Krulldimension

The Krull dimension of a topological space is a named after Wolfgang Krull dimension of topological term. This is motivated by algebraic investigations 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

Of non-empty, closed, irreducible subsets. This is referred to.

Respect to ring theory

Is a commutative ring with identity, then you look on the spectrum usually the Zariski topology. One arranges a prime ideal the set of all prime ideals to it comprehensive, we obtain a bijective relationship between and the set of all non- empty closed irreducible subsets of. Therefore, the observed in commutative algebra dimension of a ring, which is defined by the maximum length of Primidealketten, nothing other than the Krull dimension defined above its spectrum.

Examples

  • A non-empty Hausdorff space has the Krull dimension 0, since the irreducible subsets are precisely the one-point quantities.
  • Provided with the Zariski topology, that is completed are the common zeros of sets of sets of polynomials in indeterminates, has the dimension. All Zariski closed proper subsets have a smaller dimension.
  • Is a Noetherian ring, then for the polynomial ring:
  • Is an entire ring expansion, then:
  • For an arbitrary commutative unitary ring true: and for each pair of natural numbers there is a ring with and.
  • It applies to the power series ring over a Noetherian ring.
  • In a Noetherian rings true for an element that is not transcendental over.

Compared with other dimension terms

Since all Hausdorff spaces the Krull dimension 0, even this is not consistent with the Lebesgue covering dimension or the inductive dimensions. That the dimension of the above example is coincident with the Lebesgue overlap dimension only true because one considers the finer real Euclidean topology in the first case the Zariski topology and in the second case.

Is a Noetherian space with Krull dimension, so too is the cohomological dimension.

Codimension

Is a closed, irreducible subset, it is called the maximum length of all chains

Of non-empty, closed, irreducible subsets of codimension and they designated. For an arbitrary closed subset we define

As the infimum of, the irreducible components of passes.

Properties

  • The Krull dimension of a topological space is equal to the supremum of the Krull dimensions of its irreducible components.
  • Includes closed subsets, then.
Height (ring theory) Closed set Spectrum of a ring Integral element Formal power series Algebraic element Noetherian topological space
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