\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*}

The Stacks project

Lemma 10.15.4. Let $R$ be a ring. Let $M$ be a finite $R$-module. Let $\varphi : M \to M$ be a surjective $R$-module map. Then $\varphi $ is an isomorphism.

First proof. Write $R' = R[x]$ and think of $M$ as a finite $R'$-module with $x$ acting via $\varphi $. Set $I = (x) \subset R'$. By our assumption that $\varphi $ is surjective we have $IM = M$. Hence we may apply Lemma 10.15.3 to $M$ as an $R'$-module, the ideal $I$ and the endomorphism $\text{id}_ M$. We conclude that $(1 + a_1 + \ldots + a_ n)\text{id}_ M = 0$ with $a_ j \in I$. Write $a_ j = b_ j(x)x$ for some $b_ j(x) \in R[x]$. Translating back into $\varphi $ we see that $\text{id}_ M = -(\sum _{j = 1, \ldots , n} b_ j(\varphi )) \varphi $, and hence $\varphi $ is invertible. $\square$

Second proof. We perform induction on the number of generators of $M$ over $R$. If $M$ is generated by one element, then $M \cong R/I$ for some ideal $I \subset R$. In this case we may replace $R$ by $R/I$ so that $M = R$. In this case $\varphi : R \to R$ is given by multiplication on $M$ by an element $r \in R$. The surjectivity of $\varphi $ forces $r$ invertible, since $\varphi $ must hit $1$, which implies that $\varphi $ is invertible.

Now assume that we have proven the lemma in the case of modules generated by $n - 1$ elements, and are examining a module $M$ generated by $n$ elements. Let $A$ mean the ring $R[t]$, and regard the module $M$ as an $A$-module by letting $t$ act via $\varphi $; since $M$ is finite over $R$, it is finite over $R[t]$ as well, and since we're trying to prove $\varphi $ injective, a set-theoretic property, we might as well prove the endomorphism $t : M \to M$ over $A$ injective. We have reduced our problem to the case our endomorphism is multiplication by an element of the ground ring. Let $M' \subset M$ denote the sub-$A$-module generated by the first $n - 1$ of the generators of $M$, and consider the diagram

\[ \xymatrix{ 0 \ar[r] & M' \ar[r]\ar[d]^{\varphi \mid _{M'}} & M\ar[d]^\varphi \ar[r] & M/M' \ar[d]^{\varphi \bmod M'} \ar[r] & 0 \\ 0 \ar[r] & M' \ar[r] & M \ar[r] & M/M' \ar[r] & 0, } \]

where the restriction of $\varphi $ to $M'$ and the map induced by $\varphi $ on the quotient $M/M'$ are well-defined since $\varphi $ is multiplication by an element in the base, and $M'$ and $M/M'$ are $A$-modules in their own right. By the case $n = 1$ the map $M/M' \to M/M'$ is an isomorphism. A diagram chase implies that $\varphi |_{M'}$ is surjective hence by induction $\varphi |_{M'}$ is an isomorphism. This forces the middle column to be an isomorphism by the snake lemma. $\square$


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