Math 2.0

Commutative algebra Definition

Complete intersection ring

Builds on Noetherian ringLocal ringCompletion of a ringRegular local ringComplete local ringRegular sequenceGorenstein ringCohen structure theorem

Definition

Let (R,𝔪)(R, \mathfrak{m}) be a commutative Noetherian local ring and let R^\hat{R} be its 𝔪\mathfrak{m}-adic completion. Then RR is a complete intersection ring if the following holds. There are a complete regular local ring SS and an SS-regular sequence f1,…,frf_1, \ldots, f_r such that R^≅S/(f1,…,fr).\hat{R} \cong S/(f_1, \ldots, f_r).

Remarks

The completion is used because a Noetherian local ring need not be a quotient of a regular local ring, while its completion always is, by the Cohen structure theorem. If R=S/(f1,…,fr)R = S/(f_1, \ldots, f_r) with SS a regular local ring and f1,…,frf_1, \ldots, f_r an SS-regular sequence, then RR is a complete intersection. Example: K[x,y]K[x,y] localized at (x,y)(x, y) and divided by (xy)(xy) is a complete intersection, since xy≠0xy \ne 0 is a non-zero-divisor in the domain K[x,y](x,y)K[x,y]_{(x,y)}. Example: K[x]/(xn)K[x]/(x^n) is a complete intersection, with S=K[ ⁣[x] ⁣]S = K[\![x]\!] and f1=xnf_1 = x^n. Non-example: K[x,y]/(x2,xy,y2)K[x,y]/(x^2, xy, y^2) is not a complete intersection, because it is not Gorenstein, while every complete intersection is Gorenstein.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
Characteristic of a ring, concept, no evidence levelCharacteristic of a r…conceptComplete local ring, concept, no evidence levelComplete local ringconceptCompletion of a ring, concept, no evidence levelCompletion of a ringconceptDiscrete valuation ring, concept, no evidence levelDiscrete valuation ri…conceptGorenstein ring, concept, no evidence levelGorenstein ringconceptKrull dimension, concept, no evidence levelKrull dimensionconceptLocal ring, concept, no evidence levelLocal ringconceptMultivariate formal power series, concept, no evidence levelMultivariate formal p…conceptNoetherian ring, concept, no evidence levelNoetherian ringconceptRegular local ring, concept, no evidence levelRegular local ringconceptRegular sequence, concept, no evidence levelRegular sequenceconceptResidue field, concept, no evidence levelResidue fieldconceptWitt vectors, concept, no evidence levelWitt vectorsconceptCohen structure theorem, statement, no evidence levelCohen structure theor…statementComplete intersection ring, definition, no evidence levelComplete intersection…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Characteristic of a ring concept
  2. Complete local ring concept
  3. Completion of a ring concept
  4. Discrete valuation ring concept
  5. Gorenstein ring concept
  6. Krull dimension concept
  7. Local ring concept
  8. Multivariate formal power series concept
  9. Noetherian ring concept
  10. Regular local ring concept
  11. Regular sequence concept
  12. Residue field concept
  13. Witt vectors concept
  14. Cohen structure theorem statement
  15. Complete intersection ring definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Complete_intersection_ring
About this page
TagC1PFAQBCB8
URIm2://C1PFAQBCB8
Kinddefinition
WritingW0 stub
MSC13H10, 14M10, 13C40
DefinesComplete intersection ring
UsesNoetherian ring, Local ring, Completion of a ring, Regular local ring, Complete local ring, Regular sequence, Gorenstein ring, Cohen structure theorem
Changed2 November 2026

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