$\operatorname{Spec} \mathbb{Z} = \{(0), (2), (3), (5), (7), (11), \ldots\}.$

• The closed points are $(p)$ for primes $p$, with residue field $\kappa((p)) = \mathbb{F}_p$.
• The generic point $(0)$ has residue field $\kappa((0)) = \mathbb{Q}$.

The closure $\overline{\{(0)\}} = \operatorname{Spec} \mathbb{Z}$ (the generic point is dense). Think of $\operatorname{Spec} \mathbb{Z}$ as a "curve" whose points are the primes, with a fat generic point spread over the whole space.

If $k$ is a field, then $\operatorname{Spec} k = \{(0)\}$: a single point. This is the terminal object in the category of affine schemes (over $k$, $\operatorname{Spec} k$ plays the role of a point).

$\operatorname{Spec} k[x]$ for $k$ algebraically closed:

• Closed points: $(x - a)$ for each $a \in k$, with $\kappa = k$. These correspond to the "classical" points of $\mathbb{A}^1$.
• Generic point: $(0)$, with $\kappa = k(x)$ (the function field).

So $\operatorname{Spec} k[x] = \mathbb{A}^1 \cup \{\text{generic point}\}$. The generic point is dense; its closure is the entire space.

For $k = \mathbb{R}$ (not algebraically closed): $\operatorname{Spec} \mathbb{R}[x]$ has additional closed points like $(x^2 + 1)$ with residue field $\mathbb{C}$. These "non-rational points" correspond to conjugate pairs $\pm i$.

$\operatorname{Spec} k[x, y]$ for $k = \bar{k}$ has three types of points:

| Type | Prime ideal | Residue field | Dimension | |---|---|---|---| | Closed points | $(x - a, y - b)$ | $k$ | $0$ | | Generic points of curves | $(f(x,y))$, $f$ irred. | $k(x,y)/(f) = k(C)$ | $1$ | | Generic point | $(0)$ | $k(x,y)$ | $2$ |

The point $(y - x^2)$ is the "generic point of the parabola" — its closure $\overline{\{(y-x^2)\}} = V(y - x^2)$ is the parabola. Every irreducible subvariety has a unique generic point.

$\operatorname{Spec} k[\varepsilon]/(\varepsilon^2)$ has a single point $({\varepsilon})$, but the ring is not a field: it has a nilpotent element $\varepsilon \neq 0$ with $\varepsilon^2 = 0$.

This is a "fat point" or "point with tangent direction." A morphism $\operatorname{Spec} k[\varepsilon]/(\varepsilon^2) \to X$ corresponds to a point $p \in X$ together with a tangent vector at $p$. This is why scheme theory is more flexible than classical algebraic geometry: nilpotents carry infinitesimal information.

For a local ring $(R, \mathfrak{m})$, $\operatorname{Spec} R$ has a unique closed point $\mathfrak{m}$. Examples:

• $\operatorname{Spec} k[[x]]$ has two points: the closed point $(x)$ and the generic point $(0)$. Think of it as a "formal neighborhood of a point on a curve."
• $\operatorname{Spec} \mathbb{Z}_{(p)}$ has two points: $(p)$ (closed) and $(0)$ (generic). This is the "local picture of $\operatorname{Spec} \mathbb{Z}$ at the prime $p$."
• $\operatorname{Spec} k$ is the special case where $\mathfrak{m} = (0)$.

$\operatorname{Spec}(A \times B) \cong \operatorname{Spec} A \sqcup \operatorname{Spec} B$ (disjoint union). For example:

$\operatorname{Spec}(k \times k) = \{(1,0), (0,1)\} = \text{two points}.$

More interesting: by CRT, $\operatorname{Spec} \mathbb{Z}/6\mathbb{Z} \cong \operatorname{Spec}(\mathbb{Z}/2 \times \mathbb{Z}/3) = \{(2), (3)\}$, two points.

But $\operatorname{Spec} \mathbb{Z}/4\mathbb{Z}$ is a single point $(2)$ with nilpotent structure ($\bar{2}^2 = \bar{0}$). Topologically it's the same as $\operatorname{Spec} \mathbb{F}_2$, but the scheme structure remembers the multiplicity.

$\operatorname{Spec} k[x]$: the closed sets are $V(f(x))$ for $f \in k[x]$. Since $k[x]$ is a PID, every closed set is either all of $\operatorname{Spec} k[x]$ or a finite set of closed points. So the topology is the cofinite topology on the closed points, plus the generic point whose closure is everything.

This is highly non-Hausdorff: any two nonempty open sets intersect.

The closed sets of $\operatorname{Spec} \mathbb{Z}$ are: $\emptyset$, $\operatorname{Spec} \mathbb{Z}$, and finite sets of closed points $\{(p_1), \ldots, (p_n)\} = V(p_1 \cdots p_n)$. Same cofinite topology on the closed points.

The open set $D(6) = \operatorname{Spec} \mathbb{Z} \setminus \{(2), (3)\}$ consists of primes $p \neq 2, 3$ plus the generic point. As a scheme: $D(6) \cong \operatorname{Spec} \mathbb{Z}[1/6]$.

$\operatorname{Spec} k[x,y]$: closed sets include $V(f)$ for irreducible $f$ (curves), $V(x-a, y-b)$ (points), and intersections/unions. The topology is much richer than in dimension 1:

• $V(y - x^2)$ is a closed parabola.
• $V(xy)$ is the union of the two axes.
• $V(x - a, y - b) = \{(a,b)\}$ is a closed point.

The generic point of $V(y - x^2)$ is the prime $(y - x^2)$; its closure is the parabola. The generic point $(0)$ is dense in everything.

• $\mathcal{O}(\operatorname{Spec} \mathbb{Z}) = \mathbb{Z}$ (global sections).
• $\mathcal{O}(D(p)) = \mathbb{Z}[1/p]$ (integers with $p$ inverted).
• $\mathcal{O}(D(6)) = \mathbb{Z}[1/6] = \mathbb{Z}[1/2, 1/3]$.
• Stalk at $(p)$: $\mathcal{O}_{\operatorname{Spec}\mathbb{Z}, (p)} = \mathbb{Z}_{(p)}$ (integers localized at $p$).
• Stalk at $(0)$: $\mathcal{O}_{\operatorname{Spec}\mathbb{Z}, (0)} = \mathbb{Q}$ (the function field).

For $k[x]$:

• $\mathcal{O}(\mathbb{A}^1) = k[x]$.
• $\mathcal{O}(D(x)) = k[x, 1/x]$ — polynomials in $x$ and $1/x$.
• $\mathcal{O}(D(x(x-1))) = k[x, 1/(x(x-1))]$ — rational functions regular away from $0$ and $1$.
• $\mathcal{O}_{\mathbb{A}^1, (x-a)} = k[x]_{(x-a)}$ — rational functions regular at $a$.

Let $A = k[x,y]/(y^2 - x^2(x+1))$ (nodal cubic). Then $\operatorname{Spec} A$ is the affine nodal curve. At the node $\mathfrak{m} = (x, y)$:

$\mathcal{O}_{\operatorname{Spec} A, \mathfrak{m}} = A_\mathfrak{m}$

is a local ring that is not a domain (the completion $\hat{A}_\mathfrak{m} \cong k[[u,v]]/(uv)$, two branches). The local ring detects the singularity: $\dim_k \mathfrak{m}/\mathfrak{m}^2 = 2 > \dim = 1$, so the point is singular.

The surjection $k[x,y] \twoheadrightarrow k[x,y]/(y - x^2) \cong k[x]$ induces

$\operatorname{Spec} k[x] \hookrightarrow \operatorname{Spec} k[x,y] = \mathbb{A}^2$

which is the inclusion of the parabola $y = x^2$ into the plane. In general, surjections $A \twoheadrightarrow A/I$ correspond to closed immersions $V(I) \hookrightarrow \operatorname{Spec} A$.

The localization $k[x] \to k[x, 1/x]$ induces

$\operatorname{Spec} k[x, 1/x] \hookrightarrow \operatorname{Spec} k[x] = \mathbb{A}^1$

which is the inclusion of $D(x) = \mathbb{A}^1 \setminus \{0\}$. Localization maps $A \to S^{-1}A$ correspond to open immersions into distinguished opens.

In characteristic $p > 0$, the Frobenius $\varphi : A \to A$, $a \mapsto a^p$ induces $\operatorname{Spec} A \to \operatorname{Spec} A$. On $\mathbb{A}^1 = \operatorname{Spec} \mathbb{F}_p[x]$:

$\varphi^* : \operatorname{Spec} \mathbb{F}_p[x] \to \operatorname{Spec} \mathbb{F}_p[x], \quad (x - a) \mapsto (x - a^p) = (x - a)$

since $a^p = a$ in $\mathbb{F}_p$ (Fermat's little theorem). So on $\mathbb{F}_p$-points, the Frobenius is the identity! But on the scheme level it's a nontrivial degree-$p$ morphism. The fixed points of Frobenius on $\overline{\mathbb{F}_p}$-points are exactly the $\mathbb{F}_p$-rational points — this is the foundation of the Weil conjectures.

The quotient $\mathbb{Z} \twoheadrightarrow \mathbb{F}_p$ induces $\operatorname{Spec} \mathbb{F}_p \hookrightarrow \operatorname{Spec} \mathbb{Z}$, the inclusion of the closed point $(p)$. Think of this as the fiber of the "arithmetic curve" $\operatorname{Spec} \mathbb{Z}$ over the "prime $p$." The inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ gives $\operatorname{Spec} \mathbb{Q} \hookrightarrow \operatorname{Spec} \mathbb{Z}$: the generic fiber.

An equation like $y^2 = x^3 - x$ over $\mathbb{Z}$ gives a scheme over $\operatorname{Spec} \mathbb{Z}$ whose fibers are:

• Over $(0)$: the elliptic curve $E/\mathbb{Q}$.
• Over $(p)$: the reduction $E/\mathbb{F}_p$ (smooth for $p \neq 2$, singular at $p = 2$).

This "spreading out" perspective is central to arithmetic geometry.

In $\operatorname{Spec} k[x,y]$:

$(0) \leadsto (y - x^2) \leadsto (x - 1, y - 1)$

The generic point $(0)$ specializes to the generic point of the parabola, which specializes to the point $(1,1)$ on the parabola. This chain corresponds to the inclusion of primes:

$(0) \subset (y - x^2) \subset (x-1, y-1).$

The chain length is $2 = \dim \mathbb{A}^2$, illustrating that the Krull dimension equals the maximal chain length.

The generic point $\eta = (0) \in \operatorname{Spec} k[x]$ is the "point in general position." Evaluating a polynomial $f(x)$ at $\eta$ gives $f$ itself in $k(x)$. So:

• $f(\eta) = 0$ in $\kappa(\eta) = k(x)$ $\iff$ $f = 0$ as a polynomial $\iff$ $f$ vanishes everywhere.
• $f(\eta) \neq 0$ $\iff$ $f \neq 0$ $\iff$ $f$ vanishes on only finitely many closed points.

A property holds "at the generic point" iff it holds "on a dense open set." This is the scheme-theoretic formulation of "a general polynomial of degree $d$ has $d$ distinct roots."

$\operatorname{Spec} k[x]/(x^2)$ and $\operatorname{Spec} k[x]/(x)$ have the same underlying topological space (a single point $(x)$). But as schemes they differ:

• $\operatorname{Spec} k$ has ring $k$ (reduced).
• $\operatorname{Spec} k[x]/(x^2)$ has ring $k[\varepsilon]$ with $\varepsilon^2 = 0$ (non-reduced, "double point").

The scheme $V(x^2) \subseteq \mathbb{A}^1$ remembers that the origin has multiplicity 2. Similarly, $V(x^n) = \operatorname{Spec} k[x]/(x^n)$ is the origin with multiplicity $n$.

In $\mathbb{A}^2$, consider the intersection of $y = 0$ and $y = x^2$:

$\operatorname{Spec} k[x,y]/(y, y - x^2) = \operatorname{Spec} k[x]/(x^2).$

This is a double point — the intersection has multiplicity $2$ at the origin because the parabola is tangent to the $x$-axis. Compare with $y = 0 \cap y = x$:

$\operatorname{Spec} k[x,y]/(y, y - x) = \operatorname{Spec} k[x]/(x) = \operatorname{Spec} k.$

A simple (reduced) point — transverse intersection has multiplicity $1$. Scheme theory naturally computes intersection multiplicities via the tensor product of rings.