---
tag: 7N50GA46W1
kind: statement
role: theorem
name: Quillen–Suslin theorem
msc: [13C10, 19A13]
uses: [0SDC934R52, BFFA03GE3V, 05QY1VEBYD, 0DZ9DKAKZR, 0CMD0BCH8T]
about: [BFFA03GE3V, 0SDC934R52]
sources:
  - url: https://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem
same_as:
  wikidata: Q7272004
  wikipedia: Quillen–Suslin theorem
---

## Statement

Let $k$ be a field, let $n \ge 1$, and let $R = k[x_1, \ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] in $n$ variables over $k$.
Then every [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module is [[free|0DZ9DKAKZR]].

## Remarks

The theorem answers [[Serre's problem|7GG0XYHB0T]] and was proved independently by Daniel Quillen and Andrei Suslin in 1976.
Quillen's proof combines a local-global principle, which reduces the question to polynomial rings over local rings, with Horrocks' theorem on projective modules over $A[x]$ for a local ring $A$.
Suslin's proof works with unimodular rows and shows that they can be completed to invertible matrices.
Quillen's argument gives the result more generally for polynomial rings over a [[principal ideal domain|0CMD0BCH8T]].

Example: if $f, g \in R$ satisfy $af + bg = 1$ for some $a, b \in R$, the kernel $P$ of $R^2 \to R$, $(u, v) \mapsto uf + vg$, is projective, because the map splits.
The theorem says that $P$ is free; here one can see it directly, since $P$ is generated by $(-g, f)$.

The hypothesis on the ring cannot be dropped.
Over $A = \R[x, y, z]/(x^2 + y^2 + z^2 - 1)$, the kernel of $A^3 \to A$, $(u, v, w) \mapsto ux + vy + wz$, is projective but not free; this module corresponds to the tangent bundle of the $2$-sphere.

## Explanations

### L1 human

Projective modules are the algebraic version of vector bundles, families of vector spaces glued over a space.
Over affine space, a space without holes, one expects every such family to be trivial.
The theorem confirms this: over a polynomial ring over a field, every finitely generated projective module has a basis.
It took about twenty years from the question to the answer.

### L2 human

The theorem says that algebraic vector bundles on affine space $\mathbb{A}^n_k$ are trivial, the algebraic analogue of the contractibility of $\R^n$.
It gives $K_0(k[x_1, \ldots, x_n]) \cong \Z$ in a strong form: every class is represented by a free module.
In matrix terms, every unimodular row over $k[x_1, \ldots, x_n]$ is the first row of an invertible matrix.
The local-global principle in Quillen's proof is itself a widely used tool in the study of projective modules.