Lemma 10.79.3. 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 10.78.2. 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 10.23.1 we see that $P_{1, f} \to P_{2, f}$ is injective, i.e., (1)(a) holds. By Lemma 10.78.2 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 10.79.1. 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). $\square$

Comment #2813 by Dario Weißmann on

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

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