely examine two different ways of gluing \(X_0\) and \(X_1\).
First, consider the case where the isomorphism \(\varphi:(U_0, \mathcal{O}_{X_0}\vert_{U_0})\rightarrow (U_1, \mathcal{O}_{X_1}\vert_{U_1})\) comes from the isomorphism \(\mathbb{K}[\x_0,1/\x_0]\rightarrow \mathbb{K}[\x_1,1/\x_1]\) identifying \(\x_0\) and \(\x_1\). Then as a topological space \(X=X_0\cup_\varphi X_1\) is the line with double origin, and the structure sheaf is defined for any open subset \(U\subseteq X\) by the following formula:
\[\mathcal{O}_X(U)=\mathcal{O}_{X_0}(U\cap X_0)\times_{\mathcal{O}_{X_0}(U\cap U_0)\cong \mathcal{O}_{X_1}(U\cap U_1)} \mathcal{O}_{X_1}(U\cap X_1)\]In particular the global sections are
\[\Gamma(X, \mathcal{O}_X)=\mathcal{O}_{X_0}(X_0)\times_{\mathcal{O}_{X_0}(U_0)\cong \mathcal{O}_{X_1}(U_1)} \mathcal{O}_{X_1}(X_1)=\mathbb{K}[\x_0]\times_{\mathbb{K}[\x_0,1/\x_0]\cong \mathbb{K}[\x_1,1/\x_1]}\mathbb{K}[\x_1]\cong \mathbb{K}[\x_0]\]On the other hand, this time consider the case where the isomorphism \(\varphi\) comes from the isomorphism \(\mathbb{K}[\x_0,1/\x_0]\rightarrow \mathbb{K}[\x_1,1/\x_1]\) identifying \(\x_0\) and \(1/\x_1\). Then in particular, the closed point \((\x_0-\alpha)\) of \(U_0\) will correspond to the closed point \((\x_1-1/\alpha)\) of \(X_1\), and the closed point \((\x_1-\beta)\) of \(U_1\) corresponds to the closed point \((\x_0-1/\beta)\) of \(U_0\). That is, the space obtained from this will be the projective space \(\mathbb{P}^1\).
Thinking about the global sections in this case,
\[\Gamma(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1})=\mathbb{K}[\x_0]\times_{\mathbb{K}[\x_0,1/\x_0]\cong \mathbb{K}[\x_1,1/\x_1]} \mathbb{K}[\x_1]\cong \mathbb{K}\]only. The last isomorphism means, in everyday language, that among the elements of \(\mathbb{K}[\x_0]\), only constant functions remain elements of \(\mathbb{K}[\x_1]\) when \(\x_0\) is replaced by \(1/\x_1\).
The last calculation in Example 10 above shows that \(\mathbb{P}^1\) is not an affine scheme. If \(\mathbb{P}^1\) were an affine scheme, it would necessarily have to be \(\Spec \mathbb{K}\), because \(\Spec \mathbb{K}\) is a scheme consisting of a single point.
References
[Har] R. Hartshorne, Algebraic geometry. Graduate texts in mathematics. Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundation of algebraic geometry. Available online.
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