Going up and going down
Encyclopedia
In commutative algebra
, a branch of mathematics
, going up and going down are terms which refer to certain properties of chains of prime ideal
s 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 Irving S. Cohen and Abraham Seidenberg
. These are colloquially
known as the going-up and going-down theorems.
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, can be extended to the length of the chain of prime ideals in A.
and 
are prime ideal
s of A and B, respectively, such that
then we say that
lies under 
and that 
lies over 
. In general, a ring extension A⊆B of commutative rings is said to satisfy the lying over property if every prime ideal P of A lies under some prime ideal Q of B.
The extension A⊆B is said to satisfy the incomparability property if whenever Q and Q' are distinct primes of B lying over prime P in A, then Q⊈Q' and Q' ⊈Q.
is a chain of prime ideal
s of A and
(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m,
lies over 
, then the chain
can be extended to a chain
such that for each 1 ≤ i ≤ n,
lies over 
.
In it is shown that if an extension A⊆B satisfies the going-up property, then it also satisfies the lying-over property.
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,
lies over 
, then the chain
can be extended to a chain
such that for each 1 ≤ i ≤ n,
lies over 
.
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 f(A) is a ring extension, 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.
There is another sufficient condition for the going-down property:
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 Ap2⊆Bq2 is a flat extension of rings. In fact, Ap2⊆Bq2 is a faithfully flat extension of rings since the inclusion map Ap2 → Bq2 is a local homomorphism. Therefore, the induced map on spectra Spec(Bq2) → Spec(Ap2) is surjective and there exists a prime ideal of Bq2 that contracts to the prime ideal p1Ap2 of Ap2. The contraction of this prime ideal of Bq2 to B is a prime ideal q1 of B contained in q2 that contracts to p1. The proof is complete. Q.E.D.
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, going up and going down are terms which refer to certain properties of chains of prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s in integral extensions.
The phrase going up refers to the case when a chain can be extended by "upward inclusion
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
", 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 Irving S. Cohen and Abraham Seidenberg
Abraham Seidenberg
Abraham Seidenberg was an American mathematician.- Early life :Seidenberg was born on June 2, 1916 in Washington D.C.. He graduated with a B.A. from the University of Maryland in 1937. He completed his Ph.D. in mathematics from Johns Hopkins University in 1943. His Ph.D...
. These are colloquially
Colloquialism
A colloquialism is a word or phrase that is common in everyday, unconstrained conversation rather than in formal speech, academic writing, or paralinguistics. Dictionaries often display colloquial words and phrases with the abbreviation colloq. as an identifier...
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, can be extended to the length of the chain of prime ideals in A.
Lying over and incomparability
First, we fix some terminology. If and are prime idealPrime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s of A and B, respectively, such that
then we say that
lies under and that lies over . In general, a ring extension A⊆B of commutative rings is said to satisfy the lying over property if every prime ideal P of A lies under some prime ideal Q of B.The extension A⊆B is said to satisfy the incomparability property if whenever Q and Q' are distinct primes of B lying over prime P in A, then Q⊈Q' and Q' ⊈Q.
Going-up
The ring extension A⊆B is said to satisfy the going-up property if wheneveris a chain of prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
s of A and
(m < n) is a chain of prime ideals of B such that for each 1 ≤ i ≤ m,
lies over , then the chaincan be extended to a chain
such that for each 1 ≤ i ≤ n,
lies over .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 wheneveris 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,
lies over , then the chaincan be extended to a chain
such that for each 1 ≤ i ≤ n,
lies over .There is a generalization of the ring extension case with ring morphisms. Let f : A → B be a (unital) ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
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 f(A) is a ring extension, 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
Inclusion map
In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
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:- (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.
- (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 Ap2⊆Bq2 is a flat extension of rings. In fact, Ap2⊆Bq2 is a faithfully flat extension of rings since the inclusion map Ap2 → Bq2 is a local homomorphism. Therefore, the induced map on spectra Spec(Bq2) → Spec(Ap2) is surjective and there exists a prime ideal of Bq2 that contracts to the prime ideal p1Ap2 of Ap2. The contraction of this prime ideal of Bq2 to B is a prime ideal q1 of B contained in q2 that contracts to p1. The proof is complete. Q.E.D.