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

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

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

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

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

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

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

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

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.