Serre's splitting theorem

Remarks

The proof finds a unimodular element of $P$, that is, an element $p$ such that $f(p) = 1$ for some linear form $f \colon P \to R$; then $P = Rp \oplus \ker f$. Such an element is built by a general position argument. Since the rank exceeds the dimension, one can choose $p$ so that its image in $P_\mathfrak{p}/\mathfrak{p} P_\mathfrak{p}$ is nonzero at every prime $\mathfrak{p}$. Repeating the theorem shows that every finitely generated projective module of constant rank $n > d$ is $P_0 \oplus R^{n-d}$ with $P_0$ of rank $d$.

Example: for a Dedekind domain ($d \le 1$), every finitely generated projective module of rank $n \ge 1$ is $R^{n-1} \oplus I$ with $I$ of rank $1$. This is part of the description of $K_0$ of a Dedekind domain.

The rank hypothesis cannot be weakened to rank $d$. Let $A = \R[x, y, z]/(x^2 + y^2 + z^2 - 1)$, of Krull dimension $2$, and let $P$ be the kernel of $(u, v, w) \mapsto ux + vy + wz$ from $A^3$ to $A$. Then $P$ has rank $2$, and a free summand $A$ in $P$ would give a nowhere-vanishing tangent vector field on the $2$-sphere, which does not exist.

Going deeper

For a student

The theorem says that "large" projective modules carry no new information beyond a module of rank at most $d$. Combined with the Bass cancellation theorem, it shows that in rank greater than $d$, a projective module is determined by its class in $K_0(R)$. Over a ring of dimension $0$ or $1$ it reduces the classification to modules of rank at most $1$.