Dimension (commutative algebra)

The dimension or more precisely Krull dimension (according to Wolfgang Krull ) of a commutative ring with one element is the descriptive dimension of the variety assigned to it in algebraic geometry or, more generally, of the associated scheme .

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

definition

The height of a prime ideal is the maximum length of an ascending chain of prime ideals ${\displaystyle P}$

${\displaystyle P_{0}\subsetneq P_{1}\subsetneq \ldots \subsetneq P_{n}=P;}$

the height is then . If there is no maximum length, the prime ideal has an infinite height. ${\displaystyle n}$

The dimension of a ring is the supremum of the heights of its prime ideals. ${\displaystyle A}$

properties

• In a Noetherian ring , every prime ideal has a finite height. But there are Noetherian rings of infinite dimensions.
• In Noetherian local rings , the dimension is equal to the smallest possible thickness of an ideal of definition , especially finite.
• The height of a prime ideal is equal to the codimension of the corresponding closed subset of the spectrum of the ring .
• Krull's main ideal theorem states that the height of prime ideals of a Noetherian ring that are minimally above a main ideal (i.e., contain it and are minimal with regard to this property) can be at most 1. More generally, the height of prime ideals of Noetherian rings that are minimally above an ideal that can be generated by elements is at most . ${\displaystyle r}$ ${\displaystyle r}$

Examples

• ${\displaystyle \mathrm {dim} (\mathbb {Z} )=1}$. Maximum ascending chains of prime ideals have the form

${\displaystyle (0)\subsetneq (p)}$

for prime numbers . ${\displaystyle p}$

• An integrity domain is one-dimensional if and only if every non-zero prime ideal is maximal. Each dedekind ring is a one-dimensional domain of integrity.
• Solids and all other Artinian rings are zero-dimensional.
• The formula

${\displaystyle \mathrm {dim} (A[X])=\mathrm {dim} (A)+1}$

applies to Noetherian rings . In particular, the affine coordinate ring of the -dimensional affine space over a body has the dimension . ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle n}$

• The going up phrase says that

${\displaystyle \mathrm {dim} \,R=\mathrm {dim} \,S}$

if there is a whole ring expansion. ${\displaystyle S\supset R}$

Topological version

The dimension discussed here can be generalized to the Krull dimension of topological spaces by replacing the prime ideal chains with chains of closed, irreducible subsets. Then the dimension of a ring is nothing other than the Krull dimension of its spectrum .

See also

Dimension of a module

literature

• Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah , Macdonald : Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• H. Matsumura, Commutative algebra 1980 ISBN 0-8053-7026-9 .