In mathematics, especially in

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, equidimensionality is a property of a

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually cons ...

that the local

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

is the same everywhere.

Definition (topology)

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

''X'' is said to be equidimensional if for all points ''p'' in ''X'', the

at ''p'', that is dim ''p''(''X''), is constant. The

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

is an example of an equidimensional space. The

disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...

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 In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...

. For example, the affine scheme Spec k ,y,z(xy,xz), which intuitively looks like a line intersecting a plane, is not equidimensional.

Cohen–Macaulay ring

affine algebraic variety Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine com ...

whose coordinate ring is a

Cohen–Macaulay ring In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebraic geometry, algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly ...

is equidimensional.

References

Mathematical terminology Topology {{topology-stub