## 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 $\left(D(f),\ro{\OO_X}{D(f)} \right)$ 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.

a. 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.

b. 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 $\left(X,\left(\OO_X\right)_{\text {red }}\right)$ 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.

c. 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 $f^{\sharp}: \mathcal{U}_{\spec(A)} \rightarrow f_* \mathcal{O}_X$. Taking global sections we obtain a homomorphism $A \rightarrow \Gamma\left(X, \OO_X\right)$. Thus there is a natural map
$$
\alpha: \operatorname{Hom}_\Sch(X, \operatorname{Spec} A) \rightarrow \operatorname{Hom}_{\Ring} \left(A, \Gamma\left(X,\left.\OO_X\right)\right) .\right.
$$
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 $\mfm_x / \mfm_x^2$. 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 

(1) for each $i, j, \varphi_{j i}=\varphi_{i j}^{-1}$, and 
(2) 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 

(1) $\psi_i$ is an isomorphism of $X_i$ onto an open subscheme of $X$ 
(2) the $\psi_i\left(X_i\right)$ cover $X$, 
(3) $\psi_i\left(U_{i j}\right)=\psi_i\left(X_i\right) \cap \psi_j\left(X_j\right)$ 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.

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

b. 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.

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

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


### II.2.14 #to-work

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

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

c. 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.

d. 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

a. 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 $k$.

b. 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 $k$.

c. 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 $\mathcal{O}_x$.

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

b. 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 $X$.

c. 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 $A$.

d. 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

a. 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.

b. 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.[^hint_2.2.15]

[^hint_2.2.15]: 
Hint: Use (Ex. 2.4) and (Ex. 2.16d) above.

### 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.

a. 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.

b. Let $\varphi: A \rightarrow B$ be a homomorphism of rings, and let $f: Y=\operatorname{Spec} B \rightarrow X = \Spec A$ 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 $X$.

c. 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.

d. 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.[^hint_2.2.123]

[^hint_2.2.123]: 
Hint: Consider $X^{\prime}=\operatorname{Spec}(A / \operatorname{ker} \varphi)$ and use (b) and (c).

### II.2.19 #to-work

Let $A$ be a ring. Show that the following conditions are equivalent:

(i) $\operatorname{Spec} A$ is disconnected;

(ii) 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**);

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