Math 2.0

Commutative algebra Modules Problem

Serre's problem on projective modules

Builds on Polynomial ringProjective moduleFinitely generated moduleFree module

Problem

Let kk be a field, let n≥1n \ge 1, and let R=k[x1,…,xn]R = k[x_1, \ldots, x_n] be the polynomial ring in nn variables over kk. Is every finitely generated projective RR-module free?

Remarks

The problem is solved: the answer is yes, by the Quillen–Suslin theorem. For n=1n = 1, the ring k[x]k[x] is a principal ideal domain, so the answer is yes by the structure theorem for modules over such rings. Serre showed that every finitely generated projective RR-module PP is stably free: P⊕RmP \oplus R^m is free for some m≥0m \ge 0.

History

Jean-Pierre Serre raised the question in his 1955 paper on coherent algebraic sheaves. C. S. Seshadri proved the case n=2n = 2 in 1958. Daniel Quillen and Andrei Suslin solved the general case independently in 1976.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Finitely generated module, concept, no evidence levelFinitely generated mo…conceptFree module, concept, no evidence levelFree moduleconceptPolynomial ring, concept, no evidence levelPolynomial ringconceptProjective module, concept, no evidence levelProjective moduleconceptSerre's problem on projective modules, problem, no evidence levelSerre's problem on pr…problem
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. Finitely generated module concept
  2. Free module concept
  3. Polynomial ring concept
  4. Projective module concept
  5. Serre's problem on projective modules problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem
About this page
Tag7GG0XYHB0T
URIm2://7GG0XYHB0T
Kindproblem
WritingW0 stub
MSC13C10, 19A13
StatusThis copy of the site has no build data. Run m2 status 7GG0XYHB0T.
AboutProjective module, Polynomial ring
UsesPolynomial ring, Projective module, Finitely generated module, Free module
Changed2 November 2026

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