equidimensionality
{{Short description|Property of a space in which the local dimensionality is the same everywhere}}
In mathematics, especially in topology, equidimensionality is a property of a space that the local dimension is the same everywhere.{{cite book|last=Wirthmüller|first=Klaus|url=https://kluedo.ub.uni-kl.de/frontdoor/deliver/index/docId/1843/file/top_skript.pdf#page=92|title=A Topology Primer: Lecture Notes 2001/2002|page=90|archive-url=https://web.archive.org/web/20200629013828/https://kluedo.ub.uni-kl.de/frontdoor/deliver/index/docId/1843/file/top_skript.pdf|archive-date=29 June 2020|url-status=live}}
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)
{{main|Equidimensional scheme}}
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.{{cite book|last=Sawant|first=Anand P.|url=http://www.math.tifr.res.in/~anands/connectedness.pdf|title=Hartshorne's Connectedness Theorem|page=3|archive-url=https://web.archive.org/web/20150624161836/https://www.math.tifr.res.in/~anands/connectedness.pdf|archive-date=24 June 2015|url-status=dead}}{{clarify|in what sense?|date=March 2019}}