Lemma 52.18.2 (0EIZ)—The Stacks project

Lemma 52.18.2. Let $Y$ be a universally catenary Noetherian scheme. Let $Z \subset Y$ be a closed subscheme. Let $f : Y' \to Y$ be a finite type morphism all of whose fibres have dimension $\leq e$. Set $Z' = f^{-1}(Z)$. Then

\[ \delta _ Z(y) \leq \delta _{Z'}(y') + e - \text{trdeg}_{\kappa (y)}(\kappa (y')) \]

for $y' \in Y'$ with image $y \in Y$.

Proof. If $\delta _{Z'}(y') = \infty $, then there is nothing to prove. If $\delta _{Z'}(y') < \infty $, then we choose a system of integral closed subschemes

\[ V'_0 \subset W'_0 \supset V'_1 \subset W'_1 \supset \ldots \subset W'_ k \]

of $Y'$ with $V'_0 \subset Z'$ and $y'$ the generic point of $W'_ k$ such that $\delta _{Z'}(y') = \delta (V'_0, W'_0, \ldots , W'_ k)$. Denote

\[ V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_ k \]

the scheme theoretic images of the above schemes in $Y$. Observe that $y$ is the generic point of $W_ k$ and that $V_0 \subset Z$. For each $i$ we look at the diagram

\[ \xymatrix{ V'_ i \ar[r] \ar[d] & W'_ i \ar[d] & V'_{i + 1} \ar[l] \ar[d] \\ V_ i \ar[r] & W_ i & V_{i + 1} \ar[l] } \]

Denote $n_ i$ the relative dimension of $V'_ i/V_ i$ and $m_ i$ the relative dimension of $W'_ i/W_ i$; more precisely these are the transcendence degrees of the corresponding extensions of the function fields. Set $c_ i = \text{codim}(V_ i, W_ i)$, $c'_ i = \text{codim}(V'_ i, W'_ i)$, $b_ i = \text{codim}(V_{i + 1}, W_ i)$, and $b'_ i = \text{codim}(V'_{i + 1}, W'_ i)$. By the dimension formula we have

\[ c_ i = c'_ i + n_ i - m_ i \quad \text{and}\quad b_ i = b'_ i + n_{i + 1} - m_ i \]

See Morphisms, Lemma 29.52.1. Hence $c_ i - b_ i = c'_ i - b'_ i + n_ i - n_{i + 1}$. Thus we see that

\begin{align*} & c_ i + c_{i + 1} + \ldots + c_ k - b_ i - b_{i + 1} - \ldots - b_{k - 1} \\ & = c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i - n_ k + c_ k - c'_ k \\ & = c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i - m_ k \end{align*}

Thus we see that

\begin{align*} \max _{i = 0, \ldots , k} & (c_ i + c_{i + 1} + \ldots + c_ k - b_ i - b_{i + 1} - \ldots - b_{k - 1}) \\ & = \max _{i = 0, \ldots , k} (c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i - m_ k) \\ & = \max _{i = 0, \ldots , k} (c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i) - m_ k \\ & \leq \max _{i = 0, \ldots , k} (c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1}) + e - m_ k \end{align*}

Since $m_ k = \text{trdeg}_{\kappa (y)}(\kappa (y'))$ we conclude that

\[ \delta (V_0, W_0, \ldots , W_ k) \leq \delta (V'_0, W'_0, \ldots , W'_ k) + e - \text{trdeg}_{\kappa (y)}(\kappa (y')) \]

as desired. $\square$

The Stacks project

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