A scheme whose structure sheaf contains nonzero nilpotent sections, equivalently a scheme with local rings that are not reduced; geometric points may have embedded infinitesimal structure beyond the underlying topological space.

Principle

Nonreducedness captures infinitesimal or embedded algebraic structure: nilpotent elements in local rings encode thickness, multiplicity, or first‑order infinitesimal directions that are invisible on the reduced underlying variety.

Demonstration

Spec(k[x]/(x^2)) is a basic nonreduced scheme: the underlying topological space is a single point, but the coordinate ring has a nilpotent x, so functions and tangent information reflect a first‑order infinitesimal neighborhood rather than an ordinary reduced point.

Misapplication

Mistaking nonreduced for reducible is erroneous: a nonreduced scheme can be irreducible topologically while carrying nilpotent structure, and reducibility concerns decomposition of the topological space rather than presence of nilpotents.

Consequence

Nonreduced schemes require modified statements of many theorems: dimensions, multiplicities, tangent spaces, and deformation theories must account for nilpotents; intersection multiplicities and sheaf cohomology can differ from their reduced counterparts.

Reversal

A reduced scheme has no nonzero nilpotent sections in its structure sheaf; its local rings are reduced and geometric and algebraic invariants coincide with those computed on the underlying topological space.

Boundary

The notion belongs to scheme theory and algebraic geometry; it excludes ordinary varieties or schemes assumed reduced, and many familiar results assume reduction unless nilpotents are explicitly allowed and treated.

Semantic Tension

There is tension between the topological intuition of a point and the algebraic reality: a nonreduced point looks identical topologically to a reduced point but carries extra algebraic thickness; distinguishing infinitesimal structure from genuine geometric multiplicity is essential.

Synthesis

A nonreduced scheme is an algebraic space whose local rings contain nilpotents, encoding infinitesimal or multiplicity data invisible on the underlying topology and forcing refinements of dimension, tangent and intersection-theoretic notions.