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The Stacks project

Lemma 10.79.4. 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


Comments (2)

Comment #2813 by Dario Weißmann on

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


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