Equivalence of dimension notions for affine domain: Difference between revisions
(New page: ==Statement== Let <math>A</math> be an affine domain over a field <math>k</math>, i.e. a finitely generated algebra over <math>k</math>, that also happens to be an integral domain...) |
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* Proving that the Krull dimension of the polynomial ring in <math>n</math> variables, is equal to exactly <math>n</math> | * Proving that the Krull dimension of the polynomial ring in <math>n</math> variables, is equal to exactly <math>n</math> | ||
* [[Going up theorem]] | * [[Going up theorem]] | ||
* [[Going down | * [[Going down for integral extensions of normal domains]] | ||
==Proof== | ==Proof== | ||
{{fillin}} | {{fillin}} | ||
Revision as of 01:57, 17 March 2008
Statement
Let be an affine domain over a field , i.e. a finitely generated algebra over , that also happens to be an integral domain. Then, the following are equivalent:
- The Krull dimension of
- The Krull dimension of the localization of at any maximal ideal (which is the same as that obtained using the Hilbert-Samuel polynomial)
- The transcendence degree of the field of fractions of , over
Facts used
- Proving that the Krull dimension of the polynomial ring in variables, is equal to exactly
- Going up theorem
- Going down for integral extensions of normal domains
Proof
Fill this in later