##  [Complete Intersection Property](/complete-intersection-property-0) 

 Definition

A property of an ideal or a local (or graded) ring meaning it can be generated by a regular sequence whose length equals the codimension; equivalently the ring is a quotient of a regular ring by a regular sequence.

 

 

 

 

 

 





## Principle

Principle

Being cut out by a regular sequence means the singularities and homological complexity are controlled by the minimal number of equations equal to the codimension, yielding predictable periodic or finite resolutions.

 

 

 

 

 





## Demonstration

Demonstration

If S is a regular local ring and I = (f1,...,fc) is generated by a regular sequence of length c, then R = S/I is a complete intersection of codimension c. Hypersurface rings (c = 1) are the simplest complete intersections.

 

 

 

 

## Misapplication

Misapplication

Assuming that any ideal with the minimal number of generators equal to codimension is a complete intersection; the generators must form a regular sequence, not merely be minimal in cardinality.

 

 

 

 

 





## Consequence

Consequence

Complete intersections have especially simple homological invariants (periodic or finite projective resolutions in many cases), are Gorenstein, and their deformation and singularity theory is more tractable.

 

 

 

 

## Reversal

Reversal

Non‑complete intersections can have higher homological complexity, nonperiodic resolutions and failure of the special duality properties enjoyed by complete intersections.

 

 

 

 

 





## Boundary

Boundary

Typically stated for ideals in regular local rings or graded polynomial rings; for nonregular ambient rings or nonlocal contexts the notion must be adjusted and regular sequence conditions checked carefully.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with Gorenstein and Cohen–Macaulay: every complete intersection is Gorenstein (hence Cohen–Macaulay), but the converses fail; the hierarchy distinguishes increasing generality and complexity.

 

 

 

 

 





## Synthesis

Synthesis

A complete intersection is a quotient of a regular ambient ring by a regular sequence of length equal to the codimension; this exactness of cutting equations yields strong homological simplifications and controlled singularities.