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*}

Lemma 10.78.3. Let $R$ be a ring. Let $\varphi : P_1 \to P_2$ be a map of finite projective modules. Then

  1. The set $U$ of primes $\mathfrak p \in \mathop{\mathrm{Spec}}(R)$ such that $\varphi \otimes \kappa (\mathfrak p)$ is injective is open and for any $f\in R$ such that $D(f) \subset U$ we have

    1. $P_{1, f} \to P_{2, f}$ is injective, and

    2. the module $\mathop{\mathrm{Coker}}(\varphi )_ f$ is finite projective over $R_ f$.

  2. The set $W$ of primes $\mathfrak p \in \mathop{\mathrm{Spec}}(R)$ such that $\varphi \otimes \kappa (\mathfrak p)$ is surjective is open and for any $f\in R$ such that $D(f) \subset W$ we have

    1. $P_{1, f} \to P_{2, f}$ is surjective, and

    2. the module $\mathop{\mathrm{Ker}}(\varphi )_ f$ is finite projective over $R_ f$.

  3. The set $V$ of primes $\mathfrak p \in \mathop{\mathrm{Spec}}(R)$ such that $\varphi \otimes \kappa (\mathfrak p)$ is an isomorphism is open and for any $f\in R$ such that $D(f) \subset V$ the map $\varphi : P_{1, f} \to P_{2, f}$ is an isomorphism of modules over $R_ f$.

Proof. To prove the set $U$ is open we may work locally on $\mathop{\mathrm{Spec}}(R)$. Thus we may replace $R$ by a suitable localization and assume that $P_1 = R^{n_1}$ and $P_2 = R^{n_2}$, see Lemma 10.77.2. In this case injectivity of $\varphi \otimes \kappa (\mathfrak p)$ is equivalent to $n_1 \leq n_2$ and some $n_1 \times n_1$ minor $f$ of the matrix of $\varphi $ being invertible in $\kappa (\mathfrak p)$. Thus $D(f) \subset U$. This argument also shows that $P_{1, \mathfrak p} \to P_{2, \mathfrak p}$ is injective for $\mathfrak p \in U$.

Now suppose $D(f) \subset U$. By the remark in the previous paragraph and Lemma 10.22.1 we see that $P_{1, f} \to P_{2, f}$ is injective, i.e., (1)(a) holds. By Lemma 10.77.2 to prove (1)(b) it suffices to prove that $\mathop{\mathrm{Coker}}(\varphi )$ is finite projective locally on $D(f)$. Thus, as we saw above, we may assume that $P_1 = R^{n_1}$ and $P_2 = R^{n_2}$ and that some minor of the matrix of $\varphi $ is invertible in $R$. If the minor in question corresponds to the first $n_1$ basis vectors of $R^{n_2}$, then using the last $n_2 - n_1$ basis vectors we get a map $R^{n_2 - n_1} \to R^{n_2} \to \mathop{\mathrm{Coker}}(\varphi )$ which is easily seen to be an isomorphism.

Openness of $W$ and (2)(a) for $D(f) \subset W$ follow from Lemma 10.78.1. Since $P_{2, f}$ is projective over $R_ f$ we see that $\varphi _ f : P_{1, f} \to P_{2, f}$ has a section and it follows that $\mathop{\mathrm{Ker}}(\varphi )_ f$ is a direct summand of $P_{2, f}$. Therefore $\mathop{\mathrm{Ker}}(\varphi )_ f$ is finite projective. Thus (2)(b) holds as well.

It is clear that $V = U \cap W$ is open and the other statement in (3) follows from (1)(a) and (2)(a). $\square$


Comments (2)

Comment #2813 by Dario WeiƟmann on

Typo in the proof of (2)(a): "... for D(f) \subset W".


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