Quillen–Suslin theorem

Remarks

The theorem answers Serre's problem 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.

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.

Going deeper

For a student

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.