Schemes

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 Schemes

We may now 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. 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, letting \(\mathfrak{p}_x\) denote the point of \(\Spec A\) corresponding to \(x\in U\) via the above isomorphism, we have

\[\mathcal{O}_{X,x}=\varinjlim_{V\ni x} \mathcal{O}_X(V)=\varinjlim_{U\supseteq V\ni x} \mathcal{O}_X(V)=\mathcal{O}_{U, x}\cong \mathcal{O}_{\Spec A, \mathfrak{p}_x}\tag{$\ast$}\]

In general, the restriction \((U, \mathcal{O}_{\Spec A}\vert_U)\) of an affine scheme \((\Spec A, \mathcal{O}_{\Spec A})\) to an arbitrary open subset is not always an affine scheme. (Example 8) However, a 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 set \(D(f)\) is always an affine scheme via the isomorphism (\(\ast\ast\)) above.

Proof

From the third result of §Spectra, ⁋Proposition 9, we know that the injective continuous map \(\Spec\epsilon: \Spec A_f \rightarrow A\) between topological spaces gives a homeomorphism \(D(f)\cong \Spec A_f\), but to justify the above isomorphism we must verify that this is a morphism of schemes.

First, the ring homomorphism \(\epsilon: A \rightarrow A_f\) induces a scheme morphism

\[(\Spec \epsilon, (\Spec\epsilon)^\sharp): (\Spec A_f, \mathcal{O}_{\Spec A_f}) \rightarrow (\Spec A, \mathcal{O}_A)\]

and the homeomorphism between \(D(f)\) and \(\Spec A_f\) arises from the following decomposition, once we consider the canonical inclusion \(\iota: D(f)\hookrightarrow \Spec A\):

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 \(\Sh(D(f), \cRing)\) for the homeomorphism \(\Spec\epsilon\vert^{D(f)}\). 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 \((\Spec\epsilon)^\sharp:\mathcal{O}_{\Spec A} \rightarrow (\Spec \epsilon)_\ast \mathcal{O}_{\Spec A_f}\), and that this is an isomorphism follows essentially from the fact that \(\Spec A_f\) is an open subset of \(\Spec A\), as can be seen by tracing through the proof of that lemma.

In particular, by definition a scheme can be covered by affine schemes, and since these have a base 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 again 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 §Spectra, ⁋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 that \(x\in D(f_i)\) for some \(i\). Since \((D(f_i), \mathcal{O}_X\vert_{D(f_i)})\) is an affine scheme and \(x\in D(f_i)\subseteq V\), we obtain the desired result.

Definition 4 For an arbitrary 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 Schemes

In §Affine Schemes, §§Algebraic Functions on \(\Spec A\), we regarded 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 delicate, because the above procedure is not transparent for \((X, \mathcal{O}_X)\), and moreover, even if we repeat the above discussion using an affine open neighborhood of \(x\in X\), it is not obvious that the result is independent of the choice of affine open neighborhood.

To resolve this, we proceed as follows.

Definition 5 Consider a scheme \((X, \mathcal{O}_X)\), a point \(x\in X\), and the local ring \((\mathcal{O}_{X, x}, \mathfrak{m}_x)\). The residue field of \(X\) at \(x\) is defined as \(\mathcal{O}_{X,x}/\mathfrak{m}_x\), and we denote it 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 Schemes, ⁋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 matches 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 have

\[\kappa(x)\cong A_{\mathfrak{p}_x}/\mathfrak{p}_xA_{\mathfrak{p}_x}\cong \Frac(A/\mathfrak{p}_x)\]

and thus we see that Definition 5 generalizes the notion of function values on affine schemes correctly. From this expression, for any \(f\in \mathcal{O}_X(X)\), let us 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 \(x\in X_f\), choosing an affine open neighborhood \(U=\Spec A\) containing \(x\), we know from the above discussion that \(U\cap X_f\) equals the open subset \(D(f\vert_{U\cap X_f})\), and thus by §Spectra, ⁋Lemma 11 we see that \(X_f\) is an open subset of \(X\).

Meanwhile, at a point \(x\in X\), a function \(f\) can be examined not only through its value but also through its germ. Let us define the following.

Definition 6 For a scheme \((X, \mathcal{O}_X)\) and \(f\in \mathcal{O}_X(X)\), the support of \(f\) is given by

\[\supp(f)=\{x\in X\mid f_x\neq 0\text{ in $\mathcal{O}_{X,x}$}\}\]

Here \(f_x\) denotes the stalk of \(f\) at \(x\).

Then if the stalk of \(f\) at \(x\in X\) is \(0\), we can choose a suitable open neighborhood \(U=\Spec A\) of \(x\) so that \(f\) is identically \(0\) on \(U\), and thus we see 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 obvious that \(f_x\) is \(0\) in \(\kappa(x)\), so the 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 shall denote a scheme simply by \(X\) instead of \((X, \mathcal{O}_X)\), omitting the structure sheaf, and denote a scheme morphism simply by \(\varphi\) instead of \((\varphi, \varphi^\sharp)\).

Example 7 In §Spectra, §§Classical algebraic geometry, we defined the (classical) affine \(n\)-space as

\[Z(\x(\x-1))=\mathbb{A}_{\mathbb{K},\mathrm{classical}}^n=\{(x_1,\ldots, x_n)\mid x_i\in \mathbb{K}\}=\mSpec \mathbb{K}[\x_1,\ldots, \x_n]\]

Generalizing this, we shall henceforth 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 \(A\)

\[\mathbb{A}_A^n=\Spec A[\x_1,\ldots, \x_n]\]

Moving from the maximal spectrum \(\mSpec A\) to \(\Spec A\) not only makes algebra easier, but is also geometrically meaningful. Now each point of \(\mathbb{A}^n\) represents not only a point of the \(n\)-space \(\mathbb{A}_{\mathbb{K},\mathrm{classical}}^n\) in the traditional sense, but also treats closed subsets appearing as zero loci of functions on this space as points.

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. Meanwhile, functions defined on \(D(\x_1)\) and \(D(\x_2)\) are of the forms

\[\mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(D(\x_1))\cong \mathbb{K}[\x_1,\x_2]_{\x_1}=\mathbb{K}[\x_1,\x_2, \x_1^{-1}],\qquad \mathcal{O}_{\mathbb{A}_\mathbb{K}^2}(D(\x_2))\cong \mathbb{K}[\x_1,\x_2]_{\x_2}=\mathbb{K}[\x_1,\x_2, \x_2^{-1}]\]

Now functions on \(D(\x_1)\cup D(\x_2)\) are those obtained by gluing these functions, and such functions must agree when restricted to the intersection \(D(\x_1)\cap D(\x_2)=D(\x_1\x_2)\). 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 global sections, but considering the point in \(U\) 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 too, to compute sections over \(U\) we understood it as gluing \(D(\x_1)\) and \(D(\x_2)\). Before looking at more examples, let us first examine the following lemma, which will be helpful 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 unique scheme \(X\) and open subschemes \(U_i\) of \(X\) such that \(U_i\cong X_i\) and \(U_i\cap U_j\cong X_{ij}\).

The proof proceeds as follows: first, considering the topological space \(\coprod X_i\), we can define a relation \(\sim\) on it by

\[(x_i, X_i)\sim (x_j,X_j) \iff \varphi_{ij}(x_i)=x_j\]

and using the cocycle condition we can prove that this is an equivalence relation. Thus we define the topological space as \(X=\coprod_{i\in X_i}X_i\big/{\sim}\), and then it suffices to verify that sheaves can be glued in the same way as in [Topology] §Presheaves, ⁋Lemma 1. Using this, let us see the following example.

Example 10 Suppose 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

\[U_0=X_0\setminus \{(\x_0)\}=D(\x_0),\quad U_1=X_1\setminus \{(\x_1)\}=D(\x_1)\]

are 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 examine concretely two different ways of gluing \(X_0\) and \(X_1\).

First, consider the case where an isomorphism \(\varphi:(U_0, \mathcal{O}_{X_0}\vert_{U_0})\rightarrow (U_1, \mathcal{O}_{X_1}\vert_{U_1})\) comes from an isomorphism \(k[\x_0,1/\x_0]\rightarrow k[\x_1,1/\x_1]\) identifying \(\x_0\) with \(\x_1\). 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

\[\mathcal{O}_X(U)=\mathcal{O}_{X_0}(U\cap X_0)\times_{\mathcal{O}_{X_0}(U\cap U_0)\cong \mathcal{O}_{X_1}(U\cap U_1)} \mathcal{O}_{X_1}(U\cap X_1)\]

In particular, the global sections are

\[\Gamma(X, \mathcal{O}_X)=\mathcal{O}_{X_0}(X_0)\times_{\mathcal{O}_{X_0}(U_0)\cong \mathcal{O}_{X_1}(U_1)} \mathcal{O}_{X_1}(X_1)=\mathbb{K}[\x_0]\times_{\mathbb{K}[\x_0,1/\x_0]\cong \mathbb{K}[\x_1,1/\x_1]}\mathbb{K}[\x_1]\cong \mathbb{K}[\x_0]\]

Meanwhile, this time consider the case where the isomorphism \(\varphi\) comes from an isomorphism \(\mathbb{K}[\x_0,1/\x_0]\rightarrow \mathbb{K}[\x_1,1/\x_1]\) identifying \(\x_0\) with \(1/\x_1\). Then in particular, the closed point \((\x_0-\alpha)\) of \(U_0\) corresponds 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 gluing is the projective space \(\mathbb{P}^1\).

In this case the global sections are

\[\Gamma(\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1})=\mathbb{K}[\x_0]\times_{\mathbb{K}[\x_0,1/\x_0]\cong \mathbb{K}[\x_1,1/\x_1]} \mathbb{K}[\x_1]\cong \mathbb{K}\]

and nothing more. The last isomorphism means, in plain terms, that among the elements of \(\mathbb{K}[\x_0]\), only the constants 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.

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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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