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Concept
Going up and going down
Summary
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.
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.
First, we fix some terminology. If and are prime ideals of A and B, respectively, such that
(note that is automatically a prime ideal of A) 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 of A lies under some prime ideal of B.
The extension A ⊆ B is said to satisfy the incomparability property if whenever and are distinct primes of B lying over a prime in A, then ⊈ and ⊈ .
The ring extension A ⊆ B is said to satisfy the going-up property if whenever
is a chain of prime ideals of A and
is a chain of prime ideals of B with m < n and such that lies over for 1 ≤ i ≤ m, then the latter chain can be extended to a chain
such that lies over for each 1 ≤ i ≤ n.
In it is shown that if an extension A ⊆ B satisfies the going-up property, then it also satisfies the lying-over property.
The ring extension A ⊆ B is said to satisfy the going-down property if whenever
is a chain of prime ideals of A and
is a chain of prime ideals of B with m < n and such that lies over for 1 ≤ i ≤ m, then the latter chain can be extended to a chain
such that lies over for each 1 ≤ i ≤ n.
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Ontological neighbourhood
Related concepts (2)
Integral element
In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and aj in A such that That is to say, b is a root of a monic polynomial over A. The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A. If every element of B is integral over A, then we say that B is integral over A, or equivalently B is an integral extension of A.
Commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers. Commutative algebra is the main technical tool in the local study of schemes.