Going Up and Going Down Theorems

Core Idea

The going up and going down theorems describe how prime ideal chains in a ring R extend to prime ideal chains in an extension ring S. The lying-over theorem says that for an integral extension R ⊆ S, every prime of R is the contraction of some prime of S. Going up (Cohen-Seidenberg) says chains of primes in R can be lifted to chains in S preserving containment. Going down requires additional hypotheses (R integrally closed, or the extension being flat) and guarantees chains can be extended downward. These theorems are the algebraic engine behind dimension-theoretic arguments in algebraic geometry.

Explainer

The going up and going down theorems, due to Cohen and Seidenberg, describe how the prime ideal structure of a ring relates to that of an integral extension. Given a ring extension R ⊆ S, every prime ideal Q of S contracts to a prime ideal Q ∩ R of R. The fundamental question is the converse: given primes in R, can we find primes in S lying over them? And can we do this compatibly with chains?

The lying-over theorem is the starting point: if R ⊆ S is an integral extension and P is a prime ideal of R, then there exists a prime Q of S with Q ∩ R = P. The incomparability theorem adds that distinct primes of S lying over the same prime of R are incomparable under inclusion. Together, these say the map Spec S → Spec R is surjective and the fibers have no containments. The going up theorem extends this to chains: if P_1 ⊆ P_2 are primes of R and Q_1 lies over P_1, then there exists Q_2 ⊇ Q_1 lying over P_2. Going up holds for any integral extension without additional hypotheses.

The going down theorem is more delicate. It states: if R ⊆ S is an integral extension with R an integrally closed domain and S a domain, and if P_1 ⊇ P_2 are primes of R with Q_1 lying over P_1, then there exists Q_2 ⊆ Q_1 lying over P_2. The hypothesis that R is integrally closed is essential -- without it, going down can fail. There is an alternative route to going down that bypasses integrality entirely: if R → S is a flat ring homomorphism, then going down holds. Flatness-based going down is used extensively in algebraic geometry, where flat morphisms are the algebraic counterpart of "continuously varying fibers."

The consequences for dimension theory are immediate. Going up implies that dim(S) ≥ dim(R) for integral extensions (chains in R lift to chains of at least the same length in S). When going down also holds, the dimensions are equal for integral extensions of domains. Combined with Noether normalization (every finitely generated k-algebra is integral over a polynomial subring), these theorems prove that the Krull dimension of a finitely generated k-algebra equals its transcendence degree over k -- one of the foundational results connecting algebra to geometry.