Lemma 29.48.6. Let $f : Y \to X$ be a finite morphism with $X$ affine. There exists a diagram

$\xymatrix{ Z' \ar[rd] & Y' \ar[l]^ i \ar[d] \ar[r] & Y \ar[d] \\ & X' \ar[r] & X }$

where

1. $Y' \to Y$ and $X' \to X$ are surjective finite locally free,

2. $Y' = X' \times _ X Y$,

3. $i : Y' \to Z'$ is a closed immersion,

4. $Z' \to X'$ is finite locally free, and

5. $Z' = \bigcup _{j = 1, \ldots , m} Z'_ j$ is a (set theoretic) finite union of closed subschemes, each of which maps isomorphically to $X'$.

Proof. Write $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(B)$. See also More on Algebra, Section 15.21. Let $x_1, \ldots , x_ n \in B$ be generators of $B$ over $A$. For each $i$ we can choose a monic polynomial $P_ i(T) \in A[T]$ such that $P(x_ i) = 0$ in $B$. By Algebra, Lemma 10.136.9 (applied $n$ times) there exists a finite locally free ring extension $A \subset A'$ such that each $P_ i$ splits completely:

$P_ i(T) = \prod \nolimits _{k = 1, \ldots , d_ i} (T - \alpha _{ik})$

for certain $\alpha _{ik} \in A'$. Set

$C = A'[T_1, \ldots , T_ n]/(P_1(T_1), \ldots , P_ n(T_ n))$

and $B' = A' \otimes _ A B$. The map $C \to B'$, $T_ i \mapsto 1 \otimes x_ i$ is an $A'$-algebra surjection. Setting $X' = \mathop{\mathrm{Spec}}(A')$, $Y' = \mathop{\mathrm{Spec}}(B')$ and $Z' = \mathop{\mathrm{Spec}}(C)$ we see that (1) – (4) hold. Part (5) holds because set theoretically $\mathop{\mathrm{Spec}}(C)$ is the union of the closed subschemes cut out by the ideals

$(T_1 - \alpha _{1k_1}, T_2 - \alpha _{2k_2}, \ldots , T_ n - \alpha _{nk_ n})$

for any $1 \leq k_ i \leq d_ i$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03HW. Beware of the difference between the letter 'O' and the digit '0'.