Math 2.0

Commutative algebra Definition

Rank of a projective module

Builds on Commutative ringFinitely generated projective modulePrime idealLocalizationLocal ringFinitely generated projective modules over local rings are freeSpectrum of a ring

Definition

Let RR be a commutative ring and let PP be a finitely generated projective RR-module. For a prime ideal 𝔭\mathfrak{p} of RR, the localization P𝔭P_\mathfrak{p} is a finitely generated projective module over the local ring R𝔭R_\mathfrak{p}. So it is free of some finite rank r𝔭r_\mathfrak{p}. The rank of PP is the function rk P ⁣:Spec R→ℕ\mathrm{rk}\, P \colon \mathrm{Spec}\, R \to \N, 𝔭↦r𝔭\mathfrak{p} \mapsto r_\mathfrak{p}. The module PP has constant rank nn if r𝔭=nr_\mathfrak{p} = n for every prime 𝔭\mathfrak{p}.

Remarks

The rank is locally constant on the spectrum with its Zariski topology. It is additive, rk(P⊕Q)=rk P+rk Q\mathrm{rk}(P \oplus Q) = \mathrm{rk}\, P + \mathrm{rk}\, Q, and multiplicative for tensor products. For an integral domain RR with field of fractions KK, the rank is constant and equals dim⁡K(K⊗RP)\dim_K (K \otimes_R P).

Example: RnR^n has constant rank nn, since (Rn)𝔭=R𝔭n(R^n)_\mathfrak{p} = R_\mathfrak{p}^n.

Non-example of constant rank: let R=ℤ/6ℤR = \Z/6\Z and P=ℤ/2ℤP = \Z/2\Z. The spectrum has two points, (2)(2) and (3)(3). At (2)(2), R(2)≅ℤ/2ℤR_{(2)} \cong \Z/2\Z and P(2)≅ℤ/2ℤP_{(2)} \cong \Z/2\Z has rank 11; at (3)(3), P(3)=0P_{(3)} = 0 has rank 00.

Reading path

To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Commutative ring, concept, no evidence levelCommutative ringconceptFinitely generated module, concept, no evidence levelFinitely generated mo…conceptFinitely generated projective module, concept, no evidence levelFinitely generated pr…conceptFree module, concept, no evidence levelFree moduleconceptIdeal, concept, no evidence levelIdealconceptJacobson radical, concept, no evidence levelJacobson radicalconceptLocal ring, concept, no evidence levelLocal ringconceptLocalization, concept, no evidence levelLocalizationconceptModule over a ring, concept, no evidence levelModule over a ringconceptPrime ideal, concept, no evidence levelPrime idealconceptProjective module, concept, no evidence levelProjective moduleconceptResidue field, concept, no evidence levelResidue fieldconceptSpectrum of a ring, concept, no evidence levelSpectrum of a ringconceptNakayama's lemma, statement, no evidence levelNakayama's lemmastatementFinitely generated projective modules over local rings are free, statement, no evidence levelFinitely generated pr…statementRank of a projective module, definition, no evidence levelRank of a projective …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. Commutative ring concept
  2. Finitely generated module concept
  3. Finitely generated projective module concept
  4. Free module concept
  5. Ideal concept
  6. Jacobson radical concept
  7. Local ring concept
  8. Localization concept
  9. Module over a ring concept
  10. Prime ideal concept
  11. Projective module concept
  12. Residue field concept
  13. Spectrum of a ring concept
  14. Nakayama's lemma statement
  15. Finitely generated projective modules over local rings are free statement
  16. Rank of a projective module definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Projective_module
About this page
Tag46N7X1XJA3
URIm2://46N7X1XJA3
Kinddefinition
WritingW0 stub
MSC13C10, 19A49
DefinesRank of a projective module
UsesCommutative ring, Finitely generated projective module, Prime ideal, Localization, Local ring, Finitely generated projective modules over local rings are free, Spectrum of a ring
Changed3 November 2026

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