##  [Nonreduced Intersection](/nonreduced-intersection-0) 

 Definition

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

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

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

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

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

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

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

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

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.