Integral extension implies surjective map on spectra

This article gives the statement, and possibly proof, of a fact about how a property of a homomorphism of commutative unital rings, forces a property for the induced map on spectra
View other facts about induced maps on spectra

Name

This result is sometimes termed lying over, and is a precursor to the going up theorem.

Statement

Suppose $S$ is an integral extension of a ring $R$, in other words $f:R\to S$ is an injective homomorphism of commutative unital rings with the property that every element of $S$ is integral over the image of $R$. Then, the following are true.

The map:

$f^{*}:Spec(S)\to Spec(R)$

from the spectrum of $S$ to that of $R$, that sends a prime ideal of $S$ to its contraction in $R$, is surjective. In other words, every prime ideal of $R$ occurs as the contraction of a prime ideal of $S$.

This result is sometimes termed the lying over theorem.

Note that injectivity of $f$ is crucial for surjectivity of the map on spectra; this is analogous to the fact that surjective ring homomorphisms induce injective maps on spectra.

Related facts

• Integral extension implies inverse image of max-spectrum is max-spectrum
• Going up theorem: The going up theorem is a somewhat stronger version of this result, and also follows as a corollary of this result.

Proof

The goal is to prove that starting with a prime ideal $P$ of $R$, we can find a prime ideal $Q$ of $S$ such that $f^{-1}(Q)=P$.

We localize $R$ at $P$, and localize $S$ at the image of $U=R\setminus P$ to get $S^{\prime }$. Then $R_P$ is a local ring with unique maximal ideal $P^{\prime }=P{R}_P$, and $f$ induces a map $R_P\to S^{\prime }=S[U^{-1}]$.

We thus have an inclusion $f:R_P\to S^{\prime }$. Consider the image $P^{\prime }{S}^{\prime }$. This is an ideal of $S^{\prime }$. If $P^{\prime }{S}^{\prime }$ is a proper ideal, it is contained in some maximal ideal $M$, and the contraction of that maximal ideal to $R_P$ is precisely $P^{\prime }$. Contracting back along the localization, we find a prime ideal of $S^{\prime }$, whose contraction is exactly $P$. (we are using the fact that contracting a maximal ideal of $S^{\prime }$ yields a prime, though not necessarily maximal, ideal of $S$).

Thus, the main goal is to show that $P^{\prime }{S}^{\prime }\ne S^{\prime }$ (this is where we need to use integrality). The idea is to construct a $R$-subalgebra of $S^{\prime }$, called $S^{″}$, that is finite over $R_P$, and use Nakayama's lemma to derive a contradiction. Here are the steps:

• Since $S$ is integral over $R$, $S^{\prime }$ is integral over $R_P$
• If $P^{\prime }{S}^{\prime }=S^{\prime }$, then the element $1\in S^{\prime }$ can be written as a $P^{\prime }$-linear combination of finitely many elements from $S^{\prime }$
• Let $S^{″}$ be the $R_P$-subalgebra generated by these finitely many elements. Then $S^{″}$ is finitely generated and integral over $R_P$, hence it is finitely generated as a module over $R_P$. For full proof, refer: finitely generated and integral implies finite
• We thus have $P^{\prime }{S}^{″}=S^{″}$ (since $1\in P^{\prime }{S}^{″}$). Since $P^{\prime }$ is the Jacobson radical of $R_P$, Nakayama's lemma tells us that $S^{″}=0$, yielding a contradiction.