Equidimensionality
In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.[1]
Definition (topology)
[edit]A topological space X is said to be equidimensional if for all points p in X, the dimension at p, that is dim p(X), is constant. The Euclidean space is an example of an equidimensional space. The disjoint union of two spaces X and Y (as topological spaces) of different dimension is an example of a non-equidimensional space.
Definition (algebraic geometry)
[edit]A scheme S is said to be equidimensional if every irreducible component has the same Krull dimension. For example, the affine scheme Spec k[x,y,z]/(xy,xz), which intuitively looks like a line intersecting a plane, is not equidimensional.
Cohen–Macaulay ring
[edit]An affine algebraic variety whose coordinate ring is a Cohen–Macaulay ring is equidimensional.[2][clarification needed]
References
[edit]- ^ Wirthmüller, Klaus. A Topology Primer: Lecture Notes 2001/2002 (PDF). p. 90. Archived (PDF) from the original on 29 June 2020.
- ^ Sawant, Anand P. Hartshorne's Connectedness Theorem (PDF). p. 3. Archived from the original (PDF) on 24 June 2015.
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