# The Stacks Project

## Tag 0BC2

### 15.3. Stably free modules

Here is what seems to be the generally accepted definition.

Definition 15.3.1. Let $R$ be a ring.

1. Two modules $M$, $N$ over $R$ are said to be stably isomorphic if there exist $n, m \geq 0$ such that $M \oplus R^{\oplus m} \cong N \oplus R^{\oplus m}$ as $R$-modules.
2. A module $M$ is stably free if it is stably isomorphic to a free module.

Observe that a stably free module is projective.

Lemma 15.3.2. Let $R$ be a ring. Let $0 \to P' \to P \to P'' \to 0$ be a short exact sequence of finite projective $R$-modules. If $2$ out of $3$ of these modules are stably free, then so is the third.

Proof. Since the modules are projective, the sequence is split. Thus we can choose an isomorphism $P = P' \oplus P''$. If $P' \oplus R^{\oplus n}$ and $P'' \oplus R^{\oplus m}$ are free, then we see that $P \oplus R^{\oplus n + m}$ is free. Suppose that $P'$ and $P$ are stably free, say $P \oplus R^{\oplus n}$ is free and $P' \oplus R^{\oplus m}$ is free. Then $$P'' \oplus (P' \oplus R^{\oplus m}) \oplus R^{\oplus n} = (P'' \oplus P') \oplus R^{\oplus m} \oplus R^{\oplus n} = (P \oplus R^{\oplus n}) \oplus R^{\oplus m}$$ is free. Thus $P''$ is stably free. By symmetry we get the last of the three cases. $\square$

Lemma 15.3.3. Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the radical of $R$). For every finite stably free $R/I$-module $E$ there exists a finite stably free $R$-module $M$ such that $M/IM \cong E$.

Proof. Choose a $n$ and $m$ and an isomorphism $E \oplus (R/I)^{\oplus n} \cong (R/I)^{\oplus m}$. Choose $R$-linear maps $\varphi : R^{\oplus m} \to R^{\oplus n}$ and $\psi : R^{\oplus n} \to R^{\oplus m}$ lifting the projection $(R/I)^{\oplus m} \to (R/I)^{\oplus n}$ and injection $(R/I)^{\oplus n} \to (R/I)^{\oplus m}$. Then $\varphi \circ \psi : R^{\oplus n} \to R^{\oplus n}$ reduces to the identity modulo $I$. Thus the determinant of this map is invertible by our assumption on $I$. Hence $P = \text{Ker}(\varphi)$ is stably free and lifts $E$. $\square$

Lemma 15.3.4. Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the radical of $R$). Let $M$ be a finite flat $R$-module such that $M/IM$ is a projective $R/I$-module. Then $M$ is a finite projective $R$-module.

Proof. By Algebra, Lemma 10.77.4 we see that $M_\mathfrak p$ is finite free for all prime ideals $\mathfrak p \subset R$. By Algebra, Lemma 10.77.2 it suffices to show that the function $\rho_M : \mathfrak p \mapsto \dim_{\kappa(\mathfrak p)} M \otimes_R \kappa(\mathfrak p)$ is locally constant on $\mathop{\rm Spec}(R)$. Because $M/IM$ is finite projective, this is true on $V(I) \subset \mathop{\rm Spec}(R)$. Since every closed point of $\mathop{\rm Spec}(R)$ is in $V(I)$ and since $\rho_M(\mathfrak p) = \rho_M(\mathfrak q)$ whenever $\mathfrak p \subset \mathfrak q \subset R$ are prime ideals, we conclude by an elementary argument on topological spaces which we omit. $\square$

The lift of Lemma 15.3.3 is unique up to isomorphism by the following lemma.

Lemma 15.3.5. Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that every element of $1 + I$ is a unit (in other words $I$ is contained in the radical of $R$). If $P$ and $P'$ are finite projective $R$-modules, then

1. if $\varphi : P \to P'$ is an $R$-module map inducing an isomorphism $\overline{\varphi} : P/IP \to P'/IP'$, then $\varphi$ is an isomorphism,
2. if $P/IP \cong P'/IP'$, then $P \cong P'$.

Proof. Proof of (1). As $P'$ is projective as an $R$-module we may choose a lift $\psi : P' \to P$ of the map $P' \to P'/IP' \xrightarrow{\overline{\varphi}^{-1}} P/IP$. By Nakayama's lemma (Algebra, Lemma 10.19.1) $\psi \circ \varphi$ and $\varphi \circ \psi$ are surjective. Hence these maps are isomorphisms (Algebra, Lemma 10.15.4). Thus $\varphi$ is an isomorphism.

Proof of (2). Choose an isomorphism $P/IP \cong P'/IP'$. Since $P$ is projective we can choose a lift $\varphi : P \to P'$ of the map $P \to P/IP \to P'/IP'$. Then $\varphi$ is an isomorphism by (1). $\square$

The code snippet corresponding to this tag is a part of the file more-algebra.tex and is located in lines 47–175 (see updates for more information).

\section{Stably free modules}
\label{section-stably-free}

\noindent
Here is what seems to be the generally accepted definition.

\begin{definition}
\label{definition-stably-free}
Let $R$ be a ring.
\begin{enumerate}
\item Two modules $M$, $N$ over $R$ are said to be
{\it stably isomorphic} if there exist $n, m \geq 0$ such
that $M \oplus R^{\oplus m} \cong N \oplus R^{\oplus m}$
as $R$-modules.
\item A module $M$ is {\it stably free} if it is stably isomorphic
to a free module.
\end{enumerate}
\end{definition}

\noindent
Observe that a stably free module is projective.

\begin{lemma}
\label{lemma-exact-category-stably-free}
Let $R$ be a ring. Let $0 \to P' \to P \to P'' \to 0$ be a short
exact sequence of finite projective $R$-modules. If $2$ out of $3$
of these modules are stably free, then so is the third.
\end{lemma}

\begin{proof}
Since the modules are projective, the sequence is split. Thus we can
choose an isomorphism $P = P' \oplus P''$. If $P' \oplus R^{\oplus n}$
and $P'' \oplus R^{\oplus m}$ are free, then we see that
$P \oplus R^{\oplus n + m}$ is free. Suppose that $P'$ and $P$ are
stably free, say $P \oplus R^{\oplus n}$ is free and $P' \oplus R^{\oplus m}$
is free. Then
$$P'' \oplus (P' \oplus R^{\oplus m}) \oplus R^{\oplus n} = (P'' \oplus P') \oplus R^{\oplus m} \oplus R^{\oplus n} = (P \oplus R^{\oplus n}) \oplus R^{\oplus m}$$
is free. Thus $P''$ is stably free. By symmetry we get the last of the
three cases.
\end{proof}

\begin{lemma}
\label{lemma-lift-stably-free}
Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that
every element of $1 + I$ is a unit (in other words $I$ is contained
in the radical of $R$). For every finite stably free $R/I$-module $E$
there exists a finite stably free $R$-module $M$ such that $M/IM \cong E$.
\end{lemma}

\begin{proof}
Choose a $n$ and $m$ and an isomorphism
$E \oplus (R/I)^{\oplus n} \cong (R/I)^{\oplus m}$.
Choose $R$-linear maps $\varphi : R^{\oplus m} \to R^{\oplus n}$
and $\psi : R^{\oplus n} \to R^{\oplus m}$ lifting the
projection $(R/I)^{\oplus m} \to (R/I)^{\oplus n}$
and injection $(R/I)^{\oplus n} \to (R/I)^{\oplus m}$.
Then $\varphi \circ \psi : R^{\oplus n} \to R^{\oplus n}$
reduces to the identity modulo $I$. Thus the determinant of
this map is invertible by our assumption on $I$. Hence
$P = \Ker(\varphi)$ is stably free and lifts $E$.
\end{proof}

\begin{lemma}
\label{lemma-lift-projective}
Let $R$ be a ring. Let $I \subset R$ be an ideal.
Assume that every element of $1 + I$ is a unit
(in other words $I$ is contained in the radical of $R$).
Let $M$ be a finite flat $R$-module such that
$M/IM$ is a projective $R/I$-module.
Then $M$ is a finite projective $R$-module.
\end{lemma}

\begin{proof}
By Algebra, Lemma \ref{algebra-lemma-finite-flat-local}
we see that $M_\mathfrak p$ is finite free for all prime ideals
$\mathfrak p \subset R$.
By
Algebra, Lemma \ref{algebra-lemma-finite-projective}
it suffices to show that the function $\rho_M : \mathfrak p \mapsto \dim_{\kappa(\mathfrak p)} M \otimes_R \kappa(\mathfrak p)$
is locally constant on $\Spec(R)$. Because $M/IM$ is finite projective, this
is true on $V(I) \subset \Spec(R)$. Since every closed point
of $\Spec(R)$ is in $V(I)$ and since
$\rho_M(\mathfrak p) = \rho_M(\mathfrak q)$
whenever $\mathfrak p \subset \mathfrak q \subset R$
are prime ideals, we conclude by
an elementary argument on topological spaces which we omit.
\end{proof}

\noindent
The lift of Lemma \ref{lemma-lift-stably-free}
is unique up to isomorphism by the following lemma.

\begin{lemma}
\label{lemma-isomorphic-finite-projective-lifts}
Let $R$ be a ring. Let $I \subset R$ be an ideal. Assume that
every element of $1 + I$ is a unit (in other words $I$ is contained
in the radical of $R$). If $P$ and $P'$ are finite
projective $R$-modules, then
\begin{enumerate}
\item if $\varphi : P \to P'$ is an $R$-module map inducing an
isomorphism $\overline{\varphi} : P/IP \to P'/IP'$, then $\varphi$
is an isomorphism,
\item if $P/IP \cong P'/IP'$, then $P \cong P'$.
\end{enumerate}
\end{lemma}

\begin{proof}
Proof of (1). As $P'$ is projective as an $R$-module we may
choose a lift $\psi : P' \to P$ of the map
$P' \to P'/IP' \xrightarrow{\overline{\varphi}^{-1}} P/IP$.
By Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK})
$\psi \circ \varphi$ and $\varphi \circ \psi$ are surjective.
Hence these maps are isomorphisms (Algebra, Lemma \ref{algebra-lemma-fun}).
Thus $\varphi$ is an isomorphism.

\medskip\noindent
Proof of (2). Choose an isomorphism $P/IP \cong P'/IP'$.
Since $P$ is projective we can choose a lift $\varphi : P \to P'$ of the map
$P \to P/IP \to P'/IP'$. Then $\varphi$ is an isomorphism by (1).
\end{proof}

## Comments (1)

Comment #2442 by Peng Du on March 1, 2017 a 3:47 pm UTC

In Definition 15.3.1. (1), "if there exist n,m≥0" should be "if there exist m≥0" as there is no 'n'.

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