##  [Embedded Component](/embedded-component-0) 

 Definition

A component of a scheme or algebraic set whose support lies inside the support of another (minimal) component; typically realised by an associated prime that is not minimal and reflecting nonpure dimensional or nonreduced structure.

 

 

 

 

 

 





## Principle

Principle

Embedded components arise from nonminimal associated primes in primary decompositions: they indicate hidden structure supported along a larger component rather than giving a top-level irreducible piece.

 

 

 

 

 





## Demonstration

Demonstration

For the ideal I = (x^2, xy) in k[x,y], the primary decomposition can exhibit an embedded component supported at (x,y) inside the x-axis component; scheme‑theoretically this appears as an embedded associated subscheme.

 

 

 

 

## Misapplication

Misapplication

Treating embedded components as honest irreducible components for counting geometric pieces (for instance, counting embedded multiplicities as separate geometric components) — embedded components should not be conflated with minimal primes.

 

 

 

 

 





## Consequence

Consequence

The presence of embedded components affects depth, local cohomology, and the behaviour of sheaves; it signals that the space is not equidimensional or not Cohen–Macaulay at those points and complicates intersection and deformation theory.

 

 

 

 

## Reversal

Reversal

An isolated or minimal component is supported by a minimal prime and corresponds to a genuine top-level irreducible geometric piece; removing embedded components yields the primary topological decomposition.

 

 

 

 

 





## Boundary

Boundary

Embedded components are associated to nonminimal primes and are invisible in the topological support of the reduced scheme (they are scheme‑theoretic phenomena); they should not be confused with embedded points in the underlying topological space.

 

 

 

 

 





## Semantic Tension

Semantic Tension

There is tension between 'embedded component' and 'associated prime': every embedded component corresponds to a nonminimal associated prime, but not every associated prime behaves like a distinct geometric component in the reduced topology.

 

 

 

 

 





## Synthesis

Synthesis

An embedded component is a scheme‑theoretic, nonminimal associated piece supported inside a larger component: it represents nonpure dimensional or infinitesimal structure that alters algebraic and homological invariants without appearing as a separate reduced geometric component.