Equidimensionality
In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.[1]
Definition (topology)
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)
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
An affine algebraic variety whose coordinate ring is a Cohen–Macaulay ring is equidimensional.[2][clarification needed]
References
- ↑ Wirthmüller, Klaus. A Topology Primer: Lecture Notes 2001/2002. p. 90. https://kluedo.ub.uni-kl.de/frontdoor/deliver/index/docId/1843/file/top_skript.pdf#page=92.
- ↑ Sawant, Anand P.. Hartshorne's Connectedness Theorem. p. 3. http://www.math.tifr.res.in/~anands/connectedness.pdf.
 |  |
