Lemma 26.17.5. Let f : X \to S and g : Y \to S be morphisms of schemes with the same target. Points z of X \times _ S Y are in bijective correspondence to quadruples
where x \in X, y \in Y, s \in S are points with f(x) = s, g(y) = s and \mathfrak p is a prime ideal of the ring \kappa (x) \otimes _{\kappa (s)} \kappa (y). The residue field of z corresponds to the residue field of the prime \mathfrak p.
Proof.
Let z be a point of X \times _ S Y and let us construct a quadruple as above. Recall that we may think of z as a morphism \mathop{\mathrm{Spec}}(\kappa (z)) \to X \times _ S Y, see Lemma 26.13.3. This morphism corresponds to morphisms a : \mathop{\mathrm{Spec}}(\kappa (z)) \to X and b : \mathop{\mathrm{Spec}}(\kappa (z)) \to Y such that f \circ a = g \circ b. By the same lemma again we get points x \in X, y \in Y lying over the same point s \in S as well as field maps \kappa (x) \to \kappa (z), \kappa (y) \to \kappa (z) such that the compositions \kappa (s) \to \kappa (x) \to \kappa (z) and \kappa (s) \to \kappa (y) \to \kappa (z) are the same. In other words we get a ring map \kappa (x) \otimes _{\kappa (s)} \kappa (y) \to \kappa (z). We let \mathfrak p be the kernel of this map.
Conversely, given a quadruple (x, y, s, \mathfrak p) we get a commutative solid diagram
\xymatrix{ X \times _ S Y \ar@/_/[dddr] \ar@/^/[rrrd] & & & \\ & \mathop{\mathrm{Spec}}(\kappa (x) \otimes _{\kappa (s)} \kappa (y)/\mathfrak p) \ar[r] \ar[d] \ar@{-->}[lu] & \mathop{\mathrm{Spec}}(\kappa (y)) \ar[d] \ar[r] & Y \ar[dd] \\ & \mathop{\mathrm{Spec}}(\kappa (x)) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(\kappa (s)) \ar[rd] & \\ & X \ar[rr] & & S }
see the discussion in Section 26.13. Thus we get the dotted arrow. The corresponding point z of X \times _ S Y is the image of the generic point of \mathop{\mathrm{Spec}}(\kappa (x) \otimes _{\kappa (s)} \kappa (y)/\mathfrak p). We omit the verification that the two constructions are inverse to each other.
\square
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