The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

10.84 Projective modules over a local ring

In this section we prove a very cute result: a projective module $M$ over a local ring is free (Theorem 10.84.4 below). Note that with the additional assumption that $M$ is finite, this result is Lemma 10.77.4. In general we have:

Lemma 10.84.1. Let $R$ be a ring. Then every projective $R$-module is free if and only if every countably generated projective $R$-module is free.

Proof. Follows immediately from Theorem 10.83.5. $\square$

Here is a criterion for a countably generated module to be free.

Lemma 10.84.2. Let $M$ be a countably generated $R$-module. Suppose any direct summand $N$ of $M$ satisfies: any element of $N$ is contained in a free direct summand of $N$. Then $M$ is free.

Proof. Let $x_1, x_2, \ldots $ be a countable set of generators for $M$. By the assumption on $M$, we can construct by induction free $R$-modules $F_1, F_2, \ldots $ such that for every positive integer $n$, $\bigoplus _{i=1}^{n} F_ i$ is a direct summand of $M$ and contains $x_1, \ldots , x_ n$. Then $M = \bigoplus _{i = 1}^{\infty } F_ i$. $\square$

Lemma 10.84.3. Let $P$ be a projective module over a local ring $R$. Then any element of $P$ is contained in a free direct summand of $P$.

Proof. Since $P$ is projective it is a direct summand of some free $R$-module $F$, say $F = P \oplus Q$. Let $x \in P$ be the element that we wish to show is contained in a free direct summand of $P$. Let $B$ be a basis of $F$ such that the number of basis elements needed in the expression of $x$ is minimal, say $x = \sum _{i=1}^ n a_ i e_ i$ for some $e_ i \in B$ and $a_ i \in R$. Then no $a_ j$ can be expressed as a linear combination of the other $a_ i$; for if $a_ j = \sum _{i \neq j} a_ i b_ i$ for some $b_ i \in R$, then replacing $e_ i$ by $e_ i + b_ ie_ j$ for $i \neq j$ and leaving unchanged the other elements of $B$, we get a new basis for $F$ in terms of which $x$ has a shorter expression.

Let $e_ i = y_ i + z_ i, y_ i \in P, z_ i \in Q$ be the decomposition of $e_ i$ into its $P$- and $Q$-components. Write $y_ i = \sum _{j=1}^{n} b_{ij} e_ j + t_ i$, where $t_ i$ is a linear combination of elements in $B$ other than $e_1, \ldots , e_ n$. To finish the proof it suffices to show that the matrix $(b_{ij})$ is invertible. For then the map $F \to F$ sending $e_ i \mapsto y_ i$ for $i=1, \ldots , n$ and fixing $B \setminus \{ e_1, \ldots , e_ n\} $ is an isomorphism, so that $y_1, \ldots , y_ n$ together with $B \setminus \{ e_1, \ldots , e_ n\} $ form a basis for $F$. Then the submodule $N$ spanned by $y_1, \ldots , y_ n$ is a free submodule of $P$; $N$ is a direct summand of $P$ since $N \subset P$ and both $N$ and $P$ are direct summands of $F$; and $x \in N$ since $x \in P$ implies $x = \sum _{i=1}^ n a_ i e_ i = \sum _{i=1}^ n a_ i y_ i$.

Now we prove that $(b_{ij})$ is invertible. Plugging $y_ i = \sum _{j=1}^{n} b_{ij} e_ j + t_ i$ into $\sum _{i=1}^ n a_ i e_ i = \sum _{i=1}^ n a_ i y_ i$ and equating the coefficients of $e_ j$ gives $a_ j = \sum _{i=1}^ n a_ i b_{ij}$. But as noted above, our choice of $B$ guarantees that no $a_ j$ can be written as a linear combination of the other $a_ i$. Thus $b_{ij}$ is a non-unit for $i \neq j$, and $1-b_{ii}$ is a non-unit—so in particular $b_{ii}$ is a unit—for all $i$. But a matrix over a local ring having units along the diagonal and non-units elsewhere is invertible, as its determinant is a unit. $\square$


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