A scheme-theoretic intersection of subschemes that carries nilpotent elements or embedded components beyond the underlying set-theoretic intersection; the intersection as an object in the category of schemes (or rings) is nonreduced.

Principle

Scheme-theoretic operations retain algebraic structure (multiplicities, nilpotents, embedded components) that the set-theoretic locus omits; intersections must be computed in the ambient scheme’s structure sheaf rather than only on closed sets.

Demonstration

Two plane curves defined by f and g may meet with multiplicity: e.g., in Spec k[x,y], the ideal (f,g) can define a local ring with nilpotents or an associated embedded component, as in the intersection of a double line given by (x^2) with the line (x), producing nilpotent structure at x=0.

Misapplication

Replacing the scheme-theoretic intersection by the set-theoretic intersection and ignoring nilpotent or embedded pieces when computing intersection multiplicities, leading to erroneous counts of multiplicity or omitted obstruction classes.

Consequence

Correctly recognizing a nonreduced intersection records multiplicities and embedded components needed for intersection theory, deformation calculations, and for detecting excess intersection or nontransversality.

Reversal

The reversal is the reduced (or set-theoretic) intersection, where one passes to the reduced induced subscheme on the common underlying closed set, thereby discarding nilpotents and embedded structure.

Boundary

Applies to intersections in algebraic geometry and commutative algebra—schemes, subschemes, and ideals; does not arise for purely set-theoretic intersections of topological spaces or when working only with reduced schemes.

Semantic Tension

Tension between set-theoretic intersection (points as a locus) and scheme-theoretic intersection (algebraic structure with multiplicities and nilpotents); also tension with the notion of proper intersection versus excess intersection.

Synthesis

A nonreduced intersection is the scheme-theoretic intersection that encodes algebraic multiplicities and embedded components absent from the underlying set—essential when intersection multiplicities, obstructions, or nontransverse phenomena affect geometric or deformation-theoretic conclusions.