Equidimensional scheme

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In algebraic geometry, a field of mathematics, an equidimensional scheme (or, pure dimensional scheme) is a scheme all of whose irreducible components are of the same dimension. All irreducible schemes are equidimensional.[1]

In affine space, the union of a line and a point not on the line is not equidimensional. In general, if two closed subschemes of some scheme, neither containing the other, have unequal dimensions, then their union is not equidimensional.

If a scheme is smooth (for instance, étale) over Spec k for some field k, then every connected component (which is then in fact an irreducible component), is equidimensional.

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  1. ^ Dundas, Bjorn Ian; Jahren, Björn; Levine, Marc; Østvær, P.A.; Röndigs, Oliver; Voevodsky, Vladimir (2007), Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002, Springer, p. 101, ISBN 9783540458975.