Lemma 29.48.6 (03HW)—The Stacks project

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

$Y' \to Y$ and $X' \to X$ are surjective finite locally free,
$Y' = X' \times _ X Y$ ,
$i : Y' \to Z'$ is a closed immersion,
$Z' \to X'$ is finite locally free, and
$Z' = \bigcup _{j = 1, \ldots , m} Z'_ j$ 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 10.136.14 (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 $1 \leq k_ i \leq d_ i$ . $\square$

The Stacks project

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