---
tag: 942FK83S09
kind: statement
role: theorem
name: Finitely generated projective modules over local rings are free
msc: [13C10, 13H99]
uses: [MZ02BKPHJA, BFFA03GE3V, 05QY1VEBYD, 0DZ9DKAKZR, NJ3APQ8BHH, 15JHBYR6KH]
about: [BFFA03GE3V, MZ02BKPHJA, 0DZ9DKAKZR]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_module
---

## Statement

Let $R$ be a [[local ring|MZ02BKPHJA]] with maximal ideal $\mathfrak{m}$ and [[residue field|NJ3APQ8BHH]] $k = R/\mathfrak{m}$.
Let $P$ be a [[finitely generated|05QY1VEBYD]] [[projective|BFFA03GE3V]] $R$-module.
Then $P$ is [[free|0DZ9DKAKZR]], of rank $n = \dim_k P/\mathfrak{m}P$.

## Remarks

Choose $x_1, \ldots, x_n \in P$ whose classes form a basis of the $k$-vector space $P/\mathfrak{m}P$.
By [[Nakayama's lemma|15JHBYR6KH]], they generate $P$, so there is a surjection $\pi \colon R^n \to P$; it splits because $P$ is projective, so $R^n \cong P \oplus K$ with $K = \ker \pi$.
Reducing modulo $\mathfrak{m}$ gives $k^n \cong k^n \oplus K/\mathfrak{m}K$, so $K/\mathfrak{m}K = 0$.
As a direct summand of $R^n$, $K$ is finitely generated, and Nakayama's lemma gives $K = 0$; thus $P \cong R^n$.

The local hypothesis is necessary.
Over $R = \Z/6\Z$, the module $\Z/2\Z$ is a direct summand of $R$, hence projective, but it has $2$ elements, while every finitely generated free $R$-module has $6^n$ elements for some $n \ge 0$.
Kaplansky's theorem removes the finiteness hypothesis: every projective module over a local ring is free.
More generally, a finitely presented flat module over a local ring is free.

## Explanations

### L1 human

Projective modules are pieces of free modules, and in general they can be twisted.
Over a local ring there is no room for twisting: every finitely generated projective module has a basis.
Its size is read off from the vector space you get by reducing modulo the maximal ideal.

### L2 human

This is why finitely generated projective modules over any commutative ring are called locally free: their localizations at prime ideals are free.
The rank at each prime gives a locally constant function on the spectrum, and rank one projectives form the [[Picard group|DZW58R6T2Q]].
Geometrically, finitely generated projective modules are vector bundles on $\operatorname{Spec} R$, and the theorem says that a vector bundle is trivial near each point.
Whether such modules are globally free over polynomial rings is the content of the [[Quillen–Suslin theorem|7N50GA46W1]].