Going up and going down

In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions.
The phrase going up refers to the case when a chain can be extended by "upward inclusion", while going down refers to the case when a chain can be extended by "downward inclusion".
The major results are the Cohen–Seidenberg theorems, which were proved by Irvin S. Cohen and Abraham Seidenberg. These are known as the going-up and going-down theorems.
Going up and going down.
Let "A" ⊆ "B" be an extension of commutative rings.
The going-up and going-down theorems give sufficient conditions for a chain of prime ideals in "B", each member of which lies over members of a longer chain of prime ideals in "A", to be able to be extended to the length of the chain of prime ideals in "A".
Lying over and incomparability.
First, we fix some terminology. If formula_1 and formula_2 are prime ideals of "A" and "B", respectively, such that
(note that formula_4 is automatically a prime ideal of "A") then we say that formula_1 "lies under" formula_2 and that formula_2 "lies over" formula_1. In general, a ring extension "A" ⊆ "B" of commutative rings is said to satisfy the lying over property if every prime ideal formula_1 of "A" lies under some prime ideal formula_2 of "B".
The extension "A" ⊆ "B" is said to satisfy the incomparability property if whenever formula_2 and formula_12 are distinct primes of "B" lying over a prime formula_1 in "A", then formula_2 ⊈ formula_12 and formula_12 ⊈ formula_2.
Going-up.
The ring extension "A" ⊆ "B" is said to satisfy the going-up property if whenever
is a chain of prime ideals of "A" and
("m" < "n") is a chain of prime ideals of "B" such that for each 1 ≤ "i" ≤ "m", formula_20 lies over formula_21, then the latter chain can be extended to a chain
such that for each 1 ≤ "i" ≤ "n", formula_20 lies over formula_21.
In it is shown that if an extension "A" ⊆ "B" satisfies the going-up property, then it also satisfies the lying-over property.
Going-down.
The ring extension "A" ⊆ "B" is said to satisfy the going-down property if whenever
is a chain of prime ideals of "A" and
("m" < "n") is a chain of prime ideals of "B" such that for each 1 ≤ "i" ≤ "m", formula_20 lies over formula_21, then the latter chain can be extended to a chain
such that for each 1 ≤ "i" ≤ "n", formula_20 lies over formula_21.
There is a generalization of the ring extension case with ring morphisms. Let "f" : "A" → "B" be a (unital) ring homomorphism so that "B" is a ring extension of "f"("A"). Then "f" is said to satisfy the going-up property if the going-up property holds for "f"("A") in "B".
Similarly, if "B" is a ring extension of "f"("A"), then "f" is said to satisfy the going-down property if the going-down property holds for "f"("A") in "B".
In the case of ordinary ring extensions such as "A" ⊆ "B", the inclusion map is the pertinent map.
Going-up and going-down theorems.
The usual statements of going-up and going-down theorems refer to a ring extension "A" ⊆ "B":
There is another sufficient condition for the going-down property:
"Proof": Let "p"1 ⊆ "p"2 be prime ideals of "A" and let "q"2 be a prime ideal of "B" such that "q"2 ∩ "A" = "p"2. We wish to prove that there is a prime ideal "q"1 of "B" contained in "q"2 such that "q"1 ∩ "A" = "p"1. Since "A" ⊆ "B" is a flat extension of rings, it follows that "A""p"2 ⊆ "B""q"2 is a flat extension of rings. In fact, "A""p"2 ⊆ "B""q"2 is a faithfully flat extension of rings since the inclusion map "A""p"2 → "B""q"2 is a local homomorphism. Therefore, the induced map on spectra Spec("B""q"2) → Spec("A""p"2) is surjective and there exists a prime ideal of "B""q"2 that contracts to the prime ideal "p"1"A""p"2 of "A""p"2. The contraction of this prime ideal of "B""q"2 to "B" is a prime ideal "q"1 of "B" contained in "q"2 that contracts to "p"1. The proof is complete. Q.E.D.