Lemma 31.32.10. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. Let $b : X' \to X$ be the blowing up of $X$ along $Z$. Then $b$ induces an bijective map from the set of generic points of irreducible components of $X'$ to the set of generic points of irreducible components of $X$ which are not in $Z$.

**Proof.**
The exceptional divisor $E \subset X'$ is an effective Cartier divisor and $X' \setminus E \to X \setminus Z$ is an isomorphism, see Lemma 31.32.4. Thus it suffices to show the following: given an effective Cartier divisor $D \subset S$ of a scheme $S$ none of the generic points of irreducible components of $S$ are contained in $D$. To see this, we may replace $S$ by the members of an affine open covering. Hence by Lemma 31.13.2 we may assume $S = \mathop{\mathrm{Spec}}(A)$ and $D = V(f)$ where $f \in A$ is a nonzerodivisor. Then we have to show $f$ is not contained in any minimal prime ideal $\mathfrak p \subset A$. If so, then $f$ would map to a nonzerodivisor contained in the maximal ideal of $R_\mathfrak p$ which is a contradiction with Algebra, Lemma 10.25.1.
$\square$

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