Reduced Scheme

A reduced scheme is a scheme whose local rings are all reduced rings (Stacks Project n.d., Tag 01J0). An affine scheme is reduced iff its coordinate ring is a reduced ring (Stacks Project n.d., Tag 01J2).

The reduction of an affine scheme is , where is the nilradical of . It has the same underlying topological space as , since every prime ideal contains the nilradical, but its structure sheaf has no nonzero nilpotent sections (Stacks Project n.d., Tag 01IZ). For example, over a field , the affine scheme is not reduced, while its reduction is . Both have a single point, so reducedness is not a property of the underlying topological space alone.