Theorems of Cohen-Seidenberg

The going up and going down , named after Irvin Cohen and Abraham Seidenberg , two sets from the mathematical field of commutative algebra . Also known as going up and going down , they deal with prime ideal chains in ring extensions.

situation

Let be a ring expansion of two commutative rings with the same element . Are and prime ideals, one says lie above , if . ${\displaystyle S\supset R}$ ${\displaystyle P\subset S}$ ${\displaystyle p\subset R}$ ${\displaystyle P}$ ${\displaystyle p}$ ${\displaystyle p=P\cap R}$

If a prime ideal, then a prime ideal is in and is above . If a whole ring expansion and a prime ideal chain with real inclusions is in , then a prime ideal chain with real inclusions is in . Here we are looking into the question of whether, conversely, one can "lift" prime ideal chains in to such , so that the prime ideals of the chain in lie above those of the given chain in . To do this, one must first ensure that there are always prime ideals above the prime ideals . ${\displaystyle P\subset S}$ ${\displaystyle p:=R\cap P}$ ${\displaystyle R}$ ${\displaystyle P}$ ${\displaystyle p}$ ${\displaystyle S\supset R}$ ${\displaystyle P_{0}\subset \ldots \subset P_{n}}$ ${\displaystyle S}$ ${\displaystyle P_{0}\cap R\subset \ldots \subset P_{n}\cap R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle S}$

If one looks at the ring expansion and is a prime number, the main ideal generated is a prime ideal and there is no prime ideal that is above . If, however, it is a whole ring expansion, one can show that there is always a prime ideal above every prime ideal . ${\displaystyle \mathbb {Q} \supset \mathbb {Z} }$ ${\displaystyle \pi \in \mathbb {Z} }$ ${\displaystyle p=(\pi )}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle p}$ ${\displaystyle S\supset R}$ ${\displaystyle R}$ ${\displaystyle S}$

So if a whole ring expansion and a prime ideal chain is in , one can find an overlying prime ideal for each . The question now arises as to whether one can also choose them in such a way that they form an ascending chain. This is exactly the question that Cohen-Seidenberg's statements answer. ${\displaystyle S\supset R}$ ${\displaystyle p_{0}\subset \ldots \subset p_{n}}$ ${\displaystyle R}$ ${\displaystyle i}$ ${\displaystyle p_{i}}$ ${\displaystyle P_{i}\subset S}$ ${\displaystyle P_{i}}$

Going up

Let it be a whole ring expansion, a prime ideal chain in and the prime ideal lie above : ${\displaystyle S\supset R}$ ${\displaystyle p_{0}\subset \ldots \subset p_{n}}$ ${\displaystyle R}$ ${\displaystyle P_{0}}$ ${\displaystyle p_{0}}$

${\displaystyle {\begin{array}{ccccccc}P_{0}&\\\downarrow &\\p_{0}&\subset &p_{1}&\subset &\ldots &\subset &p_{n}\,.\end{array}}}$

Then there is the lying prime ideals , which form an ascending chain: ${\displaystyle p_{i}}$ ${\displaystyle P_{i}}$ ${\displaystyle i>0}$

${\displaystyle {\begin{array}{ccccccc}P_{0}&\subset &P_{1}&\subset &\ldots &\subset &P_{n}\\\downarrow &&\downarrow &&&&\downarrow \\p_{0}&\subset &p_{1}&\subset &\ldots &\subset &p_{n}\,.\end{array}}}$

Going down

If, in the situation of the going up sentence, one begins with an overlying prime ideal instead of an overlying prime ideal , additional requirements are required for an analogous statement: ${\displaystyle p_{0}}$ ${\displaystyle p_{n}}$

Let it be a whole ring expansion of integrity rings with normal , be a prime ideal chain in and the prime ideal lie above : ${\displaystyle S\supset R}$ ${\displaystyle R}$ ${\displaystyle p_{0}\subset \ldots \subset p_{n}}$ ${\displaystyle R}$ ${\displaystyle P_{n}}$ ${\displaystyle p_{n}}$

${\displaystyle {\begin{array}{ccccccc}&&&&&&P_{n}\\&&&&&&\downarrow \\p_{0}&\subset &p_{1}&\subset &\ldots &\subset &p_{n}\,.\end{array}}}$

Then there is the lying prime ideals , which form an ascending chain: ${\displaystyle p_{i}}$ ${\displaystyle P_{i}}$ ${\displaystyle i<n}$

${\displaystyle {\begin{array}{ccccccc}P_{0}&\subset &P_{1}&\subset &\ldots &\subset &P_{n}\\\downarrow &&\downarrow &&&&\downarrow \\p_{0}&\subset &p_{1}&\subset &\ldots &\subset &p_{n}\,.\end{array}}}$

meaning

Prime ideal chains play an important role in calculating the dimension of a ring. The going up sentence immediately results in a whole ring expansion . The going down phrase can be used to ${\displaystyle \mathrm {dim} \,R=\mathrm {dim} \,S}$ ${\displaystyle S\supset R}$

${\displaystyle \mathrm {dim} \,K[X_{1},\ldots ,X_{n}]=n}$

to show, where the polynomial ring is in indeterminates above the body . ${\displaystyle K[X_{1},\ldots ,X_{n}]}$ ${\displaystyle n}$ ${\displaystyle K}$

Individual evidence

1. ^ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , Sentence II.2.10 a
2. ^ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , Corollary II.2.12
3. ^ Ernst Kunz: Introduction to commutative algebra and algebraic geometry , Vieweg (1980), ISBN 3-528-07246-6 , sentence II.2.16
4. ^ Jean-Pierre Serre: Local Algebra , Springer (2000), ISBN 3540666419 , III Proposition 5