2_2x Schemes

II.2: Schemes

II.2.1 #to_work

Let \(A\) be a ring, let \(X=\operatorname{Spec} A\), let \(f \in A\) and let \(D(f) \subseteq X\) be the open complement of \(V(f)\). Show that the locally ringed space \(\left(D(f),{ \left.{{{\mathcal{O}}_X}} \right|_{{D(f)}} } \right)\) is isomorphic to \(\operatorname{spec} A_f\).

II.2.2 #to_work

Let \(\left(X, \mathcal{O}_X\right)\) be a scheme, and let \(U \subseteq X\) be any open subset. Show that \(\left(U,\left.\mathcal{O}_X\right|_U\right)\) is a scheme. We call this the induced scheme structure on the open set \(U\), and we refer to \(\left(U,\left.\mathcal{O}_X\right|_U\right)\) as an open subscheme of \(X\).

II.2.3 Reduced Schemes #to_work

A scheme \(\left(X, \mathcal{O}_X\right)\) is reduced if for every open set \(U \subseteq X\), the ring \(\mathcal{O}_X(U)\) has no nilpotent elements.

• Show that \(\left(X, \mathcal{O}_X\right)\) is reduced if and only if for every \(P \in X\), the local ring \(\mathcal{O}_{X, P}\) has no nilpotent elements.

• Let \(\left(X, \mathcal{O}_X\right)\) be a scheme. Let \(\left(\mathcal{O}_X\right)_{\text {red }}\) be the sheaf associated to the presheaf \(U \mapsto \mathcal{O}_X(U)_{\text {red }}\), where for any ring \(A\), we denote by \(A_{\text {red }}\) the quotient of \(A\) by its ideal of nilpotent elements. Show that \(\left(X,\left({\mathcal{O}}_X\right)_{\text {red }}\right)\) is a scheme. We call it the reduced scheme associated to \(X\), and denote it by \(X_{\text {red }}\). Show that there is a morphism of schemes \(X_{\text {red }} \rightarrow X\), which is a homeomorphism on the underlying topological spaces.

• Let \(f: X \rightarrow Y\) be a morphism of schemes, and assume that \(X\) is reduced. Show that there is a unique morphism \(g: X \rightarrow Y_{\mathrm{red}}\) such that \(f\) is obtained by composing \(g\) with the natural map \(Y_{\text {red }} \rightarrow Y\).

II.2.4 #to_work

Let \(A\) be a ring and let \(\left(X, \mathcal{O}_X\right)\) be a scheme. Given a morphism \(f: X \rightarrow \operatorname{Spec} A\), we have an associated map on sheaves \(f^{\sharp}: \mathcal{U}_{\operatorname{Spec}(A)} \rightarrow f_* \mathcal{O}_X\). Taking global sections we obtain a homomorphism \(A \rightarrow \Gamma\left(X, {\mathcal{O}}_X\right)\). Thus there is a natural map \begin{align*} \alpha: \operatorname{Hom}_{\mathsf{Sch}}(X, \operatorname{Spec} A) \rightarrow \operatorname{Hom}_{\mathsf{Ring}} \left(A, \Gamma\left(X,\left.{\mathcal{O}}_X\right)\right) .\right. \end{align*} Show that \(\alpha\) is bijective (cf. (I, 3.5) for an analogous statement about varieties).

II.2.5 #to_work

Describe Spec \(\mathbf{Z}\), and show that it is a final object for the category of schemes, i.e., each scheme \(X\) admits a unique morphism to Spec \(\mathbf{Z}\).

II.2.6 #to_work

Describe the spectrum of the zero ring, and show that it is an initial object for the category of schemes. (According to our conventions, all ring homomorphisms must take 1 to 1 . Since \(0=1\) in the zero ring, we see that each ring \(R\) admits a unique homomorphism to the zero ring, but that there is no homomorphism from the zero ring to \(R\) unless \(0=1\) in \(R\).)

II.2.7 #to_work

Let \(X\) be a scheme. For any \(x \in X\), let \(\mathcal{O}_x\) be the local ring at \(x\), and \(\mathfrak{m}_x\) its maximal ideal. We define the residue field of \(x\) on \(X\) to be the field \(k(x)=\mathcal{O}_x / \mathfrak{m}_x\). Now let \(K\) be any field. Show that to give a morphism of \(\operatorname{Spec} K\) to \(X\) it is equivalent to give a point \(x \in X\) and an inclusion map \(k(x) \rightarrow K\).

II.2.8 #to_work

Let \(X\) be a scheme. For any point \(x \in X\), we define the Zariski tangent space \(T_x\) to \(X\) at \(x\) to be the dual of the \(k(x)\)-vector space \({\mathfrak{m}}_x / {\mathfrak{m}}_x^2\). Now assume that \(X\) is a scheme over a field \(k\), and let \(k[\varepsilon] / \varepsilon^2\) be the ring of dual numbers over \(k\). Show that to give a \(k\)-morphism of \(\operatorname{Spec} k[\varepsilon] / \varepsilon^2\) to \(X\) is equivalent to giving a point \(x \in X\), rational over \(k\) (i.e., such that \(k(x)=k\) ), and an element of \(T_x\).

II.2.9 #to_work

If \(X\) is a topological space, and \(Z\) an irreducible closed subset of \(X\), a generic point for \(Z\) is a point \(\zeta\) such that \(Z=\{\zeta\}^{-}\). If \(X\) is a scheme, show that every (nonempty) irreducible closed subset has a unique generic point.

II.2.10 #to_work

Describe \(\operatorname{Spec} \mathbf{R}[x]\). How does its topological space compare to the set \(\mathbf{R}\) ? To \(\mathbf{C}\) ?

II.2.11 #to_work

Let \(k=\mathbf{F}_p\) be the finite field with \(p\) elements. Describe \(\operatorname{Spec} k[x]\). What are the residue fields of its points? How many points are there with a given residue field?

II.2.12 Glueing Lemma #to_work

Generalize the glueing procedure described in the text (2.3.5) as follows. Let \(\left\{X_i\right\}\) be a family of schemes (possible infinite). For each \(i \neq j\), suppose given an open subset \(U_{i j} \subseteq X_i\), and let it have the induced scheme structure (Ex. 2.2). Suppose also given for each \(i \neq j\) an isomorphism of schemes \(\varphi_{i j}: U_{i j} \rightarrow U_{j i}\) such that

• for each \(i, j, \varphi_{j i}=\varphi_{i j}^{-1}\), and
• for each \(i, j, k\), \(\varphi_{i j}\left(U_{i j} \cap U_{i k}\right)=U_{j i} \cap U_{j k}\), and \(\varphi_{i k}=\varphi_{j k} \circ \varphi_{i j}\) on \(U_{i j} \cap U_{i k}\).

Then show that there is a scheme \(X\), together with morphisms \(\psi_i: X_i \rightarrow X\) for each \(i\), such that

• \(\psi_i\) is an isomorphism of \(X_i\) onto an open subscheme of \(X\)
• the \(\psi_i\left(X_i\right)\) cover \(X\),
• \(\psi_i\left(U_{i j}\right)=\psi_i\left(X_i\right) \cap \psi_j\left(X_j\right)\) and (4) \(\psi_i=\psi_j \circ \varphi_{i j}\) on \(U_{i j}\).

We say that \(X\) is obtained by glueing the schemes \(X_i\) along the isomorphisms \(\varphi_{i j}\). An interesting special case is when the family \(X_i\) is arbitrary, but the \(U_{i j}\) and \(\varphi_{i j}\) are all empty. Then the scheme \(X\) is called the disjoint union of the \(X_i\), and is denoted \(\coprod X_i\).

II.2.13 #to_work

A topological space is quasi-compact if every open cover has a finite subcover.

• Show that a topological space is noetherian (I.1) if and only if every open subset is quasi-compact.

• If \(X\) is an affine scheme, show that \(\operatorname{sp}(X)\) is quasi-compact, but not in general noetherian. We say a scheme \(X\) is quasi-compact if \(\operatorname{sp}(X)\) is.

• If \(A\) is a noetherian ring, show that \(\operatorname{sp}(\operatorname{Spec} A)\) is a noetherian topological space.

• Give an example to show that \(\operatorname{sp}(\operatorname{Spec} A)\) can be noetherian even when \(A\) is not.

II.2.14 #to_work

• Let \(S\) be a graded ring. Show that \(\mathop{\mathrm{Proj}}S=\varnothing\) if and only if every element of \(S_{+}\)is nilpotent.

• Let \(\varphi: S \rightarrow T\) be a graded homomorphism of graded rings (preserving degrees). Let \(U=\left\{\mathfrak{p} \in\right.\left.T \mathrel{\Big|}\mathfrak{p} \nsupseteq \varphi\left(S_{+}\right)\right\}\). Show that \(U\) is an open subset of \(\mathop{\mathrm{Proj}}T\), and show that \(\varphi\) determines a natural morphism \(f: U \rightarrow \operatorname{Proj} S\).

• The morphism \(f\) can be an isomorphism even when \(\varphi\) is not. For example, suppose that \(\varphi_d: S_d \rightarrow T_d\) is an isomorphism for all \(d \geqslant d_0\), where \(d_0\) is an integer. Then show that \(U=\operatorname{Proj} T\) and the morphism \(f: \operatorname{Proj} T \rightarrow \operatorname{Proj} S\) is an isomorphism.

• Let \(V\) be a projective variety with homogeneous coordinate ring \(S\) (See I.2). Show that \(t(V) \cong \operatorname{Proj} S\).

II.2.15 #to_work

• Let \(V\) be a variety over the algebraically closed field \(k\). Show that a point \(P \in t(V)\) is a closed point if and only if its residue field is \(k\).

• If \(f: X \rightarrow Y\) is a morphism of schemes over \(k\), and if \(P \in X\) is a point with residue field \(k\), then \(f(P) \in Y\) also has residue field \(k\).

• Now show that if \(V, W\) are any two varieties over \(k\), then the natural map is bijective. (Injectivity is easy. The hard part is to show it is surjective.)

II.2.16 #to_work

Let \(X\) be a scheme, let \(f \in \Gamma\left(X, \mathcal{O}_X\right)\), and define \(X_f\) to be the subset of points \(x \in X\) such that the stalk \(f_x\) of \(f\) at \(x\) is not contained in the maximal ideal \(\mathfrak{m}_x\) of the local ring \(\mathcal{O}_x\).

• If \(U=\operatorname{Spec} B\) is an open affine subscheme of \(X\), and if \(\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu \in B=\Gamma\left(U,\left.\mathcal{O}_X\right|_U\right)\) is the restriction of \(f\), show that \(U \cap X_f=D(\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu)\). Conclude that \(X_f\) is an open subset of \(X\).

• Assume that \(X\) is quasi-compact. Let \(A=\Gamma\left(X, \mathcal{O}_X\right)\), and let \(a \in A\) be an element whose restriction to \(X_f\) is 0 . Show that for some \(n>0, f^n a=0\).

Hint: Use an open affine cover of \(X\).

• Now assume that \(X\) has a finite cover by open affines \(U_i\) such that each intersection \(U_i \cap U_j\) is quasi-compact. (This hypothesis is satisfied, for example, if \(\operatorname{sp}(X)\) is noetherian.) Let \(b \in \Gamma\left(X_f, \mathcal{O}_{X_f}\right)\). Show that for some \(n>0, f^n b\) is the restriction of an element of \(A\).

• With the hypothesis of (c), conclude that \(\Gamma\left(X_f, \mathcal{O}_{X_f}\right) \cong A_f\).

II.2.17 A Criterion for Affineness #to_work

• Let \(f: X \rightarrow Y\) be a morphism of schemes, and suppose that \(Y\) can be covered by open subsets \(U_i\), such that for each \(i\), the induced map \(f^{-1}\left(U_i\right) \rightarrow U_i\) is an isomorphism. Then \(f\) is an isomorphism.

• A scheme \(X\) is affine if and only if there is a finite set of elements \(f_1, \ldots, f_r \in\) \(A=\Gamma\left(X, \mathcal{O}_X\right)\), such that the open subsets \(X_{f_i}\) are affine, and \(f_1, \ldots, f_r\) generate the unit ideal in A. ¹

II.2.18 #to_work

In this exercise, we compare some properties of a ring homomorphism to the induced morphism of the spectra of the rings.

• Let \(A\) be a ring, \(X=\operatorname{Spec} A\), and \(f \in A\). Show that \(f\) is nilpotent if and only if \(D(f)\) is empty.

• Let \(\varphi: A \rightarrow B\) be a homomorphism of rings, and let \(f: Y=\operatorname{Spec} B \rightarrow X = \operatorname{Spec}A\) be the induced morphism of affine schemes. Show that \(\varphi\) is injective if and only if the map of sheaves \(f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y\) is injective. Show furthermore in that case \(f\) is dominant, i.e., \(f(Y)\) is dense in \(X\).

• With the same notation, show that if \(\varphi\) is surjective, then \(f\) is a homeomorphism of \(Y\) onto a closed subset of \(X\), and \(f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y\) is surjective.

• Prove the converse to (c), namely, if \(f: Y \rightarrow X\) is a homeomorphism onto a closed subset, and \(f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y\) is surjective, then \(\varphi\) is surjective. ²

II.2.19 #to_work

Let \(A\) be a ring. Show that the following conditions are equivalent:

• \(\operatorname{Spec} A\) is disconnected;

• there exist nonzero elements \(e_1, e_2 \in A\) such that \(e_1 e_2=0, e_1^2=e_1, e_2^2=e_2\), \(e_1+e_2=1\) (these elements are called orthogonal idempotents);

• \(A\) is isomorphic to a direct product \(A_1 \times A_2\) of two nonzero rings.