Lemma 10.79.4. 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