##  [Nonreduced Scheme](/nonreduced-scheme-0) 

 Definition

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

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

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

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

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

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

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

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

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.