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

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

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

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

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

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

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

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

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.