Krull's main ideal theorem

The Krullsche principal ideal set is a key set of the dimension theory of Noetherian rings in commutative algebra , which after Wolfgang Krull is named and published by him 1928th

formulation

Be a Noetherian ring, a non-unity and minimal among the prime ideals that contain the main ideal . ${\displaystyle R}$ ${\displaystyle x\in R\setminus R^{\ast }}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle Rx}$

Then the height of the prime ideal is at most . ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle 1}$

Generalization to any ideals

The statement of Krull's main ideal theorem can be generalized from main ideals to any ideals . It is then also referred to as Krull's height theorem .

Be a Noetherian ring, a real ideal generated by elements and minimally among the prime ideals that contain the ideal . Then the height of the prime ideal is at most . ${\displaystyle R}$ ${\displaystyle I\subsetneq R}$ ${\displaystyle m}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle I}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle m}$

Significance for algebraic geometry

Since the dimension of an affine algebraic variety is obtained as the Krull dimension of the associated coordinate ring, Krull's main ideal theorem provides direct estimates of the dimensions of certain varieties. This gives something like the following statement:

Are irreducible projective varieties in the -dimensional projective space above the body . Then one obtains the estimate for an irreducible component ${\displaystyle V,W\subseteq \mathbb {P} ^{n}(K)}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle Z\subseteq V\cap W}$

${\displaystyle \dim(Z)\geq \dim(V)+\dim(W)-n}$.

Individual evidence

1. ^ Eisenbud: Commutative Algebra. 1995, p. 231.
2. ^ Krull: Prime ideal chains in general ring areas. 1928, §3.
3. ^ Eisenbud: Commutative Algebra. 1995, theorem 10.1.
4. ↑ Kunz: Introduction to Algebraic Geometry. 1997, sentence 5.1.
5. ^ Atiyah, Macdonald: Introduction to Commutative Algebra. 1969, Corollary 11-17.
6. ↑ Markus Brodmann: Algebraic Geometry: An Introduction. Birkhäuser, Basel 1989, ISBN 978-3-7643-1779-9 , p. 143.
7. ^ Eisenbud: Commutative Algebra. 1995, theorem 10.2.
8. ↑ Kunz: Introduction to Algebraic Geometry. 1997, sentence 5.4.
9. ↑ Kunz: Introduction to Algebraic Geometry. 1997, sentence 5.9.

literature

• Wolfgang Krull: Prime ideal chains in general ring areas . In: Report of the meeting of the Heidelberg Academy of Sciences . 7th treatise, 1928.
• Ernst Kunz : Introduction to algebraic geometry (=  advanced course in mathematics ). 14th edition. Vieweg, Braunschweig / Wiesbaden 1997, ISBN 978-3-528-07287-2 , VI.Dimensional Theory, §5, doi : 10.1007 / 978-3-322-80313-9 .
• David Eisenbud : Commutative Algebra . with a View Toward Algebraic Geometry (=  Graduate Texts in Mathematics . No. 150 ). Springer, New York 1995, ISBN 0-387-94268-8 , 10. The Principal Ideal Theorem an Systems of Parameters.
• Michael Francis Atiyah , Ian Grant Macdonald : Introduction to Commutative Algebra . Westview Press, New York 1969, ISBN 0-201-00361-9 , 11 Dimension Theory.