Krull dimension

The Krull dimension of a topological space is a topological dimension concept named after Wolfgang Krull . This is motivated by algebraic studies of rings in algebraic geometry and is closely related to the dimension of a ring.

definition

Be a topological space . The Krull dimension (or combinatorial dimension ) is the supremum of all lengths of chains ${\displaystyle X}$ ${\displaystyle n}$

${\displaystyle X_{0}\subsetneqq X_{1}\subsetneqq \ldots \subsetneqq X_{n}}$

of non-empty, closed , irreducible subsets. This is denoted by. ${\displaystyle \mathrm {dim} \,X}$

Relation to ring theory

If a commutative ring with a single element is used, the Zariski topology is usually considered on the spectrum . If one assigns to a prime ideal the set of all the prime ideals that encompass it, one obtains a bijective relationship between and the set of all non-empty closed irreducible subsets of . Therefore the dimension of a ring considered in commutative algebra , which is defined by the maximum length of prime ideal chains, is nothing other than the Krull dimension of its spectrum defined above. ${\displaystyle R}$ ${\displaystyle \mathrm {Spec} (R)}$ ${\displaystyle \mathrm {Spec} (R)}$ ${\displaystyle \mathrm {Spec} (R)}$

The Krull dimension of a Noetherian ring has the following properties: ${\displaystyle A}$

• ${\displaystyle \dim(A)=\sup \left\{\dim A_{\mathfrak {m}}\colon {\mathfrak {m}}{\mbox{ Maximalideal in }}A\right\}}$
• ${\displaystyle \dim k\left[x_{1},\ldots ,x_{n}\right]=n}$
• if is a domain of integrity and a finitely generated -algebra, then is the degree of transcendence and holds for every prime ideal . ${\displaystyle A}$ ${\displaystyle k}$ ${\displaystyle \dim A}$ ${\displaystyle Trg(A:k)}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \dim A/{\mathcal {P}}+\dim A_{\mathcal {P}}=\dim A}$

Examples

• A non-empty Hausdorff space has the Krull dimension 0, because the irreducible subsets are exactly the one-point sets.
• ${\displaystyle \mathbb {C} ^{n}}$provided with the Zariski topology, i.e. the common sets of zeros of sets of polynomials in indeterminates are closed, has the dimension . All Zariski-closed real subsets have a smaller dimension. ${\displaystyle n}$ ${\displaystyle n}$
• If a Noetherian ring , then the following applies to the polynomial ring : ${\displaystyle R}$ ${\displaystyle R[X_{1},\ldots ,X_{n}]}$

${\displaystyle \quad \mathrm {dim} (R[X_{1},\ldots ,X_{n}])=\mathrm {dim} (R)+n}$

• If there is a whole ring expansion , then: ${\displaystyle R\hookrightarrow S}$ ${\displaystyle \dim(R)=\dim(S)}$
• For any commutative unitary ring : and for every pair of natural numbers with there is a ring with and . ${\displaystyle R}$ ${\displaystyle \dim(R)+1\leq \dim(R[X])\leq 2\cdot \dim(R)+1}$ ${\displaystyle (n,m)}$ ${\displaystyle n+1\leq m\leq 2n+1}$ ${\displaystyle R}$ ${\displaystyle \dim(R)=n}$ ${\displaystyle \dim(R[X])=m}$
• It applies to the power series ring over a Noetherian ring : . ${\displaystyle R}$ ${\displaystyle \dim(R[[X]])=\dim(R)+1}$
• In a Noether rings applies to an element that does not transcendental over is . ${\displaystyle R}$ ${\displaystyle \alpha }$ ${\displaystyle R}$ ${\displaystyle \dim(R[\alpha ])\in \{\dim(R)-1,\dim(R)\}}$

Comparison with other dimension terms

Since all Hausdorff spaces have the Krull dimension 0, this does not match the Lebesgue cover dimension or the inductive dimensions . It is only correct that the dimension of the in the above example corresponds to the Lebesgue cover dimension, because in the first case the Zariski topology and in the second case the really finer Euclidean topology is considered. ${\displaystyle \mathbb {C} ^{n}}$

If a Noetherian space is Krull dimension , so is the cohomological dimension . ${\displaystyle X}$ ${\displaystyle \leq n}$ ${\displaystyle \leq n}$

Codimension

If a closed, irreducible subset is called the maximum length of all chains ${\displaystyle Y\subset X}$

${\displaystyle Y=X_{0}\subsetneqq X_{1}\subsetneqq \ldots \subsetneqq X_{n}}$

of non-empty, closed, irreducible subsets the codimension of and denotes it with . For any closed subset one defines ${\displaystyle Y}$ ${\displaystyle \mathrm {codim} _{X}Y}$ ${\displaystyle A\subset X}$

${\displaystyle \mathrm {codim} _{X}A}$as the infimum of where the irreducible components of passes through. ${\displaystyle \mathrm {codim} _{X}Y}$ ${\displaystyle Y}$ ${\displaystyle A}$

properties

• The Krull dimension of a topological space is equal to the supremum of the Krull dimensions of its irreducible components.
• Is with closed subsets , so is . ${\displaystyle X=A_{1}\cup \ldots \cup A_{n}}$ ${\displaystyle A_{i}}$ ${\displaystyle \mathrm {dim} \,X=\sup\{\mathrm {dim} \,A_{1},\ldots ,\mathrm {dim} \,A_{n}\}}$

Individual evidence

1. ↑ Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry (= Vieweg Studies. Vol. 46). Vieweg, Braunschweig et al. 1980, ISBN 3-528-07246-6 , definition II, 1.1.
2. ^ Klaus Hulek : Elementary Algebraic Geometry. Basic terms and techniques with numerous examples and applications. Vieweg, Braunschweig et al. 2000, ISBN 3-528-03156-5 , Chapter III: Smooth points and dimensions.
3. ↑ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry (= Vieweg Studies. Vol. 46). Vieweg, Braunschweig et al. 1980, ISBN 3-528-07246-6 , sentence II, 3.11 (b).
4. ^ Jacob Lurie : Higher Topos Theory (= Annals of Mathematics Studies. 170). Princeton University Press, Princeton NJ et al. 2009, ISBN 978-0-691-14049-0 , Corollary 7.2.4.10.
5. ↑ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry (= Vieweg Studies. Vol. 46). Vieweg, Braunschweig et al. 1980, ISBN 3-528-07246-6 , rules II, 1.2.