A topological spaceX 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.
A schemeS 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.
equidimensionality, mathematics, especially, topology, equidimensionality, property, space, that, local, dimension, same, everywhere, contents, definition, topology, definition, algebraic, geometry, cohen, macaulay, ring, referencesdefinition, topology, edita,. In mathematics especially in topology equidimensionality is a property of a space that the local dimension is the same everywhere 1 Contents 1 Definition topology 2 Definition algebraic geometry 3 Cohen Macaulay ring 4 ReferencesDefinition topology EditA 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 EditMain article 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 EditAn affine algebraic variety whose coordinate ring is a Cohen Macaulay ring is equidimensional 2 clarification needed References Edit Wirthmuller 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 nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Equidimensionality amp oldid 1140987333, wikipedia, wiki, book, books, library,
