Going Up and Going Down - Going-up and Going-down Theorems

Going-up and Going-down Theorems

The usual statements of going-up and going-down theorems refer to a ring extension A⊆B:

1. (Going up) If B is an integral extension of A, then the extension satisfies the going-up property (and hence the lying over property), and the incomparability property.
2. (Going down) If B is an integral extension of A, and B is a domain, and A is integrally closed in its field of fractions, then the extension (in addition to going-up, lying-over and incomparability) satisfies the going-down property.

There is another sufficient condition for the going-down property:

• If A⊆B is a flat extension of commutative rings, then the going-down property holds.

Proof: Let p₁⊆p₂ be prime ideals of A and let q₂ be a prime ideal of B such that q₂ ∩ A = p₂. We wish to prove that there is a prime ideal q₁ of B contained in q₂ such that q₁ ∩ A = p₁. Since A⊆B is a flat extension of rings, it follows that A_p2⊆B_q2 is a flat extension of rings. In fact, A_p2⊆B_q2 is a faithfully flat extension of rings since the inclusion map A_p2 → B_q2 is a local homomorphism. Therefore, the induced map on spectra Spec(B_q2) → Spec(A_p2) is surjective and there exists a prime ideal of B_q2 that contracts to the prime ideal p₁A_p2 of A_p2. The contraction of this prime ideal of B_q2 to B is a prime ideal q₁ of B contained in q₂ that contracts to p₁. The proof is complete. Q.E.D.