Integral Extensions - Main Theorem

The lying over and going up theorems describe how prime ideals behave under integral extensions, fundamental for understanding morphisms of schemes.

TheoremLying Over Theorem

Let $R \subseteq S$ be an integral extension and $\mathfrak{p} \in \text{Spec}(R)$ . Then there exists $\mathfrak{P} \in \text{Spec}(S)$ such that $\mathfrak{P} \cap R = \mathfrak{p}$ .

Moreover, for any prime $\mathfrak{q} \subseteq \mathfrak{p}$ and $\mathfrak{Q} \in \text{Spec}(S)$ with $\mathfrak{Q} \cap R = \mathfrak{q}$ , there exists $\mathfrak{P}$ over $\mathfrak{p}$ with $\mathfrak{Q} \subseteq \mathfrak{P}$ .

This ensures that the morphism $\text{Spec}(S) \to \text{Spec}(R)$ is surjective, a key property of integral extensions geometrically corresponding to finite morphisms being surjective.

TheoremGoing Up Theorem

Let $R \subseteq S$ be an integral extension. Suppose $\mathfrak{p}_1 \subseteq \mathfrak{p}_2$ are primes in $R$ and $\mathfrak{P}_1 \in \text{Spec}(S)$ lies over $\mathfrak{p}_1$ . Then there exists $\mathfrak{P}_2 \in \text{Spec}(S)$ lying over $\mathfrak{p}_2$ with $\mathfrak{P}_1 \subseteq \mathfrak{P}_2$ .

This implies chains of primes in $R$ lift to chains in $S$ , preserving inclusion relations.

ExampleApplication to Dimension

The going up theorem implies $\dim(S) \geq \dim(R)$ for integral extensions. In fact, if $S$ is finitely generated over $R$ , then $\dim(S) = \dim(R)$ .

For $\mathbb{Z} \subseteq \mathbb{Z}[i]$ : both have dimension 1, reflecting that integral extensions preserve dimension in nice cases.

TheoremIncomparability Theorem

Let $R \subseteq S$ be an integral extension. If $\mathfrak{P}_1, \mathfrak{P}_2 \in \text{Spec}(S)$ both lie over the same prime $\mathfrak{p} \in \text{Spec}(R)$ , then $\mathfrak{P}_1$ and $\mathfrak{P}_2$ are incomparable (neither contains the other).

Distinct primes over the same prime are unrelated, preventing "collapsing" of the prime spectrum structure.

Remark

For field extensions $k \subseteq K$ , lying over is trivial (only $(0)$ exists), but for rings of integers $\mathbb{Z} \subseteq \mathcal{O}_K$ , lying over describes how primes factor in extensions—the foundation of algebraic number theory.

TheoremGoing Down (for Normal Domains)

If $R \subseteq S$ are integral domains with $R$ integrally closed and $S$ integral over $R$ , then going down holds: given $\mathfrak{p}_1 \supseteq \mathfrak{p}_2$ in $R$ and $\mathfrak{P}_1$ over $\mathfrak{p}_1$ , there exists $\mathfrak{P}_2$ over $\mathfrak{p}_2$ with $\mathfrak{P}_2 \subseteq \mathfrak{P}_1$ .

Going down requires additional hypotheses (like integrally closed) and has geometric significance for flat morphisms.

ExampleFailure of Going Down

Consider $k[x^2, x^3] \subseteq k[x]$ . Let $\mathfrak{p}_1 = (0)$ and $\mathfrak{p}_2 = (x^2)$ in $k[x^2, x^3]$ . Taking $\mathfrak{P}_1 = (0)$ in $k[x]$ over $\mathfrak{p}_1$ , there is no prime strictly below $(0)$ in $k[x]$ , so going down fails.

The failure occurs because $k[x^2, x^3]$ is not integrally closed.

TheoremFiber Structure

For $R \subseteq S$ integral and $\mathfrak{p} \in \text{Spec}(R)$ , the fiber over $\mathfrak{p}$ is: $\text{Spec}(S \otimes_R \kappa(\mathfrak{p}))$

where $\kappa(\mathfrak{p}) = R_\mathfrak{p}/\mathfrak{p}R_\mathfrak{p}$ is the residue field. For integral extensions, fibers are finite (discrete spaces in Zariski topology).

Remark

The discriminant and different ideals measure ramification in integral extensions of Dedekind domains. A prime $\mathfrak{p}$ ramifies in $S$ if some prime over $\mathfrak{p}$ appears with multiplicity $> 1$ in the factorization of $\mathfrak{p}S$ .

ExampleRamification in Number Fields

For $\mathbb{Z} \subseteq \mathbb{Z}[i]$ :

$(2) = (1+i)^2$ ramifies (multiplicity 2)
Primes $p \equiv 1 \pmod{4}$ split: $(p) = \mathfrak{p}\overline{\mathfrak{p}}$ with distinct primes
Primes $p \equiv 3 \pmod{4}$ remain prime: $(p)$ is prime in $\mathbb{Z}[i]$

Lying over, going up, and ramification theory describe this arithmetic completely.

These theorems provide the foundation for understanding how geometric and arithmetic properties transfer along integral extensions.