Proposition 4

Let A and B be two finitely generated k-algebras and $\rho : A \to B$ an algebra homomorphism. Suppose that A is integral and that the A-module B is without torsion, and denote by K the field of fractions of A. One has

Suppose first that B is integral. The algebra $B \otimes_A K$ is then a ring of fractions of B defined by a multiplicative subset not containing 0. It therefore has as its field of fractions the field of fractions L of B. By th. 3, one has

Now, by the corollary of A, V, p. 106

whence the result in this case.

Let us pass to the general case. Every minimal prime ideal $p$ of $B$ is formed of zero divisors in $B$, and is therefore above the ideal 0 of $A$. It follows that the mapping $p \mapsto p . (B \otimes_A K)$ is a bijection from the set of minimal prime ideals of $B$ onto the set of minimal prime ideals of $B \otimes_A K$. The proposition therefore follows from the first part of the proof and from prop. 6, c) of § 1, no 3.