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

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

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

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

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

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

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

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 $R_ f$.

## Comments (2)

Comment #2813 by Dario Weißmann on

Comment #2916 by Johan on